118 DS17 Abstracts

IP1 ations from this limit can only be due to very rare events Some Geometrical and Physical Aspects of Mor- (Large Deviations). The net result is that the deviations phogenesis from the limit behavior can be observed only on time scales that are huge (for example, the exponential of number of How is living matter organized in space and time during particles/units in the system). However, in several natural morphogenesis ? A comparative view across animal and instances deviations happen on a much shorter time scale. plant species suggests that the answer may lie in reusing After discussing this important issue and its practical rel- just a few geometric cellular principles and organ-sculpting evance in a general framework, I will focus on mean field motifs. Using examples, I will discuss the quantitative ba- synchronization models for which 1. we can sharply trace sis for three of these motifs: elongation, lumenization and the time scale of validity of the deterministic large scale folding. PDE; 2. we can describe in detail what happens after this time scale. The talk is based on papers written by sub- L Mahadevan sets of the following authors: Lorenzo Bertini, Eric Lu¸con, Harvard University Christophe Poquet, & myself. [email protected] Giambattista Giacomin Universit´eParisDiderot IP2 giambattista.giacomin@univ-paris-diderot.fr Pattern Formation in the Drylands: Vegetation Patterns in Mathematical Models and in Satellite Images of the Horn of Africa IP4 Computational Approach Elucidating the Mecha- An awe-inspiring example of spontaneous pattern forma- nisms of Cardiovascular Diseases tion appears in the distribution of vegetation in some dry- land environments. Examples from Africa, Australia and Cardiovascular diseases such as aortic aneurysms and aor- the Americas reveal vegetation congregated in stripe-like tic dissections persist as life-threatening hazards. Patient- bands, alternating with striking regularity with bands of specific simulations are now common in biomedical engi- bare soil. A typical length scale for such patterns is 100 m; neering. Although they are extremely useful for grasping they may be readily surveyed in Google Maps. The typical the flow/stress distributions and for patient-specific treat- time scale for pattern evolution, however, is 100 years, so ment planning, they remain insufficient to elucidate the investigations of dynamics are a bit thwarted, with only a general mechanisms of a targeted disease. Several mathe- few early data points provided by aerial photographs from matical viewpoints should play important roles in this con- the 1950s. These ecosystems represent some of Earths most text. For instance, we introduce a geometrical characteri- vulnerable under threats to desertification, and ecologists zation of blood vessels, which vary widely among individ- have suggested that the patterns, easily monitored by satel- uals. Differences in the vessel morphology can produce lites, may tell us something about a regions possible stage different flow characteristics, stress distributions, and ulti- of collapse. This is an attractive idea, especially to us mately different outcomes. Therefore, the characterization pattern formation researchers who want to work on prob- of the morphologies of these vessels poses an important lems that have potential impact. My talk consists of two clinical question. Through close collaboration with medi- parts. In the first, I describe how one proposed early warn- cal doctors, these analyses can yield greater understanding ing sign, related to vegetation pattern morphology, can be leading to better risk assessments. framed mathematically in terms of an equivariant bifur- cation problem, and then probed by a systematic analysis Hiroshi Suito of models. In the second, I turn to investigations of image Okayama University and data, focusing on our preliminary studies of vegetation pat- Tohoku University, Japan terns in the Horn of Africa. The goal of this presentation [email protected] is to highlight the potential for models to be confronted by data, and vice versa. IP5 Mary Silber Dynamics, Mixing, and Coherence University of Chicago Dept. of Engineering Sciences and Applied Mathematics Coherent regions in geophysical flows play fundamental [email protected] roles by organising fluid flow and obstructing transport. For example, in the ocean, coherence impacts dynamics from global scales down to scales of at least tens of kilome- IP3 tres, and strongly influences the transportation of heat, Random Long Time Dynamics for Large Systems salt, nutrients, phytoplankton, pollution, and garbage. of Interacting Oscillators I will describe some recent mathematical constructions, ranging across dynamical systems, probability, and geome- A considerable amount of effort has been put into under- try, which enable the accurate identification and tracking of standing large scale dynamics of interacting particles or, such structures, and the quantification of associated mix- more generally, interacting “units” (like cells or individ- ing and transport properties. I will present case studies uals). This large scale limit, leading often to a partial from a variety of geophysical settings. differential equation, is in most cases taken under the as- sumption of finite time horizon. This of course raises the Gary Froyland crucial issue of the relevance of long time dynamics for the UNSW Australia limit PDE when dealing with the long time dynamics of [email protected] the original system, which is made of a finite number of particles or units. This issue has been successfully tack- led for stochastic systems (the large scale limit is in this IP6 case a law of large numbers) for the cases in which devi- Stochastic Arnold Diffusion of Deterministic Sys- DS17 Abstracts 119

tems and study all the macroscopic attractors and bifurcations of these systems. In 1964, V. Arnold constructed an example of a nearly integrable deterministicsystem exhibiting instabilities. In Edward Ott the 1970s, physicist B. Chirikov coined the term for this University of Maryland phenomenon Arnold diffusion, where diffusion refers to Inst. for Research in Electronics and Applied Physics stochastic nature of instability. One of the most famous [email protected] examples of stochastic instabilities for nearly integrable systems is dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. They were discovered numerically by as- CP0 tronomer J. Wisdom. During the talk we describe a class Mentoring Program of nearly integrable deterministic systems, where we prove stochastic diffusive behavior. Namely, we show that distri- See session abstract. butions given by deterministic evolution of certain random initial conditions weakly converge to a diffusion process. Conference Participants This result is conceptually different from known mathe- n/a matical results, where existence of diffusing orbits is shown. n/a This work is based on joint papers with O. Castejon, M. Guardia, J. Zhang, and K. Zhang. CP0 Vadim Kaloshin Student Icebreaker University of Maryland at College Park, USA and ETH Z¨urich On Saturday immediately before the opening reception, [email protected] there will be an informal Student Icebreaker session. Through structured activities, the primary goal is for at- tendees to leave feeling comfortable about the meeting, IP7 and having made new connections to depend on as friendly Interactions, Deformations and Bifurcations of Sin- faces during the conference. There is no additional cost, gular Patterns but in order to get a count on how many people to expect, we request that you register for this session when you reg- Singular patterns appear in multiscale systems. These sys- ister for the conference. tems exhibit the rich behavior of general systems, their singular nature provides a structure by which this may Chad M. Topaz be understood. Moreover, many natural phenomena are Dept. of Mathematics, Statistics, and Computer Science modeled by such systems. I will discuss the strong cross- Macalester College fertilization between applications and the development of [email protected] mathematical theory. Unravelling the nature of patterns exhibited by specific chemical or ecological models goes hand in hand with uncovering novel generic destabiliza- CP1 tion mechanisms as the ‘Hopf dance’. Understanding real- Control of Complex Network Dynamics and Syn- istic patterns requires analytical descriptions of deforma- chronization by Symbolic Regression tions, bifurcations and annihilation of interacting localized structures – from an ecological point of view preferably un- The control of dynamical systems is a relevant and im- der varying (climatological) circumstances. Through this, portant issue in engineering applications, like turbulence, mathematics may explain why desertification sometimes is but also for experimental designs in natural sciences and in a sudden catastrophic event, while it is a gradual process medicine, where drugs and technical devices are more and in other situations. more fine-tuned to enhance the quality of life. Large-scale systems are typically hard to control: due to the many de- Arjen Doelman grees of freedom and related possible instability of modes, Mathematisch Instituut linear control is often not effective to control a complex [email protected] system such that it remains on a well-defined orbit. We use symbolic regression to find analytical control laws for the described situations. Here, we use coupled oscillator SP1 networks as a general model for a complex network, as Juergen Moser Lecture - Emergent Behavior in e.g., the brain. We investigate in particular the important Large Systems of Many Coupled Oscillators situation where synchronization is forced/suppressed. The results are analyzed and compared with existing bench- Large systems of many coupled dynamical units are of cru- marks. As a result, we conclude that individual control cial interest in a host of physical, biological and techno- is feasible - an extremely important and useful feature for logical settings. Often the dynamical units that are cou- medicine applications. pled exhibit oscillatory behavior. The understanding and analysis of these large, complex systems offers many chal- Markus W Abel lenges. In this talk I will introduce this topic, give some Universit¨at Potsdam examples, and describe a technique for analyzing a large [email protected] class of problems of this type. The results I will discuss will reduce the complicated, high dimensional, microscopic Julien Gout dynamics of the full system to that or a low dimensional Ambrosys GmbH, Potsdam, Germany system governing the macroscopic evolution of certain ’or- [email protected] der parameters’. This reduction is exact in the limit of large system size, i.e., N going to infinity, where N is the Markus Quade number of coupled units, and can be employed to discover University of Potsdam 120 DS17 Abstracts

Ambrosys GmbH of the adjacency matrix can impact the dynamics in epi- [email protected] demiological, neuronal and genetic networks. Furthermore, while many results exist regarding the spectral radius for Robert K. Niven undirected graphs, very few results have been proven for he University of New South Wales, Canberra, Australia. random directed graphs. We first consider the spectral [email protected] radii distribution for directed Chung-Lu graphs, where the probability two nodes share a directed edge is proportional Kamran Shafi to the product of the respective expected in and out degrees UNSW, ADFA, Canberra BC ACT 2610, AUSTRALIA of the two nodes. We then use a path counting argument k.shafi@adfa.edu.au to provide novel concentration bounds on the likelihood that the spectral radius of a network of fixed size devi- ates from the value suggested in the asymptotic theory. CP1 Subsequently, we extend our argument to a more general random graph model that allows for community structure, The Importance of Assortativity to Improve Dy- where we partition our adjacency matrix into submatrices. namical Robustness in Complex Networks In particular, we prove that if we partition our adjacency The tolerance to failure of networked systems whose func- matrix with m square blocks on the diagonal, then under tions are maintained by collective dynamical behavior modest constraints, the spectral radius of a given realiza- of the network units has recently been analyzed in the tion will asymptotically converge to the spectral radius of 2 × 2 framework called dynamical robustness of complex net- an m m matrix. Ultimately, we anticipate that our works. Understanding this network robustness against fail- novel spectral radius results will not only help us better ures is useful for preventing large-scale breakdowns and understand how network structure can influence the dy- damages in real-world networked systems. However, al- namics for real world directed networks, but also provide though the effect of network structure on the dynamical new techniques for estimating the distribution of spectral robustness has been examined with various types of net- radii for other random graph models as well. work topology, the role of network assortativity is still un- clear. Here we study [Sasai T, Morino K, Tanaka G, Al- David Burstein mendral JA, Aihara K. LoS ONE 10(4):e0123722 (2015). Swarthmore College doi:10.1371/journal.pone.0123722] the dynamical robust- [email protected] ness of correlated networks, assortative and disassortative, consisting of diffusively coupled oscillators. To see the ef- fect of the network assortativity, we have fixed the degree CP1 distribution of the networks and changed the correlations Spatiotemporal Feedback and Network Structure of the degrees between the connected nodes using two edge- Drive and Encode C. Elegans Locomotion rewiring methods. We show that the network assortativ- ity enhances the dynamical robustness and the network Using a computational model of Caenorhabditis elegans disassortativity can have a positive or negative impact on connectome dynamics, we show that proprioceptive feed- the dynamical robustness depending on the edge-rewiring back is necessary for sustained dynamic responses to exter- method used. Precisely, we find that the disassortative nal input. This is consistent with the lack of biophysical networks with the same assortativity coefficient can have evidence for a central pattern generator, and recent ex- qualitatively different types of topology, leading to the dif- perimental evidence that proprioception drives locomotion. ference in the dynamical robustness. We show that the low-dimensional functional response of the C. elegans network of neurons to proprioception-like Juan A. Almendral feedback is optimized by input of specific spatial wave- Rey Juan Carlos University lengths which correspond to the spatial scale of real body Madrid, Spain shape dynamics. Furthermore, we find that the mo- [email protected] tor subcircuit of the network is responsible for regulat- ing this response, in agreement with experimental expec- Takeyuki Sasai, Kai Morino tations. To explore how the connectomic dynamics pro- Graduate School of Information Science and Technology duces the observed two-mode, oscillatory limit cycle be- University of Tokyo, Tokyo, Japan havior from a static fixed point, we probe the fixed point’s takeyuki [email protected], low-dimensional structure using Dynamic Mode Decompo- [email protected] sition. This reveals that the nonlinear network dynam- ics encode six clusters of dynamic modes, with timescales Gouhei Tanaka spanning three orders of magnitude. Two of these six dy- Aihara Laboratory, Institute of Industrial Science namic mode clusters correspond to previously-discovered The University of Tokyo behavioral modes related to locomotion. These dynamic [email protected] modes and their timescales are encoded by the network’s degree distribution and specific connectivity. This suggests Kazuyuki Aihara that behavioral dynamics are partially encoded within the JST/University of Tokyo, Japan connectome itself, the connectivity of which facilitates pro- Dept of Mathematical Sciences prioceptive control. [email protected] James M. Kunert-Graf University of Washington CP1 [email protected] Convergence of the Spectral Radii for Random Di- rected Graphs with Community Structure Joshua L. Proctor Institute for Disease Modeling Prior literature has demonstrated how the spectral radius [email protected] DS17 Abstracts 121

Steven Brunton h : Rk → Rk is the coupling function, i labels the vertices University of Washington of a graph, and N(i) is the set of neighbors of the vertex i. [email protected] The novel aspect of the present work is the restriction of the admissible vector fields on the coupled cell network to Nathan Kutz various special classes. The first special case is the set of University of Washington systems with h(0) = 0. In this case, the subspace of fully Dept of Applied Mathematics phase-locked states, with xi = xj , is a robustly invariant [email protected] subspace. For some graphs, there are many invariant sub- spaces that are not a consequence of the symmetry alone. We also consider case where f and h are odd, and the case CP1 where f is odd and h is linear. We characterize the invari- Nonlinear Control of a Repulsively-Coupled Triad ant subspaces for each class of systems, and give examples Network with Multiple Attractors of interesting invariant subspaces for several graphs. We apply the results to coupled van der Pol oscillators. Complex networks are ubiquitous in the natural and man- made world. The ensemble behavior of networks depends James Swift on both the nature of the coupling, and the structure of the Department of Mathematics network. We use the Belousov-Zhabotinsky (BZ) reaction Northern Arizona University as a model oscillator and control topology by confining the [email protected] chemistry to micro-fabricated PDMS wells. We report on a theoretical framework that leverages the light sensitivity of the BZ catalyst to switch a ring of three inhibitory cou- CP2 pled wells between two equally strong dynamical attrac- Synchronization of Reaction Diffusion Neural Net- tors. The method, unlike typical control schemes, tem- works Via Passivity Theory and Its Direct Appli- porarily alters chemistry to rapidly alter the fixed points cation to Image Encryption of the system and their basins of attraction. By chang- ing the underlying dynamical landscape of the system, the We study synchronization in dynamical neural networks change from one attractor to the other is spontaneous, ac- consisting of N linearly and diffusively coupled identical complished without the need for closed-loop active control, reaction diffusion neural networks. In order to make the underscoring the robustness that can be achieved by judi- model more general we have considered the effect of time cious network design. The ability to switch readily between varying delays into the network model. Then, we derive a stable dynamical attractors is intriguing and provides the sufficient condition through constructing the suitable Lya- fundamental dynamical unit for ”gait” switching as well as punov function that ensures the synchronization between pointing towards information storage via dynamic states. the controlled and uncontrolled network model. Further, different types of controllers such as feedback control, sam- MichaelM.Norton pled data control and passivity control are applied into Brandeis University the addressed system and the advantages of each controller [email protected] is discussed through numerical approach. Moreover, local stability, global stability, bifurcation analysis, existence of Camille Girabawe chaos are also revealed through these control techniques. Brandeis University Besides that, the proposed network model is directly ap- Physics Department plied into the image secure communications that is encryp- [email protected] tion/decryption process. Based on the solutions obtained from the addressed system are utilized into the process. Thomas Litschel Statistical analyses such as Histogram, Peak-Signal-Noise- Brandeis University Ratio, Mean-Average-Error, and so on are also done to Department of Biochemistry prove the effectiveness of the proposed encryption algo- [email protected] rithm. Prakash Mani Seth Fraden National Post Doctoral Fellow Brandeis University [email protected] Department of Physics [email protected] Lakshmanan Shanmugam Deakin University, Australia CP1 [email protected] Invariant Subspaces for Pair-Coupled Networks CP2 We characterize invariant subspaces, also called synchrony subspaces, for coupled cell networks with the simplest type Dynamic Mode Decomposition of Resting State of architecture, for several subsets of the admissible vector Eeg Data - a Dynamical Systems Approach to Iden- fields described by Golubitsky, Stewart and others. We tifying Epilepsy Characteristics consider the case where all the cells have the same internal dynamics, and all the coupling is two-way and identical. Electroencephalography (EEG) is commonly used to diag- Thus, the architecture of the network is a graph. More nose epilepsy. However, such diagnoses are at best dif- specifically, we define uniform pair-coupled systems of the ficult unless the EEG is performed during or shortly af- form ter an epileptic seizure. Recent studies have suggested that the functional connectivity in the resting state (eyes x˙ i = f(xi)+ h(xi − xj ), closed but not sleeping) is altered in patients with tem- j∈N(i) poral lobe epilepsy (TLE). We report how such an alter- where f : Rk → Rk controls the internal dynamics, ation in the network can be identified by applying the re- 122 DS17 Abstracts

cently developed method called dynamic mode decomposi- mitter (NT), or increased through facilitation, where the tion (DMD). Unlike other methods DMD can extract the probability of release of NT is increased by prior activa- spatio-temporal dynamics which characterise large scale tion. These competing effects can create a complicated EEG data. We show how from such an analysis novel nu- and subtle range of time dependent connectivity. Here we merical measures can be obtained to identify network dif- investigate the probabilistic properties of facilitation and ferences and characteristics for TLE and control subjects. depression (FD) for a presynaptic neuron that is receiv- ing a Poisson spike train of input. We use a model of FD that was parameterized with experimental data from Karin Mora a basket cell and pyramidal cell connection (roughly 8-10 University of Paderborn synapses were counted), for fixed frequency input spikes. [email protected] Hence our results will apply to micro circuits in the hip- pocampus that are responsible for the gamma rhythms as- Michael Dellnitz sociated with learning and memory. University of Paderborn, Germany [email protected] Emily F. Stone Dept. of Mathematical Sciences The University of Montana Solveig Vieluf, Claus Reinsberger [email protected] Department of Sports and Health University of Paderborn, Germany [email protected], Elham Bayat-Mokhtari [email protected] University of Montana - Missoula [email protected]

CP2 CP2 Topological Changes in Chaotic Neuron Models Transient Chaos and Switching Behavior in Two Spike-adding bifurcations are special bifurcations that are Synaptically Coupled Layers of Morris-Lecar Neu- commonintheslow-fastdynamicsofneuronmodels.Un- rons derstanding how to generate and control a burst of spikes Spatiotemporal chaos collapses to either a rest state or a in neuron cells, and how chaotic behavior can appear in propagating pulse solution in a ring network of diffusively such systems are some of the most fundamental questions coupled Morris-Lecar neurons. The synaptic coupling of in neuroscience. The key questions that we want to ad- two such ring networks reveals system intrinsic switching dress at this presentation are: how this chaotic behavior of patterns between the layers. E.g., the collapse of spa- is organized and how spike-adding bifurcations influence tiotemporal chaos to a pulse in one layer is accompanied chaotic behavior. We will show how the orbit-flip (OF) in the other layer by the initiation of spatiotemporal chaos codimension-2 bifurcation points, placed in homoclinic bi- from a pulse. Other observed transient behavior includes furcation curves and related with the spike-adding bifur- erratic pulse propagation, pulse distance maximization, or cations, originate countable pencils of period-doubling and spontaneous systemwide neuron activity. saddle-node (of limit cycles) bifurcation lines, but also of symbolic-flip bifurcations. These bifurcations appear in- Renate A. Wackerbauer terlaced and generate the different symbolic sequences of University of Alaska Fairbanks periodic orbits. These periodic orbits become unstable, Department of Physics constituting the skeleton of the different chaotic attractors [email protected] and determining their topological structure.

Sergio Serrano, Roberto Barrio Harrison Hartley University of Zaragoza, SPAIN UAF Physics Department [email protected], [email protected] [email protected]

Marc Lefranc CP3 PhLAM/Universit´e Lille I, Hidden Structures of Information Transport Un- France derlying Spiral Wave Dynamics [email protected] A spiral wave is a macroscopic dynamics of excitable media Angeles Martinez that plays an important role in several distinct systems, Departamento de Matematica Aplicada and IUMA. CoDy including the Belousov-Zhabotinsky reaction, seizures in group. the brain, and lethal arrhythmia in the heart. Because University of Zaragoza, Spain spiral wave dynamics can exhibit a wide spectrum of be- [email protected] haviors, its precise quantification can be challenging. Here we present a hybrid geometric and information-theoretic approach to quantifying spiral wave dynamics. We demon- CP2 strate the effectiveness of our approach by applying it Effect of Neuromodulation on Connectivity to numerical simulations of a two-dimensional excitable Properites at Hippocampal Synapses medium with different numbers and spatial patterns of spi- ral waves. We show that, by defining information flow over Neurons in a microcircuit connected by chemical synapses the excitable medium, hidden coherent structures emerge can have their connectivity affected by the prior activity that effectively quantify the information transport under- of the cells. The number of synapses available for releas- lying spiral wave dynamics. Most importantly, we find that ing neurotransmitter can be decreased by repetitive acti- some coherent structures become more clearly defined over vation through depletion of readily releasable neurotrans- a longer observation period. These findings provide valid- DS17 Abstracts 123

ity with our approach to quantitatively characterize spiral University of California, Berkeley wave dynamics by focusing on information transport. Our [email protected] approach is computationally efficient and is applicable to many excitable media of interest in distinct physical, chem- Bastian Last Name, Isidora Araya, Marcel Clerc ical and biological systems. Our approach could ultimately Universidad de Chile contribute to an improved therapy of clinical conditions [email protected], [email protected], such as seizures and cardiac arrhythmia by identifying po- marcel@dfi.uchile.cl tential targets of interventional therapies. Claudio Falcon Hiroshi Ashikaga Departamento de F´ısica Johns Hopkins University School of Medicine Universidad de Chile, Santiago, Chile [email protected] [email protected] Ryan G. James Edgar Knobloch University of California, Davis University of California at Berkeley [email protected] Dept of Physics [email protected] CP3 Spectrum of Singularities in Gravito-Capillary CP3 Waves: A Phillips Spectrum Proxy Stretching and Speed in Excitable Advection- We report the observation of the spectrum of singularities Reaction-Diffusion Systems of gravity surface waves driven by a horizontally moving wave maker interacting with Faraday waves which serve An advection-diffusion-reaction system is said to be ex- as a Phillips’ spectrum proxy. We measure the tempo- citable if growth occurs only when the local concentration ral fluctuations of the surface wave amplitude at a given exceeds a critical threshold. Advection (flow) can either location and we show that, for a wide range of forcing pa- start or stop reaction by driving the concentration above or rameters, they display a power-law spectrum that greatly below that threshold. I will present evidence that stretch- differs from the one predicted by the WT theory but coin- ing (the largest eigenvalue of the right Cauchy-Green strain cides with Phillips’ spectrum for gravity waves. We com- tensor) quantifies that effect, and that an optimal range of pute the probability density function of the local surface stretching maximizes reaction. I will also discuss ongoing height increments, which show that they change strongly work distinguishing the effects of stretching from the ef- across time scales. The structure functions of these in- fects of speed and precisely measuring the range of optimal crements are shown to display power-laws as a function stretching. of the time lag, with exponents that are not linear with Douglas H. Kelley the order of the structure function, thus showing that the University of Rochester wave field is intermittent. We argue that the origin of this 218 Hopeman Engineering Building Rochester, NY scale-invariant intermittent spectrum is the Faraday wave 14627-0132 pattern breakup due to its advection by the propagating [email protected] gravity waves, which can be related directly to the Phillips’ spectrum phenomenology.

Claudio Falcon, Gustavo Castillo CP3 Departamento de F´ısica Analysis of Pattern Emergence in Turing Systems Universidad de Chile, Santiago, Chile with Inhomogeneity in Reaction Term [email protected], [email protected] Turing’s reaction-diffusion model is widely used as a model for describing self-organized spatial pattern formation and CP3 it is well studied in the case of constant coefficients. The Slanted Snaking of Localized Faraday Waves spatial dependence yields a wider range of possible patterns but at the same time significantly complicates the analy- We perform extensive experiments on parametrically ex- sis. Here, we will consider a small spatial dependency in cited waves on the surface of a water-surfactant mixture in the coefficient of the linear term of the activator kinetics a vertically vibrated Hele-Shaw cell. The experiments re- and analyse the effect of this perturbation on the pattern veal the presence of both spatially extended and spatially and the conditions of pattern emergence. At first, we dis- localized standing oscillations called Faraday waves. The cuss which steady state will be designated as a pattern and experiments show that localized Faraday waves are found which not, then we apply stability analysis and carry out both inside and outside the hysteresis loop between spa- some supportive numerical simulations. Combining both tially extended Faraday waves and the flat interface, and approaches we determine conditions for the emergence of that they display a snaking structure that is slanted. We the pattern in such system. Moreover, we obtain a tool for attribute this behavior to the presence of a conserved quan- formation of patterns with a spatially varying frequency. tity, the liquid volume, and show that a forced complex These results contribute to complete picture of Turing’s Swift–Hohenberg equation coupled to a conserved mode model and its possible applicability to real situations. provides an excellent qualitative model for describing the experimental observations, including the observed slanted Michal Kozak snaking and the presence of localized waves outside the hys- Dept of Mathematics, FNSPE, teresis loop. General properties of the model are studied. Czech Technical University in Prague [email protected]

Punit R. Gandhi Vaclav Klika 124 DS17 Abstracts

Czech Technical University in Prague Massachusetts Institute of Technology vaclav.klika@fjfi.cvut.cz [email protected]

Eamonn Gaffney Mathematical Institute CP4 University of Oxford Understanding the Geometry of Transport: Diffu- gaffney@maths.ox.ac.uk sion Maps for Lagrangian Trajectory Data Unravel Coherent Sets.

CP4 Dynamical systems often exhibit the emergence of long- Numerical Simulation of Immiscible and Misci- lived coherent sets, which are regions in state space that ble Component-Based Co2-Oil-Rock Interaction to keep their geometric integrity to a high extent and thus Enhance Oil Recovery in Heterogeneous Reservoirs play an important role in transport. In this talk, a method for extracting coherent sets from possibly sparse A compositional reservoir simulation was constructed to Lagrangian trajectory data is discussed. Our method can model the fluid flow through porous media at the CO2 be seen as an extension of diffusion maps to trajectory Flooding. The CO2 is injected in heterogeneous reservoirs space, and it allows us to construct ’dynamical coordinates’ through vertical injection wells to enhance oil recovery. Oil which reveal the intrinsic low-dimensional organization of is produced through horizontal wells at the bottom of the the data with respect to transport. The only a priori knowl- reservoir. That model tracks the CO2-Oil-Rock chemical edge about the dynamics that we require is a locally valid interactions in reservoir conditions based on their compo- notion of distance, which renders our method highly suit- nents. Due to the gravity segregation, the gas is accumu- able for automated data analysis. We show convergence to lating at the top of the reservoir to formulate a gas cap the analytic transfer operator framework of coherence in and oil drain down the reservoir towards horizontal pro- the infinite data limit, and illustrate potential applications duction wells. The efficiency of CO2 sweep process de- on several two- and three-dimensional examples as well as pends on the difference between the injection and reservoir real world data. pressure. Higher pressure results in miscible flooding and more interaction between CO2 and oil and higher sweep Ralf Banisch efficiency. In order to show the effectiveness of immiscible Freie Universit¨at Berlin and miscible processes to reach promising levels of oil re- [email protected] covery, the compositional model was implemented for a 25 year of prediction period. The incremental recovery fac- Peter Koltai tor through immiscible and miscible cases reach to 30% Free University Berlin and 42%, respectively. In addition, the production rates [email protected] during the entire prediction period of the miscible flooding process can also be identified as higher than the immisci- ble case. However, the miscible continuous CO2 injection CP4 leads to a much higher Gas-Oil Ratio than the immiscible case as fast gas flooding cause early gas breakthrough into Lagrangian Chaos in a Differentially-Heated Three- the oil production wells. Dimensional Cavity

Watheq J. Al-Mudhafar, Dandina N. Rao Natural convection plays a key role in fluid dynamics Louisiana State University owing to its ubiquitous presence in nature and indus- [email protected], [email protected] try. Buoyancy-driven flows are prototypical systems in the study of thermal instabilities and pattern formation. The differentially-heated cavity problem has been widely stud- CP4 ied for the investigation of buoyancy-induced oscillatory Internal Wave Bolus Detection and Analysis by La- flow. However, far less attention has been devoted to the grangian Clustering three-dimensional Lagrangian transport properties in such flows. This study seeks to address this by investigating The shoaling of vertical mode internal waves on a conti- Lagrangian transport in the steady flow inside a cubic cav- nental slope produces boluses, which are trapped regions ity differentially-heated from the side. The theoretical and of fluid that travel up the slope with the wave. Unlike a numerical analysis expands on previously reported similar- propagating solitary wave, these boluses transport mate- ities between the current flow and lid-driven flows. The rial with the wave containing oxygen depleted water and Lagrangian dynamics are controlled by the P´eclet number induce rapid changes in temperature both of which have (Pe) and the Prandtl number (Pr). Pe controls the be- potential ramifications for marine biology. We extend a havior qualitatively in that growing Pe progressively per- number of two-layer studies by investigating bolus genera- turbs the integrable state (Pe→0), thus paving the way tion and material transport in continuously stratified flu- to chaotic dynamics. In general, increasing Pe promotes ids. Laboratory experiments are conducted in a 4 m long the onset of chaos. Pr plays a quantitative role, it acts tank and are complemented by 2-dimensional numerical as the control parameter for the relative contribution of simulations of the Navier-Stokes equations. The bound- fluid inertia and convection: “small’ and “large’ Pr cor- aries of the bolus are identified using a Lagrangian based respond to inertia-dominated and convection-dominated coherent structure method relying on trajectory cluster- flows, respectively. The transition from inertia-dominated ing. We investigate how to appropriately implement fuzzy to convection-dominated flow triggers a bifurcation in the c-means clustering to objectively identify the transported Lagrangian flow topology. material forming the bolus and measure the properties of the bolus as a function of the pycnocline thickness. Sebastian Contreras Osorio Eindhoven University of Technology Michael Allshouse TU/e Dept. of Mechanical Engineering [email protected] DS17 Abstracts 125

Michel Speetjens [email protected] Laboratory for Energy Technology, Dept. Mech. Eng. Eindhoven University of Technology [email protected] CP5 Dynamical Properties of Viral Defective Interfer- Herman Clercx ing Particles Fluid Dynamics Laboratory, Dept. Applied Physics Eindhoven University of Technology Defective Interfering Particles (DIPs) are non viable viral [email protected] particles which occur rarely in Nature and have the capa- bility of displacing wild type viruses, slowing down or even preventing infection. Their occurrence has been shown in CP4 several RNA viral species, including Polio, Ebola and HIV. The dynamical properties of these particles, and their co- Angular Dynamics of Non-Spherical Particles Set- evolutionary dynamic with co-infecting viral species, have tling in Turbulence not been widely investigated. We will present a multiscale model which accounts for the competition between DIPs We study the angular dynamics of small spheroidal par- and wild type viruses. Deterministic and stochastic simu- ticles settling in homogeneous isotropic turbulence. Our lations of DIPs replication models and analysis of experi- simulations of the angular dynamics using direct numeri- mentally viable parameter phase space allows for effective cal simulation of turbulence show that turbulence induces design of DIPs as therapeutic. A set of experimental re- an orientation bias: disk-like particles tend to settle edge sults on Poliovirus will be presented to support the model. first, and rod-like particles settle tip first. This bias de- pends sensitively on the turbulent dissipation rate, and on Simone Bianco the particle size and shape. We explain the underlying IBM Research mechanisms using a perturbative analysis of a statistical [email protected] model. Our results apply to small ice crystals settling in turbulent clods, small enough so that the effect of fluid inertia can be neglected. Igor Rouzine University of California San Francisco [email protected] Bernhard Mehlig Gothenburg Univ Gothenburg, Sweden CP5 [email protected] Impact of Heart Tissue Anisotropy and Ischemic Heterogeneities on Unpinning and Termination of Pinned Spirals: Insights from Simulations. CP4 Lagrangian Chaos and Transport in Geophysical Reentries are strictly related to the complex structure of Fluid Flows cardiac tissue, characterized by multi-sized heterogeneities. Spiral waves pinned to an heterogeneity can either undergo a process of spontaneous termination or be unpinned (and Advances in our understanding of the dynamics of terminated) using electric far field pulses recruiting the mesoscale eddies and vortex rings of the Agulhas Current heterogeneity as a virtual electrode. We implement a bido- Retroflection, and the Gulf Stream in the Middle Atlantic main formulation of the phase I of the Luo and Rudy model Bight have been based on modons which are nonlinear in 2D under ischemia. We investigate how size of ischemic Rossby solitary wave solutions of quasi-geostrophic equa- heterogeneities and tissue anisotropic properties may af- tions. This article focuses primarily on geometric singu- fect the evolution of reentrant dynamics in the presence lar perturbation (GSP) theory and Melnikov techniques to and in the absence of far field pacing (FFP). In absence address the problem of adiabatic chaos and transport for of FFP, the main findings show that: a) for the stability translating and rotating modons to the quasi-geostrophic of the waves, changes of conductivity in the intracellular potential vorticity system which is relevant, e.g., a La- space are more critical than alterations in the extracellu- grangian and Eulerian analysis of a geophysical fluid flow. lar space; b) the maintenance or the self-termination of A very general and central question is what hypotheses on pinned spirals is strongly dependent not only on the size of the equations and singular solutions guarantee that the the heterogeneities but also on the degree of intracellular solutions approximate some solutions for the perturbed anisotropy. Unpinning and termination of pinned spirals quasi-geostrophic potential vorticity system. We present under FFP are better understood in isotropic media than a geometric approach to the problem which gives more in anisotropic substrates. Therefore, to study the impact of refined a priori energy type estimates on the position of anisotropy, we compare the success of sequences of far field the invariant manifold and its tangent planes as the man- pulses in the isotropic and in the anisotropic cases. Our ifold passes close to a normally hyperbolic piece of a slow results clearly indicate that the range of pacing parameters manifold. We apply Melnikov technique to show that the resulting in successful termination of pinned spiral waves Poincare map associated with modon equations has trans- is larger in anisotropic than in isotropic tissues. verse heteroclinic orbits. We appeal to the Smale-Birkhoff Homoclinic Theorem and assert the existence of an invari- Edda Boccia ant hyperbolic set which contains a countable infinity of Max Planck Institute for Dynamics and Self-Organization unstable periodic orbits, a dense orbit and infinitely many [email protected] heteroclinic orbits. Stefan Luther, Ulrich Parlitz Maleafisha Stephen Tladi Max Planck Institute for Dynamics and Self-Organization University of Limpopo, South Africa Research Group Biomedical Physics 126 DS17 Abstracts

[email protected], [email protected] We discuss the application of our geometric framework to models for regulated gene expression that involve addi- tional stages. CP5 Transovarial Transmission in Vector-Borne Relaps- Frits Veerman, Nikola Popovic ing Diseases University of Edinburgh [email protected], [email protected] In this talk we consider a relapsing disease in which there are two modes of transferring the disease throughout the Carsten Marr host and vector populations: interaction between vectors Institute for Computational Biology and hosts, and the transmission of the disease to the Helmholtz Centre Munich progeny of a vector. We examine how each of these mech- [email protected] anisms drives the persistence of the disease and give con- ditions for determining which mechanism is the dominant driver of disease spread. We then discuss how this infor- mation informs control strategies for the disease. CP5 Dynamic Responses of a Steroidogenic Regulatory Cody Palmer, Erin Landguth, Tammi Johnson Network to Physiological Perturbations University of Montana [email protected], [email protected], [email protected] The stress response is mediated by glucocorticoid hormones (CORT) secreted by the adrenal glands upon stimulation by adrenocorticotropic hormone (ACTH) from the pitu- CP5 itary. These hormones exhibit a complex pattern of highly correlated ultradian oscillations that becomes altered dur- Global Stability of Zika Virus Mathematical Model ing inflammation. In particular, the role of intra-adrenal with Recurrence Based on Treatment control mechanisms on the dynamic dissociation of these In this paper, a mathematical model for dynamic of zika hormones is poorly understood. We explore the origins of virus with recurrence based on treatment is proposed and this dissociated dynamics through a mathematical model analyzed for stability of infected states. The virus free of the intra-adrenal regulatory network controlling the syn- and endemic equilibrium are locally stable if they are less thesis of CORT, accounting for both genomic and non- than unity and unstable if greater than unity. A thresh- genomic processes occurring at different time scales within old parameter is obtained to check the spread of the virus the network. We test the model through computer sim- qualitatively and quantitatively. It has been shown that ulations of ACTH perturbations, and successfully predict the timely treatment to zika virus lead to the effective con- responses to acute stressors, as judged from experiments trol of zika virus. Sensitivity analysis is also carried out to in rats injected with single pulses of ACTH and a bacterial see the effects of zika virus on environment. toxin (LPS) that triggers an inflammatory response. Our results suggest that a negative feedback loop, genomic and Ram Singh non-genomic processes occurring at different time scales, Baba Ghulam Shah Badshah University and post-transcriptional and post-translational regulation singh [email protected] are essential control mechanisms for the network to mount a fast, transient, and controlled CORT response to ACTH. We also show that cross-talk between the immune pathway CP5 and the steroidogenic regulatory network may underlie the Geometric Analysis of Fast-Slow Models for ACTH/CORT dissociated dynamics, and suggest the pres- Stochastic Gene Expression ence of novel regulatory mechanisms that can be tested experimentally. Stochastic models for gene expression frequently exhibit dynamics on several different scales. One such time-scale Eder Zavala separation is caused by significant differences in the life- University of Exeter times of mRNA and protein; the ratio of the two degrada- [email protected] tion rates appears as a small parameter in the associated Chemical Master Equation, allowing for the application Francesca Spiga of perturbation techniques. We present a framework for University of Bristol the analysis of a family of fast-slow models for gene ex- [email protected] pression, based on geometric singular perturbation theory. We demonstrate the approach by giving a complete char- Jamie Walker acterisation of a standard two-stage model which assumes University of Exeter transcription, translation, and degradation to be ?rst-order [email protected] reactions. We present a systematic expansion procedure for the probability-generating function that can be taken to any order in the perturbation parameter, allowing for Zidong Zhao an approximation of the corresponding propagator prob- University of Bristol abilities to that order. Focussing on biologically relevant [email protected] parameter regimes that induce translational bursting, as well as those in which mRNA is frequently transcribed, Stafford Lightman we find that the first-order correction can significantly im- University of Bristol, United Kingdom prove the steady-state probability distribution. Similarly, Staff[email protected] in the time-dependent scenario, inclusion of first-order fast asymptotics yields an approximation for the propagator John Terry probabilities that is superior to the slow dynamics alone. University of Exeter. DS17 Abstracts 127

[email protected] [email protected]

CP6 CP6 When Chaos Meets Hyperchaos: a Computer- The Search for the Smallest Chimera Assisted Proof on the 4D Rossler Model We demonstrate that chimera behavior can be observed It has recently been reported that it is quite difficult to dis- in small networks consisting of three identical oscillators, tinguish between chaos and hyperchaos in numerical sim- with mutual all-to-all coupling. Three different types of ulations which are frequently “noisy’. In this presentation chimeras, characterized by the coexistence of two coherent we show that, for the classical 4D Rossler model, the coex- oscillators and one incoherent oscillator, (i.e. rotating with istence of two invariant sets with different nature (a global another frequency) have been identified, where the oscilla- hyperchaotic invariant set and a chaotic attractor) and the tors show periodic (two types) and chaotic (one type) be- homoclinic and heteroclinic connections between their un- haviors. Typical bifurcations at the transitions from full sy- stable periodic orbits give rise to long hyperchaotic tran- chronization to chimera states and between different types sient behavior, and therefore it provides a mechanism for of chimeras have been identified. Parameter regions of syn- noisy simulations. The Computer-assisted proof of this chronization for the chimera states are obtained in the form coexistence of chaotic and hyperchaotic behaviors com- of Arnold tongues, issued from a singular parameter point. bines topological and smooth methods with rigorous nu- Our analusis suggests that chimera states can observed in merical computations. The existence of (hyper)chaotic sets small networks, relevant to various real-world systems. is proved by the method of covering relations and cone con- ditions. Tomasz Kapitaniak Roberto Barrio Technical University of Lodz University of Zaragoza, SPAIN Dept of Dynamic [email protected] [email protected]

Angeles Martinez CP6 Departamento de Matematica Aplicada and IUMA. CoDy group. Chimeras and Chaotic Mean Field Dynamics in University of Zaragoza, Spain Two Populations of Phase Oscillators with Hetero- [email protected] geneous Phase-Lag

Sergio Serrano The simplest network of coupled phase-oscillators exhibit- University of Zaragoza, SPAIN ing chimera states is given by two populations with dis- [email protected] parate intra- and inter-population coupling strengths. We explore the effects of heterogeneous coupling phase-lags be- Daniel Wilczak tween the two populations. Such heterogeneity arises nat- Jagiellonian University, Krakow, POLAND urally in various settings in nature and technology. Break- [email protected] ing the phase-lag symmetry results in a variety of states with uniform and non-uniform synchronization, includ- ing in-phase and anti-phase synchrony, full incoherence, CP6 chimeras with 0 or π phase separation between popula- tions, and states where both populations remain desyn- Bounding Time Averages Rigorously Using chronized. These desynchronized states exhibit stable, Semidefinite Programming oscillatory, and even chaotic dynamics. We identify the bifurcations through which chimera and desynchronized I will describe computer-assisted methods for proving states emerge and analyze how chaotic mean field dynam- bounds on infinite-time averages in differential dynamical ics arises. While these chaotic dynamics are attractors in systems. The methods rely on the construction of nonneg- the Ott-Antonsen equations, the finite size system displays ative polynomials with certain properties, similarly to the only transient chaos – thus we address a question raised by way nonlinear stability can be proven by the construction Ott and Antonsen. We discuss how associated decay times of Lyapunov functions. Nonnegativity is enforced by re- scale with system size and are affected by oscillator het- quiring the polynomials to be sums of squares, a condition erogeneity. These findings elucidate previous experimental which is then formulated as a semidefinite program (SDP) results in a mechanical oscillator network of and provide that can be solved computationally. Although such compu- insight into the breakdown of synchrony in biological sys- tations are subject to numerical error, rigorous results can tems. be obtained in one of two ways: using interval arithmetic to control the error of an approximate SDP solution, or Erik A. Martens finding exact analytical solutions to relatively small SDPs. University of Copenhagen I will illustrate these methods using the Lorenz equations, Max Planck Institute for Dynamics & Self-Organization for which they produce novel bounds on time averages of a [email protected] number of different quantities. Some of these bounds, such as the upper bound on the average of z3, are perfectly sharp and depend analytically on the control parameter r.Many Christian Bick other bounds are within 1% of being sharp, which would University of Exeter, UK be nearly impossible without computer assistance. [email protected]

David Goluskin Mark J. Panaggio Deparment of Applied Mathematics Engineering Science and Applied Mathematics Columbia University Northwestern University 128 DS17 Abstracts

[email protected] Physics [email protected]

CP6 Pseudorandom Number Generation Using Chaotic CP7 True Orbits of the Bernoulli Map on Algebraic In- PhaseSpaceTransportinHydrogeninCrossed tegers Fields: Role of the Local Surface of Section

We introduce two methods for generating pseudorandom Ionization of hydrogen in crossed electric and magnetic binary sequences using true orbits of the Bernoulli map fields has been of scientific interest for many years. Not x → 2x mod 1. The characteristic of these methods is only is this a beautiful example of a chaotic system, but it that they exactly compute chaotic orbits of the Bernoulli is also widely considered a stepping stone to understanding map on algebraic integers of degree two and three. In ad- the escape in the classic gravitational three body problem dition, we develop ways to properly select initial points The electrons classical motion resembles the motion of the (seeds), which can distribute initial points almost uni- Moon in the Sun-Earth-Moon three body system. One of formly (equidistantly) in the unit interval and which can the major challenges in studying chaotic ionization of hy- avoid mergers of the orbits starting from them. We also drogen in crossed electric and magnetic fields is the ability demonstrate through statistical testing that the generated to define a Poincare surface of section that captures all the sequences have good randomness properties. (See [A. Saito allowed electron trajectories, but at the same time does not and A. Yamaguchi, Chaos 26, 063122 (2016)] for details of suffer from tangencies. We will present a prescription for the pseudorandom number generation using algebraic in- defining a Poincare return map that defines a local surface tegers of degree two.) of section which governs the ionization process.

Asaki Saito Korana Burke Future University - Hakodate UC Davis [email protected] [email protected]

Akihiro Yamaguchi Kevin A. Mitchell Fukuoka Institute of Technology University of California, Merced aki@fit.ac.jp Physics [email protected] CP6 Homotopy Method for Characterizing Topological CP7 Chaos in Three Dimensions Internal Transport Barriers in Plasmas with Re- versed Plasma Flow In a wide variety of physical systems, the dynamics of topo- logically robust structures underlie a powerful class of tools We investigate the influence of the radial electric field pro- for characterizing complexity. While many such topologi- file on internal particle transport barriers in plasma dis- cal techniques have been applied to great effect in 1D and charges with reversed plasma flow. In this investigation 2D dynamics, extending their reach to 3D systems has we apply a bidimensional symplectic drift wave map that proven to be quite difficult. We successfully modify the describe the plasma particle transport and allows the in- homotopic lobe dynamics (HLD) technique to work for 3D tegration of the particle drift in the presence of a given volume preserving maps. Specifically, we use intersecting electrostatic turbulence spectrum. With this procedure we two-dimensional stable and unstable invariant manifolds show that the reversed flow gives rise to shearless invariant to construct a symbolic representation of the topological lines, acting as transport barriers inside the plasma. More- dynamics. This symbolic representation can be used to over, by varying the radial electric field profile, we observe compute measures of complexity, such as the topological the formation and destruction of these internal transport entropy, and discover distinct regions of the system which barriers. Applicability of our results are discussed for the have qualitatively different behavior. We apply the 3D Texas Helimak, a toroidal plasma device in which the ra- HLD technique to an explicit numerical example in fluid dial electric field can be changed by application of bias dynamics: a time-periodic perturbation of Hill’s spherical potential. vortex, modified to break both rotational symmetry and integrability. Importantly, the 3D HLD topological tech- Ibere L. Caldas nique is able to detect a distinction between the topolog- Institute of Physics ically forced 2D stretching rate of material surfaces and University of Sao Paulo the 1D stretching rate of material curves, illustrating the [email protected] truly 3D nature of our approach. Additionally, the sym- bolic dynamics point towards an intriguing duality between Rafael Ferro the regions participating in 2D and 1D stretching. Institute of Physics, University of Sao Paulo [email protected] Spencer A. Smith Mount Holyoke College [email protected] CP7 Whole-Building Fault Detection: A Scalable Ap- Joshua Arenson proach Using Spectral Methods University of California Merced [email protected] Rules-based fault detection (i.e. expert systems) have be- come ubiquitous as an approach to designing a decision- Kevin A. Mitchell making support system. In particular, building manage- University of California, Merced ment systems (BMS) make extensive use of rules-based DS17 Abstracts 129

techniques to monitor telemetry data generated that is gen- an electric potential difference and a radial shear. Due to erated by the large array of devices that are present in com- the Smectic A nature of the liquid crystal, the fluid can be mercial buildings. In this talk, an extension to rules-based considered two-dimensional and is modelled using the 2- fault detection is demonstrated utilizing properties of the D incompressible Navier-Stokes equations coupled with an Koopman operator. The Koopman operator is an infinite- equation for charge continuity. A Newton-Krylov method dimensional, linear operator that captures nonlinear, finite is implemented to identify the transitions of the flow that dimensional dynamics. The definition of the Koopman op- result due to changes in the model parameters. The pri- erator enables algorithms that can evaluate the magnitude mary transition from axisymmetric flow to rotating waves and coincidence of time-series data. Using properties of and the secondary transition from rotating waves to am- this operator, diagnostic rule signals from building man- plitude vacillation (i.e. amplitude-modulated waves) are agement system (BMS) trend data can be decomposed into investigated. The rotating waves are relative equilibria of components that allow the capture of device behavior at the system, and we discuss the effectiveness of using this varying time-scales and to a granular level. As it relates characteristic to simplify the computations. to the implementation of fault detection (FDD), this ap- proach creates additional spatial and temporal characteri- Greg Lewis, Jamil Jabbour zations of rule signals providing additional data structure UOIT and increasing effectiveness with which classification tech- [email protected], [email protected] niques can be applied to the analysis process. The ap- proach permits a knowledge base to be applied in a similar Mary Pugh manner to that of a rules-based approach, but the intro- University of Toronto duced extensions also facilitate the definition of new kinds Department of Mathematics of diagnostics and overall provide increased analysis poten- [email protected] tial. Stephen Morris Michael Georgescu University of Toronto, Canada University of California, Santa Barbara [email protected] [email protected]

CP7 CP7 Rotating Stall in Turbomachinery Compressors: a Numerical Study of Mixed Convection Effects in a Dynamical Systems Approach Ventilated Room with Different Locations of Ther- mosolutal Block Turbomachinery compressors experience rotating stall as the operating conditions for maximum efficiency are ap- A mixed convection analysis is performed over a ventilated proached. Rotating stall consists of a spatially localized domain to find the thermal and solutal effects over a ther- pulse of flow blockage that covers a few blade passages and mosoluted block by supplying mechanically driven cold air rotates at a fraction of the rotating speed of the compres- at the inlet. The block is maintained with higher temper- sor. The prediction and control of stall onset is of great in- ature and concentration than that of inlet fluid and the terest for the Turbomachinery industry since it has a strong walls are considered as impermeable and adiabatic to heat negative impact in the performance of the compressor, and and solute. To study the density influence over the fluid, it also generates large unsteady loads that increase the vi- the cold fluid is performed at the lower left vertical wall bration level of the compressor blades, and can ultimately and flushes out through the upper section of the opposite lead to blade failure. Rotating stall can be regarded as a wall. The mixed air distribution and thermosolutal effec- solitary wave that propagates from blade to blade around tive rates due to the adiabatic source is analysed by chang- of the compressor rotor. In this presentation we will give ing the block position to obtain the appropriate position and introductory overview of the stall phenomena in turbo- for maximum cooling/heating. The efficient cooling and machinery compressors, which is a very interesting example maximum variation of average temperature is obtained for of a localized propagating state that appears in a realistic higher Richardson number and Reynolds number for a fixed industrial situation. We will present some numerical sim- Prandtl number (Pr = 0.71) with buoyancy ratio (Br=1.0). ulations of rotating stall in a simplified 2D linear cascade, and we will also discuss the possibility and advantages of using the tools from Dynamical Systems to analyze stall Neha Gupta onset and propagation. Research Scholar Department of Mathematics, IIT Roorkee Carlos Martel [email protected] Universidad Politecnica de Madrid [email protected] Ameeya Nayak Indian Institute Of Technology Roorkee [email protected] CP8 Stabilized Coexistence Among Mutual Cheaters in Cyclic Public Goods Games with Optimized Taxa- CP7 tion Newton-Krylov Continuation for Annular Electro- convection We study the problem of stabilized coexistence in a three- species public goods game in which each species simultane- We investigate the flow transitions in sheared annular ously contributes to its own public good while freeloading electroconvection using matrix-free numerical bifurcation off another species’ public good. We assume population methods. In particular, we study a model that simulates growth is governed by absolute success as a function of the the flow of a liquid crystal film in the Smectic A phase sus- profit from a species’ own public good as well as the return pended between two annular electrodes, and subjected to from free-loading off another public good. We show that 130 DS17 Abstracts

proportional population growth is governed by a replicator nal electrochemical motors, with periodic motions arising dynamic with at most one interior unstable fixed point; i.e. from a Hopf bifurcation dynamic instability predicted by that the population becomes dominated by a single species. a model of a double elastic filament with axonemal con- We then show that applying an externally imposed “tax’ nections. The rich dynamics of cilia-like structures against on success can stabilize the interior fixed point, allowing for the relatively simple internally driven mechanism, which is the symbiotic coexistence of all species. We show that the based on differential axial displacements of the filaments interior fixed point is the point of globally minimal total at the common base, make them suitable for robotic ap- population growth (in number of species) in both the taxed plications that aim at miniaturization and bio-affinity. By and un-taxed cases and discuss a criterion under which the considering mechanical coupling through the base rather populations will collapse. We also show that these dy- than through environment hydrodynamics, we analyze the namics do not admit limit cycles through a diffeomorphic collective dynamics of arrays of double elastic filaments mapping to traditional rock-paper-scissors. Finally, we for- with axonemal connections. Synchronised patterns solu- mulate an optimal taxation problem, and show that it ad- tion such as metachronal waves hint at the possibility of mits a quasi-linearization that results in novel necessary using the model for bio-inspired terrestrial locomotion. conditions for the optimal control. In particular, the op- timal control problem governing the tax rate must solve a Davide Spinello certain second order ordinary differential equation. University of Ottawa [email protected] Christopher H. Griffin Applied Research Laboratory Penn State University CP8 griffi[email protected] Fish Schooling

Andrew Belmonte Fish schooling, one of animal swarming, is a commonly Department of Mathematics, Pennsylvania State observed phenomenon that is coherently performed by in- University tegration of interactions among constituent fish. This re- [email protected] markable phenomenon has already attracted interests of researchers from diverse fields including biology, physics, mathematics, computer engineering. In this talk, we will CP8 present Pedestrian Dynamics from Social Force Models (1) our fish schooling model. The model is performed by stochastic differential equations. Numerical simulation of pedestrians moving as mass points under the action of social forces and reacting to forces (2) quantitative investigations for the model. from obstacles provides a way to study crowd phenomenae. Many such social force models with inflexible parameters (3) pattern formation for fish schooling. Four obstacle suffer from limited validity when they are applied to vari- avoiding patterns of school have been observed, i.e., ous scenarios. We give examples of this, and demonstrate Rebound, Pullback, Pass and Reunion, and Separa- pathological cases of unrealistic stationary points arising tion which are performed just by tuning modeling pa- in models with potential forces. To overcome this prob- rameters. lem we propose, and mathematically formalize, a novel hy- (4) a scientific definition of the school cohesiveness. brid modeling approach including dissipation via a friction term, where pedestrian behavior is situation-dependent, i.e., switches between equations of motion according to the Ton V. Ta,LinhNguyen relative location to an obstacle. A number of scenarios are Kyushu University studied to illustrate the advantages of such a hybrid model [email protected], [email protected] approach. Atsushi Yagi Poul G. Hjorth Osaka University Technical Univ of Denmark [email protected] Department of Applied Mathematics and Computer Science [email protected] CP8 Mapping of Large-Scale Biological Dynamical Net- works CP8 Dynamics of Arrays of Coupled Cilia and Flagella- The unprecedented accumulation of high-throughput tem- like Structures Driven by Axonemal Beating poral biological data (e.g. gene expression, brain imaging data) provides an essential opportunity for researchers to Cellular beating protrusions such as cilia and flagella are study the dynamics of biological systems. At the same comprised of bundles of filaments internally connected by time, it raises new questions and grand challenges for ap- rod-like tubular elements called axonemes. Cilia are used plied mathematicians in view of the key characteristics of by cell-like structures for swimming and generally locomote these data that include high-dimensionality, multi-scale, in fluids and analogous physical media. Recent work fo- and heterogeneity. In response, we recently developed a cused on the dynamics of cilia and cilia arrays has revealed computational framework to model the temporal biological important features such as array alignments, two-phase data by identifying and mapping of large-scale dynamical asymmetric beating of individual filaments, and the emer- networks in a data-driven approach. The topological and gence of metachronal coordination with constant phase dif- statistical tools that we developed to encode and map the ference between contiguous cilia. The main underlying dynamical networks work very efficiently. We applied this mechanisms for these individual and cooperative behaviors method in the analysis of brain development gene networks have been identified as hydrodynamic coupling and inter- and provided novel insights into the field of mathematical DS17 Abstracts 131

biology. orbits. In this talk, I will present numerical methods for computing unstable manifolds of these orbits and show Lin Wan examples of such calculations on Kuramoto-Sivashinsky Chinese Academy of Sciences equation in one space dimension, and three-dimensional [email protected] Navier-Stokes equations.

Nazmi Burak Budanur CP8 School of Physics and Center for Nonlinear Science How NF-κB Oscillations Drive Stochastic Gene Ex- Georgia Institute of Technology, Atlanta GA 30332 pression [email protected]

Understanding how cells process information by translating external stimuli into an adequate transcriptional response CP9 is a fundamental question in molecular biology. Transcrip- Exploring Relations Between Smooth and Piece- tion factors play a key role in this process and for this rea- wise Smooth Models son they are typically tightly regulated by complex genetic circuits. This is the case of the transcription factor NF- Piecewise smooth and slow-fast systems are frequently used κB, whose deregulation can lead to chronic inflammation as alternative mathematical representations of the same and cancer: its activity is regulated by negative feedbacks phenomenon. From an applied scientists perspective, hav- that lead to damped oscillations upon external stimuli. Us- ing a choice between two languages can be useful, for in- ing population-level observations, we have shown that such stance to pick the most appropriate formalism for a given oscillations produce different dynamics of gene expression numerical analysis tool, as long as the differences in the for different genes. However, we are far from understand- results that can be expected are well understood. Unfor- ing how NF-κB modulates stochastic gene expression in tunately, the differences in the behaviour of a smooth and single cells. We have used a simple mathematical model of piecewise smooth system that were built to be close are the NF-κB regulatory system with just one negative feed- rarely well understood. This problem has been addressed, back loop- to show that the stochastic patterns of gene with some success, in the literature. What we propose here activation-inactivation are determined by a precise combi- is a small extension of this branch of research, mostly built nation of the timescales of the model. We will present here on classical results available in Filippov’s book. Our main our first insights on how such patterns are affected when results can be roughly stated as follows: Statement 1. The additional layers of complexity are added to the model. Fi- dynamics of a piecewise smooth system is a superset of nally, we will also present our first experimental data on the dynamics of the smooth system it represents (i.e., of how NF-κB regulates transcription in single cells, which its smoothing). Statement 2. The bifurcation diagram of suggest that unexpected mechanisms of regulation of gene a piecewise smooth system contains all transitions in the activity different to those included in the mathematical bifurcation diagram of its smoothing, but some of these models- might be in place. transitions can take place simultaneously. Alessandro Colombo Samuel Zambrano Politecnico di Milano San Raffaele University [email protected] [email protected]

Nacho Molina CP9 Institut de G´en´etique et Biologie Mol´eculaire Cellulaire Internally Delayed Oscillator in Coupling Illkirch, France [email protected] While delay effects in coupled dynamical systems are fre- quently caused by travel time within the network, there Davide Mazza, Alessia Loffreda may also be delay within the individual units, due to infor- Experimental Imaging Center Ospedale San Raffaele mation processing and reaction time. These internal delay Milan, Italy effects may cause oscillations in individual units where os- [email protected], loff[email protected] cillatory behavior would not otherwise occur. In the con- text of the coupled system, this may also bring additional Marco Bianchi frequencies and resonances into consideration. San Raffaele University As an example, the single first-order autonomous delay dif-  − − − 3 Milan, Italy ferential equation x (t)= x(t T ) (x(t)) exhibits a [email protected] stable limit cycle for large enough delay. This delay limit cycle oscillator has been shown to behave similarly to ODE oscillator models in some situations. Using perturbation Alessandra Agresti methods and bifurcation theory, I will explore the behav- San Raffaele Research Institute, Milan (Italy) ior of this delay oscillator model when it is coupled to an [email protected] oscillator defined by an ODE.

Lauren Lazarus CP9 Harvey Mudd College Unstable Manifolds of Relative Equilibria and Rel- [email protected] ative Periodic Orbits

Many nonlinear partial differential equations of practical CP9 interest, such as Navier-Stokes equations, are equivariant Dynamics of Some Nonanalytic Singular Perturba- under continuous symmetries. These systems tend to have tions of z2 + c high-dimensional time-invariant sets such as relative equi- libria (traveling and rotating waves) and relative periodic We study the dynamics of several families of maps of the 132 DS17 Abstracts

real plane which are perturbations of the well-studied com- [email protected] plex quadratic map z2 + c. All maps are subfamilies of 2 α β the following family: z + c + zd1 + zd2 . Comparisons are made in both phase space and parameter space to the CP10 unperturbed family z2 + c, as well as to the analytic ana- Solution Surfaces of PDE’s As Families of Orbit logues (with β = 0) studied over the last two decades by Segments R. L. Devaney [Devaney, “Singular Perturbations of Com- plex Polynomials’, Bull. Amer. Math. Soc. 50 (2013), We consider first order quasilinear partial differential equa- 391-429], coauthors, and others over the last two decades. tions of the form

P (x, y, z)zx + Q(x, y, z)zy = R(x, y, z), Bruce B. Peckham ∈ 3 Dept. of Mathematics and Statistics with (x, y, z) R . These PDEs arise in many applications University of Minnesota Duluth ranging from traffic flow and gas dynamics to high energy [email protected] physics and birth/death processes. A solution 3 S = {(x, y, z) ∈ R | z = f(x, y)} CP9 of this PDE corresponds to an integral surface of the asso- Dynamics of Iterated Holomorphic Function Sys- ciated vector field tems ⎧ ⎨ x˙ = P (x, y, z), In this talk I focus on the dynamics of backward iterated X : ⎩ y˙ = Q(x, y, z), function systems corresponding to the sequences of holo- z˙ = R(x, y, z), morphic functions from the unit disk into a subdomain of it. Lorentzen and Gill showed that if the subdomain is rel- i.e., the vector field X is tangent to S in every point. Given atively compact in the unit disk, then every iterated func- suitable initial data, the method of characteristics allows tion system has a unique limit function which is constant. one to obtain a functional expression for S. However, in In other words, they showed that relative non-compactness a general context, this task may demand hard algebraic of X is necessary to have a boundary point as a limit func- work from X; and even if one manages to get a formula tion. Keen and Lakic used the notion of hyperbolic Bloch by this method, its practical usefulness is not assured. In domain, first introduced by Beardon et al., and showed this talk, we take a geometric approach for the initial value that if X is not Bloch in D, every boundary point of X is problem rather than a formulaic one: We compute the so- a limit function of some iterated function system. In this lution surfaces S as families of orbit segments of the vector talk I generalize this result and show that relative non- field X obtained as solutions of a two-point boundary value compactness of X in D is a sufficient condition to have a problem solved by continuation methods. This novel ap- boundary point as a limit function. This result has impor- proach allows us to get useful insight into the geometrical tant applications in studying the degeneracy of the phe- properties of the surface S and, ultimately, of the solution nomena which are modeled by the above iterated function z = f(x, y). We illustrate this method with an example systems. that arises in the study of one-dimensional neutron trans- port theory. Kourosh Tavakoli Oklahoma City University Pablo Aguirre [email protected] Universidad T´ecnica Federico Santa Mar´ıa Departamento de Matem´atica [email protected] CP9 Generalizations of Conley Index and Relevant Ap- plications. CP10 Snaking and Localized Patterns with Twisted In- Conley index, as a homotopy index, is a useful tool to study variant Manifolds the unstability of invariant sets of autonomous dynamical systems. In view of its continuation, Conley index has We analyze the bifurcation diagrams associated with spa- been applied to solve some bifurcation problems of invari- tially localized stationary patterns, often known as local- ant sets, such as dynamic or global bifurcations.The gener- ized rolls. In a wide variety of contexts, such diagrams alization of Conley index can be considered in three ways. exhibit a phenomenon known as snaking, due to their in- One is extending the phase space to more common ones. tertwined S-shaped bifurcation curves. When the local- Rybakowski has built up the theory on non-local compact ized patterns satisfy a reversible, conservative ODE in R4, spaces, and we rebuilt this theory in a more simple way. snaking has previously been rigorously analyzed by con- Another is generalizing the homotopy index to shape index, necting the existence of the localized rolls with the bifur- since shape is a generalization of homotopy type. We have cation structure of fronts that connect the rolls to the triv- defined a kind of compact generated shape index of com- ial state. In previous work, important assumptions were pact invariant sets in non-local compact spaces. The third made that ensured the rolls had real Floquet exponents, one is to consider more complicate nonautonomous dynam- and additionally that their stable and unstable manifolds ical systems. However, this generalization is far from a were not twisted. In this work, we show that such twist- trivial one and I am focusing on this issue now. In this lec- ing produces a topological obstruction to snaking behavior, ture, I will introduce the work I have done and am about while isolas and ladders can still appear in the bifurcation to do in this topic. diagram.

Jintao Wang Paul Carter Huazhong University of Science and Technology Department of Mathematics Wuhan, China University of Arizona DS17 Abstracts 133

[email protected] embedding, in order to utilise for time-series prediction. In short, mathematical theories of the Hilbert-Schmidt inte- Tarik Aougab gral operator and the corresponding Sturm-Liouville eigen- Department of Mathematics value problem underlie the framework. Using those math- Brown University ematical theories, one can derive error estimates of mode tarik [email protected] decomposition obtained by the present method and can obtain the phase-space reconstruction by using the leading Margaret Beck modes of the decomposition. In this talk, we will show Boston University some results for some numerical and experimental datasets [email protected] to validate the present method. Naoto Nakano Surabhi Desai JST PRESTO Mathematical Institute Hokkaido University University of St. Andrews n [email protected] [email protected]

Bjorn Sandstede CP10 Division of Applied Mathematics Mild Solutions for Multi-Term Time-Fractional Im- Brown University pulsive Differential Equations with Non Local Ini- bjorn [email protected] tial Conditions

Melissa Stadt In this paper, we are concerned with the existence of the Department of Mathematics mild solutions for multi-term time-fractional differential University of Washington equations with not instantaneous impulses. We obtain [email protected] some sufficient conditions for the existence and uniqueness results using a generalization of the semigroup theory for Aric Wheeler linear operators, fractional calculus techniques and some Department of Mathematics suitable fixed point theorems under appropriate hypothe- UNC Chapel Hill ses. An example is also given to illustrate the applications [email protected] of the established results. Vikram Singh, Dwijendra N Pandey CP10 IIT ROORKEE [email protected], [email protected] Fractional Integrated Semi Groups and Non Local Cauchy Problem for Abstract Nonlinear Fractional Differential Equations CP11 Some classes of fractional abstract differential equations Bifurcations of Invariant Tori in 3-Dimensional with α-integrated semi groups are studied in Banach space. Piecewise Smooth Maps The existence of a unique solution of the nonlocal Cauchy problem is studied. Many physical systems are modeled by piecewise smooth maps. The theory of bifurcations in such maps is well de- Mahmoud M. El-Borai veloped in the context of 1D and 2D systems. In this work Professor of Mathematics Faculty of Science Alexandria we investigate the dynamics of 3D piecewise smooth maps Unive represented by a normal form, and report many atypical Alexandria University Egypt bifurcations that can happen in invariant closed curves. m m [email protected] Some of these can be explained based on Lyapunov bun- dles and eigenvalues of saddle cycles, while some demand new insights. CP10 Revisiting Delay-Embedding by Using Hilbert- Soumitro Banerjee Schmidt Integral Operator Theory for Dynamical Indian Institute of Science education & Research, Reconstruction Kolkata, India [email protected] Delay embedding is well-known for non-linear time-series analysis, and it is used in several research fields. Takens Mahashweta Patra theorem ensures validity of the delay embedding analysis: Indian Institute of Science education & Research embedded data preserves topological properties, which the Kolkata, India original dynamics possesses, if one embeds into some phase [email protected] space with sufficiently high dimension. This means that, for example, an attractor can be reconstructed by the delay coordinate system topologically. However, configuration of CP11 embedded data may easily vary with the delay width and Multiple Winner-Take-All Solutions in the Star- the delay dimension, namely, ”the way of embedding”. In Like System of Phase Oscillators with Parameter a practical sense, this sensitivity may cause degradation of Adaptation reliability of the method, therefore it is natural to require robustness of the result obtained by the embedding method We study a star-like system of phase oscillators with adap- in certain sense. In this study, we investigate the math- tation and apply this system for modelling of cognitive ematical structure of the framework of delay-embedding functions (e.g. visual search, attention focus, perceptual analysis to provide Ansatz to choose the appropriate way of switching). The system includes a Central Oscillator (CO) 134 DS17 Abstracts

and multiple Peripheral Oscillators (POs). The oscilla- [email protected]; [email protected] tors are described as generalised Kuramoto type oscillators with parameter adaptation. The frequency of CO is adap- tive and reflects the average frequency of POs. Connection CP11 strengths from POs to the CO are also adaptable. Inter- Detecting Phase Transitions in Collective Behavior actions in the network are organised in such a way that Using Manifold’s Curvature POs compete for synchronisation with the CO. We use the bifurcation analysis to investigate the dynamical modes of If a given behavior of a multi-agent system restricts the the system. In particular, we are interested in winner-take- phase variable to an invariant manifold, then we define all regimes where one of the POs runs in-phase with the a phase transition as a change of physical characteristics CO and the adaptation mechanism allows suppression of such as speed, coordination, and structure. We define all other POs. In particular, the results of mathematical such a phase transition as splitting an underlying mani- and computational studies have been accounted for reac- fold into two sub-manifolds with distinct dimensionalities tion times in visual search experiments. It is postulated around the singularity where the phase transition physi- that those POs which are synchronised by the CO are in- cally exists. Here, we propose a method of detecting phase cluded in the focus of attention. The probability for an transitions and splitting the manifold into phase transi- object to be included in the focus of attention is deter- tions free sub-manifolds. Therein, we firstly utilize a re- mined by its saliency that is described in formal terms as lationship between curvature and singular value ratio of the strength of the connection from the PO representing pointssampledinacurve,andthenextendtheassertion the target to the CO. It is shown that the model can re- into higher-dimensions using the shape operator. Secondly, produce reaction times in visual search tasks of various we attest that the same phase transition can also be ap- complexities. proximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the Phase Transition Detection (PTD) method using one Roman M. Borisyuk particle simulation and three real world examples. School of Computing, Electronics and Math Plymouth University Kelum D. Gajamannage, Erik Bollt [email protected] Clarkson University [email protected], [email protected] Yakov Kazanovich Institute of Mathematical Problems in Biology Russian Academy of Sciences CP11 [email protected] Elliptic Bubbles in Moser’s Quadratic Maps

Oleksandr Burylko In 1994 Moser generalized H´enon’s two-dimensional Institute of Mathematics quadratic maps to the four-dimensional, symplectic case. Ukranian academy of Sciences He showed that these maps can be written as the compo- [email protected] sition of an affine symplectic map with a quadratic shear or “jolt” map, or more explicitly, for (x, y) ∈ R2 × R2,as

−T (x, y)=M(x, y)=(C (−y + ∇V (x)),Cx) CP11 where C is a 2 × 2matrixandV a cubic potential. The Explicit Symmetry Breaking and Hopf Bifurcations resulting family has six free parameters and two discrete moduli. We show that there is a codimension three bifurca- Group theoretic means have been used before to study the tion that corresponds to the simultaneous creation of four effects of weak explicit symmetry breaking on steady state fixed points. Two of these can be center-center points and bifurcations of reaction-diffusion systems. We perform an two center-saddles leading to the formation of a double- analogous model-independent analysis of the effects of un- bubble of two-tori. We propose that this bifurcation is an derlying anisotropies and inhomogeneities on systems un- organizing center for the dynamics of saddle-center bifur- dergoing pattern forming Hopf bifurcations. cations in 4D maps, and in particular is a normal form for the formation of accelerator modes in maps such as the Timothy K. Callahan Froeschl´e family. Embry-Riddle Aeronautical University Department of Physics James D. Meiss [email protected] University of Colorado Dept of Applied Mathematics [email protected] CP11 Arnd Baecker On Stability of Coupled Systems with Delay and Institut f¨ur Theoretische Physik Hysteresis TU-Dresden-Germany [email protected] In this work , two coupled systems with transmission and response delay along with memory of history leading to CP12 hysteresis are studied. Stability analysis is carried out. Nu- Using Modeled Dynamics for the Control of Au- merical simulation of one illustrative problem is presented. tonomous Vehicles Conditions for chaotic behavior are obtained. The primary goal of optimal planning and control algo- Prasad G. Chhapkhane rithms is to lay out behaviors which conserve time and/or Devchand College Arjunnagar fuel, and then apply these findings to real-world systems. DS17 Abstracts 135

Traditional, continuous-time control theory often has as a dant wealth via a potentially nonlinear and discontinuous prerequisite that some technical assertions about the sys- competitiveness function. From this model we demonstrate tem dynamics be made, often in the form of explicit equa- how to infer such a competitiveness function from invest- tions. For nonlinear and high-order systems, this can be a ments, along with geometrical criteria to determine indi- significant challenge. Another approach involves using or- vidual decisions. Finally, we investigate the stability of a bit primitives combined with stochastic search algorithms, wealth distribution, both to local perturbations and the but these searches sample infeasible trajectories and are introduction of new agents with no wealth. We prove that generally are not robust to model or state uncertainty. To there is a fundamental relationship between high levels of address this, we wish to develop an intuitive, computation- individual investment levels and the existence of a wealth ally efficient method that is robust to uncertainty. In this trap, which traps otherwise identical agents at a lower level work, we present an approach to both the planning and of wealth. control of lightweight ground vehicles that makes use of physical simulation and machine learning to perform these Joel D. Nishimura tasks concurrently and in real-time. As in our previous Arizona State University work [Keivan, Sibley; Realtime simulation-in-the- loop con- [email protected] trol for agile ground vehicles; TAROS 2014], we apply this physical simulator to test the feasibility of planned paths with respect to modeled vehicle dynamics and to generate CP12 control functions. The use of physical simulation within Users Dynamics on Internet Platforms the planning and control loops provides an extremely dense feedforward model for vehicle and terrain dynamics. Fur- I will discuss a dynamical system, which models two-sided thermore, the approach allows for considering arbitrarily Internet platform. The flow of the system corresponds to sophisticated models of these features. the volume of users from each side of the platform as a function of time. I will present theorems describing the Christoffer R. Heckman long-term behavior of the volume of users, equilibriums and Autonomous Robot & Perception Group other characteristics. The results were obtained with the University of Colorado at Boulder help of classical theory of dynamical systems. These results christoff[email protected] help to propose some strategies for platform’s regulations that optimize the benefits of platform users and owners. Volume of users on platforms, equilibrium and other char- CP12 acteristics have mainly been studied with game-theoretic Analysis of Entropy Generation Due to Time Peri- approach and from platform users point of view. However, odic Heating in a MHD Porous Enclosure growing number of Internet platforms with large amount of data allows to construct dynamical systems models and In this work, we analyze the hydro-magnetic mixed convec- to study these models from the platform owners point of tion inside a nanofluid filled porous enclosure. The study view. The dynamical system’s approach to the study of focuses on the nanofluid flow behavior, heat transfer and two-sided platform allows natural generalization to multi- entropy generation due to various flow governing param- sided platforms (MSP), where one can utilize the theory eters. A portion of side walls is thermally activated us- of multidimensional dynamical systems, and obtain results ing time periodic heat sources. The effect of variable heat for MSP with high number of dimensions. source length is investigated for optimum heat transfer with least entropy generation. The results are presented Victoria Rayskin for a wide range of flow governing parameters including Penn State University Grashof number, Hartmann number, Darcy number and [email protected] volume fraction of nanoparticles. The thermodynamic op- timization is discussed by using Nusselt number and en- tropy generation and the dominance of entropy generation CP12 due to heat transfer over entropy generation due to fluid The Emergence of Power-Law Scalings in Large- friction and magnetic field is discussed with the help of Scale Systems of Weakly Correlated Units Bejan number. Characterizing the dynamical regime within which the Sumit Malik brain operates is central to understanding its function. One INDIAN INSTITUTE OF TECHNOLOGY ROORKEE attractive possibility is that neuronal networks operate at Indian Institute of Technology Roorkee a critical state. This view is supported by experiments [email protected] in neuronal cultures showing a power-law scaling of firing events size and duration, as well as the existence of a uni- Ameeya Nayak versal scaling function, in agreement with canonical models Indian Institute Of Technology Roorkee at criticality. In this talk, we will explore alternative phe- [email protected] nomena accounting for such dynamics. First, coming back to a classical network of sparsely connected integrate-and- fire neurons network, we will show that similar statistics CP12 emerge in the network away from any critical transition A Model of Intergenerational Wealth Dynamics and in regimes where there is no evidence of critical slow- and Intergenerational Wealth Traps ing down. Statistical physics theory indicates that for large network sizes, neurons in these networks are very weakly An important question about the distribution of wealth is correlated, a decorrelation also observed in experiments. how the proportion of national wealth a person controls We will show that the power-law statistics and scalings can depends on the wealth of their parents. We develop a sim- be directly associated to this low correlation, and persists ple model of intergenerational wealth dynamics where in- in networks of independent Poisson units with a common dividuals balance current consumption with investment in firing rate. We will demonstrate that these statistics can a single descendant. Investments then determine descen- rigorously be found in these systems, although microscop- 136 DS17 Abstracts

ically, the system is stochastic. These results can reconcile Parameters Based on Information Theory apparently conflicting experimental observations such as the lack of evidence for SOC in neurons measured in the In the modeling time-dependent data set with minimal awake brain. They also put caution on the interpretation dimension ODEs, recent researchers tend to consider the of scaling laws found in nature. parameters matrix to be sparse with just a few elements that project the system dynamics with respect to basis Jonathan D. Touboul functions. The uncertainty in constructing the parameters The Mathematical Neuroscience Laboratory sparse matrix can lead to high error margins, being the College de France & INRIA Paris optimally sparse representation from data may be highly [email protected] over-sparsed, and consequently sparse optimization is brit- tle in the sense that significant system identification er- Alain Destexhe rors can result. In this paper, we introduce an information CRNS Gif sur Yvette theoretic method to detect sparsity structure that may be [email protected] sparse but not too sparse, just right, by a principle of cau- sation entropy principle, which when then used as a mask in the estimation step, radically reduce the error in esti- CP12 mated parameters. Several examples are presented for low Life-Detection and Through-Wall Imaging Using and high dimensions chaotic systems. Ultra-Wideband Chaos Radar Abd Alrahman R. Almomani Life-detection and through-wall radar technology, with the Clarkson University purpose of identifying vital signs and position of life-beings Department of Mathematics located behind an obstacle, has attracted great interest in [email protected] various fields, such as urban warfare, antiterrorism, and rescue for earthquake, snow disaster or fire victims. In this paper, we propose an ultra-wideband (UWB) chaos radar CP13 for through-wall imaging and life-detection behind barrier. The UWB chaos radar transmits a wideband chaotic signal Least Action Methods and Noise Induced Transi- modulated by a single-tone sinusoidal wave. Using a split tions in Periodically-Forced Systems ring antenna integrated solid state sensor and lock-in am- plifier, the radar can identify the frequency of breathing We present a study of the metastability of periodic orbits and heartbeat by detecting the phase information of the for 1-D periodically forced systems perturbed by weak ad- reflected sinusoidal wave from the human object. Further, ditive noise. It is well known that noise can introduce dra- human location is realized by correlating the chaotic echo matic changes to the dynamics of the system, e.g. stochas- signal with its delayed duplicate and combining the syn- tic resonance, noise induced transitions between determin- thetic aperture technology. Experimental results demon- istic stable states etc. We ask the question: can noise strate the proposed radar can detect the humans vital signs induced transitions be completely understood using least- and location information through the obstacles with high action principles and how do these results compare with range resolution and large detection range. The range res- classic results from large deviations? In particular, while olution is 15 cm, benefiting from the 1 GHz-bandwidth pure noise induced transitions between metastable states of the chaotic signal. The detection range through 20 cm occur on exponentially long time scales (Kramers rate) the thick wall is about 5 m for the respiration signal detection frequency of the forcing introduces an additional time scale and 3.4 m for the heartbeat signal detection. (inverse of the Floquet exponent) and a preferred phase of transition. Using least action principles, we show that the Hang Xu the preferred phase and expected time of transition depend Taiyuan University of Technology crucially on the scaling of these parameters. [email protected] John Gemmer Li Liu, Jianguo Zhang Wake Forest University Taiyuan University of Technology Department of Mathematics Institute of Optoelectronic Engineering [email protected] [email protected], [email protected] Yuxin Chen Jingxia Li Northwestern University Taiyuan University of Technology [email protected] [email protected] Alexandria Volkening Bingjie Wang, Anbang Wang Division of Applied Mathematics Taiyuan University of Technology Brown University Institute of Optoelectronic Engineering alexandria [email protected] [email protected], [email protected] Mary Silber Yuncai Wang University of Chicago Taiyuan University Dept. of Engineering Sciences and Applied Mathematics Institute of Optoelectronic Engineering [email protected] [email protected] CP13 CP13 Perron-Frobenius Meet Monge-Kantorovich: A Deterministic Method to Identify Sparse Matrix of Set-Oriented Graph-Based Approach to Optimal DS17 Abstracts 137

Transport previous concerns regarding the traditional entropy of re- currences and reproduce classical results with advantages. We study the problem of optimally steering measures in In our conclusions we discuss some future perspectives, spe- phase space for nonlinear systems. For discrete-time case, a cially in the application to non-stationary data, real data switching approach to optimal transport is adopted. Here, and the evaluation of false neighbors. dynamics are propagated by Perron-Frobenius operator, and the discrete-time perturbations are obtained as a result Thiago L. Prado of Monge-Kantorovich optimal transport between interme- Instituto de Engenharia, Ciˆencia e Tecnologia diate measures on a graph. In the continuous-time case, Universidade Federal dos Vales do Jequitinhonha e we present optimal transport formulation of nonlinear sys- Mucuri tems with prescribed control vector fields. Our approach [email protected] allows us to recover provably optimal control laws for steer- ing agents from prescribed initial to final distributions in Gilberto Corso finite time for the case of control-affine systems, includ- Departamento de Biof´ısica e Farmacologia ing the non-holonomic case. The action of the controlled Universidade Federal do Rio Grande do Norte vector field is approximated by a continuous-time flow on [email protected] a graph obtained by discretizing the phase space. The problem is then reduced to a modified Monge-Kantorovich Gustavo Lima optimal transport on this graph via use of infinitesimal gen- science and technology school erators. We apply our method to examples from chaotic Federal University of Rio Grande do Norte fluid dynamics and vehicle dynamics. By combining the [email protected] infinitesimal-generator approach with the theory of opti- mal transport, we obtain a graph based version of optimal transport for nonlinear systems (OT-NS). Sergio Lopes Departamento de F´ısica Piyush Grover Universidade Federal do Paran´a Mitsubishi Electric Research Laboratories lopes@fisica.ufpr.br [email protected] CP14 Karthik Elamvazhuthi Arizona State University Coupled System of Electrical Oscillators and Their [email protected] Solutions in Perspective of Fractional Derivatives We study the system of fractional order coupled oscillators CP13 associated with electric circuit in view of various types of Forecasting Chaotic Business Cycles Perturbed by fractional derivatives. Further, the analytical solutions of Noise these equations have been established using tools of frac- tional calculus. For some types of fractional derivatives, The failure of the Federal Reserve to timely raise the federal the solutions have been found very close to classical so- funds rate to its “normal’ expectation causes chaos. By lutions as the order of fractional derivative approaches to applying a Sprott nonlinear dynamical system, it becomes two. possible to forecast aperiodic business cycles. When the short-term interest rate is prudently targeted, the economy Naseer Ahmad Asif mean-reverts, like a Langevin equation perturbed by noise. University of Management and Technology, Lahore, Then business cycles no longer exist. Pakistan [email protected] James M. Haley Muhammad Imran Jamil Point Park Universtiy University of Management and Technology [email protected] Lahore, Pakistan [email protected] CP13 Entropy of Recurrence Plot Configurations CP14 Recurrence plots are binary matrices where the trajecto- Existence Results for Two-Term Time Fractional ries of a dynamical system in the phase space are eval- Differential Equations with Nonlocal Conditions uated against another embedded trajectory. Due to the exponential increasing of the amount of data available due In this paper, we employ the method of lower and upper the advent of the information era, this graphical methodol- solutions coupled with the monotone iterative technique to ogy based on visual qualification of patterns in the matrix set up the existence of the mild solutions for the following has suffered natural limitations. Great part of these re- two-term time fractional differential equations with nonlo- strictions were overhauled by the recurrence quantification cal conditions in a Banach space X analysis (RQA). In this scenario the necessity on the devel- α+1 β opment of suitable tools to quantify recurrence estates is a D u(t)+ηD u(t) − Au(t)=f(t, u(t)),t∈ I =[0,T],T>0,; important topic. In our work we propose a new recurrence u(0) = u0 + g(u),ut(0) = u1 + h(u), analysis tool based on the concept of information entropy, where micro-states are defined by all possible patterns dis- where 0 <α≤ β ≤ 1, Dα denotes the Caputo fractional played by a K×K size block of a recurrence plot (RP). Our derivative of order α. η ≥ 0. A : D(A) ⊂ X → X is a methodology is a new RQA that evaluates the entropy of closed linear operator which generates a strongly continu- configurations of recurrences. This method has a weak de- ous family of bounded linear operators on X. Functions pendence on parameters, is simple to evaluate, fixes several f : I × X → X, g : X → X and h : X → X are given 138 DS17 Abstracts

functions satisfies monotonicity and measure of noncom- [email protected] pactness conditions. u0, u1 ∈ X. At last, an example is given to show the availability of main results. MS1 Renu Chaudhary, Dwijendra N Pandey Isochrons for Saddle-Type Periodic Orbits in IIT ROORKEE Three-Dimensional Space [email protected], [email protected] Many physical systems feature stable oscillations. Such systems are often modelled by ordinary differential equa- CP14 tions, such that the stable oscillation is represented by an attracting periodic orbit. Each point in the basin of at- Inverse Problem for Dynamical Systems with Un- traction of this periodic orbit will converge to it with a certain Data particular phase; all points that synchronise with the same phase are said to lie on an isochron. Isochrons are smooth The problem of estimation of parameters of a dynamical manifolds that have one dimension less than that of the system from discrete data can be formulated as the prob- basin of attraction, and the family of isochrons associated lem of inverting the map from the parameters of the sys- with all of the phase points which constitute the periodic tem to points along a corresponding trajectory. Presented orbit make up that entire basin. Isochrons are used in ap- will be an overview of recent results in this area, specifi- plications to study the effects of a perturbation, in terms cally, (i) conditions for identifiability of linear and linear- of the relative phase at which the perturbed point returns in-parameter systems from a single trajectory, (ii) analyt- to the periodic orbit. There are very few systems for which ical and numerical estimates of the maximal permissible isochrons can be computed analytically; as soon as the uncertainty in the data for which the qualitative features dynamics become interesting one has to make use of nu- of the inverse problem solution for linear systems, such as merical methods. Previously, methods to compute these existence, uniqueness, or attractor properties, persist, and isochrons have focused on planar systems. I will present (iii) techniques for parameter estimation of ensemble mod- an extension of this method to compute isochrons for both els, i.e., models of deterministic processes with uncertain orientable and non-orientable saddle-type periodic orbits parameters. in three-dimensions which will lie on that periodic orbit’s two-dimensional invariant manifolds. This constitutes a David Swigon first step towards computing isochrons in higher dimen- Department of Mathematics sions, and showcases the intricacies of their geometry. University of Pittsburgh [email protected] James Hannam Department of Mathematics Shelby Stanhope University of Auckland Temple University [email protected] [email protected] Hinke M. Osinga, Bernd Krauskopf Jonathan E. Rubin University of Auckland University of Pittsburgh Department of Mathematics Department of Mathematics [email protected], [email protected] [email protected] MS1 CP14 Time-Optimal Control of Biological Oscillators us- ing Isochrons and Isostables Understanding Mixing Processes by Transfer Op- erator We have developed an optimal control algorithm to change the period of a periodic orbit using a minimum energy Industrial and chemical mixing mixing processes of various input, which also minimizes the controlled trajectory’s kinds occur throughout nature and are vital in many tech- transversal distance to the uncontrolled periodic trajec- nological applications. In the context of discrete dynamical tory. Our algorithm uses the augmented phase reduc- systems, the transfer operator approach has been shown as tion, a recently developed two-dimensional reduction tech- a powerful tools from both theoretic and numerical view- nique based on both isochrons and isostables. To demon- point. In this talk, I will use a toy model (i.e., the one strate the effectiveness of the algorithm, we apply it along dimensional stretch and fold map) as an example to pro- with a previous optimal control algorithm based on stan- vide a brief introduction on the relationships between the dard phase reduction (one-dimensional reduction based on spectral properties of the associated transfer operator and isochrons) to biological dynamical systems where a change the estimations of the optimal mixing rate of the mixing in time period of the periodic orbit holds practical rele- process. Moreover, I will address how the optimal mix- vance. We show that the control algorithm based on the ing rate varies according to the stretch and fold map has augmented phase reduction is effective even when a large “cutting and shuffling’ behaviour (i.e., composing with a change in time period is required or when the nontrivial permutation). If time permits, I will also talk about how Floquet multiplier of the periodic orbit is close to one; in to interpret this problem to the eigenvalue estimations for such cases, the control algorithm based on the standard the Random bi-stochastic matrices (free probability the- phase reduction fails. ory) and the locations of poles of the dynamical zeta func- tion Bharat Monga Mechanical Engineering Yiwei Zhang UC Santa Barbara huazhong university of science and technology [email protected] DS17 Abstracts 139

Jeff Moehlis [email protected] Dept. of Mechanical Engineering University of California – Santa Barbara [email protected] MS2 Pairwise Network Information and Direct Connec- tivity MS1 From interacting populations of earthquake faults to the Isostable Reduction for Stable Limit-Cycling Sys- nerve cells in the brain, many complex systems can be rep- tems resented as a collection of coupled dynamical units. In many practical applications, however, this network struc- A systematic reduction method of the limit-cycling dynam- ture cannot be directly observed or reliably inferred from ics using the isochrons has been a powerful framework for the observed activity by current methods. Instead, effec- analyzing rhythmic activities in many fields. Recently, the tive or functional networks are then considered: units are Koopman operator-theoretic approach has revealed that connected by a link if their relationship satisfies some cri- the properties of the transient dynamics around attrac- teria. The idea that a given system can be well described tors can be well-characterized by the notion called isosta- by a functional network, and the pairwise measurements bles. We have developed an isostable reduction method and they are built from, is a simplifying assumption typically a bi-orthogonalization method to obtain the infinitesimal used in these cases. Here, we introduce a novel, efficient, isostable response curves for stable limit cycle oscillators. and general method to (i) quantify how good the assump- The validity of the proposed method is confirmed by com- tion of pairwise interactions is, and (ii), if it is a good as- paring the results to direct numerical simulations of the sumption, to reliably infer the backbone of the direct net- full dynamical system. We also illustrate the utility of the work connectivity between the interacting units. At the proposed reduction framework by estimating optimal in- core, our method is a novel entropy maximization scheme jection timing of external input that efficiently suppresses that is based on conditioning on entropies and mutual in- deviations of the system state from a limit cycle. formations. We establish that our method is particularly well suited for the typical case of large nonlinear systems Sho Shirasaka, Wataru Kurebayashi where the dynamical units have more than two states and Tokyo Institute of Technology in the undersampled regime, both situations where pre- [email protected], [email protected] vious methods failed. The advantages of the proposed method are documented using phase oscillator networks Hiroya Nakao and a resting-state human brain network as generic rele- Graduate School of Information Science and Engineering, vant examples. Tokyo Institute of Technology [email protected] Jaroslav Hlinka Institute of Computer Science Academy of Sciences of the Czech Republic MS1 [email protected] Defining the “Phase” of a Stochastic Oscillator Joern Davidsen, Elliot Martin A stable, finite-period limit cycle in a deterministic system Complexity Science Group admits a flow-invariant system of Poincar´esectionswith University of Calgary the properties (i) all trajectories initiated on a given leaf [email protected], [email protected] converge to a common trajectory on the limit cycle; (ii) all trajectories initiated on a given leaf pass through the same MS2 leaf after an interval equalling one period. The leaves are the isochrons, and the timing of their movement defines the Inferring Directed Interactions from Data: An asymptotic phase of points converging to the limit cycle. Overview In a Markovian stochastic setting, two generalizations of the asymptotic phase and the isochron foliation have been Complex networks are increasingly used in various scien- proposed, one based on a spectral decomposition of the tific fields to characterize spatially extended dynamical sys- generator of the Markov process, the other on a foliation tems. The reliable inference of, in particular, directed net- defined by a uniform mean first passage time property. We works from empirical data, however, continues to represent will discuss the (non)equivalence of these two generaliza- a challenging issue. This talk provides an overview of anal- tions of the phase of a noisy oscillator. ysis techniques that are frequently used to infer directed interactions from pairs of time series, discusses their pros Peter J. Thomas and cons, and points to possible improvements. Case Western Reserve University Klaus Lehnertz [email protected] Department of Epileptology University of Bonn, Germany Alexander Cao [email protected] Department of Mathematics, Applied Mathematics, and Statisti Case Western Reserve University MS2 [email protected] Tackling Indirect Directional Couplings in Large Oscillator Networks: Partial Or Non-Partial Phase Benjamin Lindner Analysis? Bernstein Center for Computational Neuroscience Berlin Department of Physics, Humboldt University, Berlin, We investigate the relative merit of the evolution map ap- Germany proach and its partialized extension for inferring direc- 140 DS17 Abstracts

tional couplings in complex networks of weakly interact- sal, i.e. coupling in the network of habitats/communities, ing dynamical systems from multivariate time-series data. can support supersaturation in the network of networks We simulate several experimental situations and study to even in the case when not all patches provide resource con- which extent the approaches correctly infer the network ditions in which more species than limiting resources can topology in the presence of indirect directional couplings survive. Our principle result states that the lower bound in coupled model systems. In addition, we investigate on the initial fraction of supersaturated patches needed to whether the partialized extension allows for additional or obtain supersaturation in the whole network, can be opti- complementary indications of directional interactions in mized by dispersal structure. Further, we identify crucial multichannel electroencephalographic recordings, for which links in the dispersal topology which, if broken, lead to the both direct and indirect directional couplings can be ex- collapse of global supersaturation. pected. Overall, our findings indicate that particularly in larger networks the partialized extension does not provide Anshul Choudhary information about directional couplings extending the in- Universit¨at Oldenburg formation gained with the evolution map approach. [email protected]

Thorsten Rings Ramesh Arumugam Dept. of Epileptology {Department of Mathematics, Indian Institute of University of Bonn, Germany Technology, [email protected] Ropar, India [email protected] Klaus Lehnertz Department of Epileptology Partha Sharathi Dutta University of Bonn, Germany Department of Mathematics, Indian Institute of [email protected] Technology, Ropar, India MS2 [email protected] Coupling Function Decomposition Ulrike Feudel Interactions of dynamical systems are often assessed only University of Oldenburg as the net coupling between the systems. Here we show ICBM, Theoretical Physics/Complex Systems how one can go beyond this and decompose the net into [email protected] direct and indirect coupling components. By modelling the interacting dynamical systems we decompose the net coupling function into its functional components which are MS3 partially grouped as unities that constitute the direct from Indirect Targeting in Networks one system, or the common indirect influence from both of the systems. The method is presented by Fourier coupling Traffic jams, genetic diseases, financial crises, ecosystem function decomposition of phase dynamics from interact- collapse. These are just a few examples of far-reaching dis- ing oscillators. The coupling function components are es- ruptions in complex systems that can nonetheless be caused timated with dynamical Bayesian method for inference of by abnormal activity in only a handful of componentsroads, time-evolving dynamics in presence of noise. The technique genes, banks, or invasive species. Yet quite often, remedy- is used to determine the effect of general anaesthesia on the ing these system-level failures is intractable precisely be- neural-cardio-respiratory coupling function components. cause the few malfunctioning components are those most unamenable to direct intervention. Here, we develop a net- Tomislav Stankovski work solution to this broadly significant problem. By ex- Skopje, Macedonia ploiting the phase space structure of the systems (nonlin- Ss. Cyril and Methodius University, Skopje, Macedonia ear) dynamics, we show that it is in general possible to [email protected] manage, activate or even disable a node indirectly, via ma- nipulations confined to the other nodes in the network. Us- ing food-web and boolean networks as model systems, we MS3 demonstrate that this concept of indirect targeting could, Networks of Networks: Dispersal induced Super- for example, allow elimination of invasive species by acting saturation in Meta-Communities on the populations of other (indigenous) species, or tog- gling the activity of disease-causing genes by modulating Metacommunity models describe ecological systems con- other, less entrenched components. Moreover, our simula- sisting of several patches or habitats in which several pop- tions show that these feats can often be accomplished by ulations interact with each other as competitors or via acting on only a small minority of the rest of the network, predator-prey relationships in form of food webs. Those suggesting a high potential for indirect intervention in real food webs are interconnected by migration corridors. Such systems. Altogether, these results provide a path forward metacommunities can be considered as networks of net- for the design of focused, minimally-intrusive control inter- works. Competition among species for resources is in most ventions that can tune otherwise inaccessible nodes. ecological models determined only by one limiting resource. We study metacommunities in which multiple resource lim- Sean P. Cornelius, Adilson E. Motter itation is taken into account leading to non-equilibrium Northwestern University coexistence of species. For a multi-species community, su- [email protected], [email protected] persaturation is a state of coexistence, where the number of species competing for resources exceeds the number of limiting resources. In this work, we study the interplay MS3 between resource conditions in the patches and dispersal structure. Specifically, we analyze to what extent disper- Evaluating the Sensitivity of Dynamical Systems to DS17 Abstracts 141

Interaction Network Topology Peter J. Mucha University of North Carolina Chapel Hill A network’s reliability was defined by Shannon and Moore [email protected] in terms of the probability that the dynamics of a specific dynamical system defined on the network exhibits specific properties. A straightforward extension of this concept MS4 to general network dynamical systems provides a useful Spectral Notions of Aperiodic Order paradigm for studying the sensitivity of dynamics to net- work topology. An understanding of this sensitivity can The characterization of aperiodic structures usually em- be used for characterization or topological control of the ploys spectral notions and properties. In particular, this dynamical system. The discrete nature of a network in- applies to Fourier analysis in the form of diffraction mea- troduces issues beyond well-known ones such as local vs. sures. These are directly linked to experimental observa- global sensitivity. In particular, evaluating the network tions, for instance in the structure analysis of quasicrys- reliability exactly is known to be NP-hard in most cases. talline materials. While diffraction provides information Approximating it is also computationally hard. Moreover, about the order in a system, we are still far from any clas- approximating sensitivity to network topology requires de- sification of systems displaying some form of long-range tecting very small differences in reliability. We define eval- order. From a mathematical perspective, many of the ape- uating sensitivity to network topology as a formal opti- riodic structures that are considered give rise, under trans- mization problem, introduce two constraints that permit lation action, to ergodic dynamical systems. There is then a greedy approach, and exhibit a provably efficient, com- a natural spectral measure associated to this dynamical putationally stable algorithm for finding approximate solu- system, which is the dynamical spectrum. While the con- tions with controllable error. We compare the algorithm’s nection between the dynamical and diffraction spectra for performance with that of several heuristics that have been the pure point case has been known and used for a while, proposed for related problems such as finding community the more general relationship between these spectral no- structure. tions has only been elucidated recently. The talk will dis- cuss the two spectral notions and explain their relation by Stephen Eubank means of simple explicit examples. Biocomplexity Institute of VA Tech [email protected] Uwe Grimm Department of Mathematics & Statistics The Open University Yihui Ren [email protected] Virginia Bioinformatics Institute [email protected] MS4 Madhurima Nath Two Decades of Multiple Scale Patterns: From Physics Dept. Faraday Waves to Soft Quasicrystals Virginia Tech [email protected] Abstract Not Available At Time Of Publication.

Ron Lifshitz MS3 School of Physics and Astronomy Transitivity Reinforcement in Co-Evolving Net- Tel Aviv University work Models [email protected]

One of the fundamental structural property of complex MS4 networks is transitivity. Its influence in models with co- evolving (adaptive) network topology is not well explored. Growth Rules for Icosahedral Quasicrystalline Existing coevolving network models use rewiring rules that Tilings randomize away this property. We will introduce a model Tiling models provide a useful conceptual framework for for epidemic spread and a model for opinion formation, addressing fundamental questions about quasicrystal sta- which explicitly reinforces and maintains transitivity. We bility and growth. The physical processes that produce will discuss impacts of reinforcement of transitivity in these highly ordered icosahedral samples, both manufactured two models. We will illustrate a semi-analytic method, ap- and naturally occurring, suggest the importance of growth proximate master equation (AME) for studying these mod- kinetics in the formation of icosahedral quasicrystals. Thus els. And show that this technique can predict the dynami- we would like to know whether perfectly ordered icosahe- cal behavior of these models for various parameter settings. dral quasicrystals can, in principle, be reached through a kinetic growth process without requiring annealing of pha- Nishant Malik son fluctuations. A recently developed growth algorithm Department of Mathematics based exclusively on local rules for sequential addition of Dartmouth College tiles to a cluster succeeds in producing an icosahedral qua- [email protected] sicrystal with a vanishing density of defects, but only when the growth proceeds from a suitable seed containing a de- Hsuan-Wei Lee fect. The same principles of growth apply to two different Department of Sociology, University of Nebraska Lincoln tilings, Ammann’s rhombohedral tiling and Socolar and [email protected] Steinhardt’s zonohedral tiling. These two tilings are very closely related, the vertices of the latter being a subset of Bill Shi vertices of the former. Nevertheless, the geometry of the UNIVERSITY OF CHICAGO growing clusters is quite different in the two cases. I will ex- [email protected] plain what we know about the two tilings, the nature of the 142 DS17 Abstracts

growth algorithm, and the features of the seeds required to [email protected] initiate infinite growth. [Work done in collaboration with C. Hann, P. Steinhardt, and X. Ma.] MS5 Joshua Socolar Feature-Based Parameter Estimation in a Model of Duke University Anterior Pituitary Cell Electrical Activity Physics Department [email protected] Electrical activity of endocrine pituitary cells is highly het- erogeneous in individual cells and across populations. This motivates fitting of models to single cells, but makes fit- MS4 ting directly to voltage trace data difficult. We take the Spatially Localized Quasicrystals approach of reducing data to distributions of features that represent the cell’s spontaneous and stimulated activity, The formation of quasicrystalline structures arising in di- then fitting model parameters based on these features. verse soft matter systems such as dendritic-, star-, and Simulated experimental data is used to suggest stimuli and block co-polymers can be analysed using a phase field crys- associated features that help to constrain the model param- tal model. Direct numerical simulations combined with eters. The feature-based optimization is accelerated using weakly nonlinear analysis highlight the parameter values a programmable graphics processing unit (GPU). When where the quasicrystals are the global minimum energy coupled with the dynamic clamp technique, this approach state and help determine the phase diagram. By locating can be used to rapidly fit a model to an individual cell, parameter values where multiple patterned states possess then test model predictions in the same cell. the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a qua- Patrick A. Fletcher sicrystal) is present in the background of another (for ex- Laboratory of Biological Modeling ample, the homogeneous liquid state). In a bifurcation di- National Institutes of Health agram such localized states fall on solution branches that patrick.fl[email protected] snake and these can be obtained through numerical con- tinuation. From the bifurcation diagram, we locate dy- namically unstable spatially localized patterns from which MS5 quasicrystals can be grown in experiments. Markov Chain Monte Carlo Estimation of Conductance-Based Model Parameters Priya Subramanian University of Leeds Formulating models of cellular electrical activity from bio- [email protected] physical principles introduces numerous parameters: mem- brane capacitance, maximal ion channel conductances, cur- Andrew Archer rent reversal potentials and time constants, etc. Tradi- Department of Mathematical Sciences tionally, laborious experimental manipulations designed Loughborough University to isolate each parameter have been used to calibrate [email protected] such models. Here, we consider finding an optimal fit of model parameters to limited data, such as the membrane’s current-voltage relation or voltage response to a varying Edgar Knobloch applied current, using a Markov Chain Monte Carlo ap- University of California, Berkeley - proach. Such a stochastic optimization procedure has the Nonlinearity, Institute of Physics advantage of fitting numerous parameters while potentially [email protected] avoiding convergence to local extrema of a chosen cost func- tion. Alastair M. Rucklidge Department of Applied Mathematics Joseph Mckenna University of Leeds Department of Mathematics [email protected] Florida State University [email protected]

MS5 Variational Approach for Estimation of Parameters MS5 in Hodgkin-Huxley Models Data Assimilation and Electrophysiological Model- ing of Mammalian Circadian Clock Neurons We use biophysically motivated Hodgkin-Huxley (HH) models, experimentally obtained recordings of the volt- Recently there has been interest in the application of data age of avian HVCI neurons and mouse CA1 neurons, and assimilation tools to the improvement of neuronal mod- our data assimilation approach to infer the full set of pa- els. Often, the only data one has access to is the measured rameters reproducing the observed waveform information. voltage from a current-clamp experiment with a prescribed Our data assimilation algorithm formulates the problem injected current. Our work aims to improve understand- of inferring parameters and unknown states in a dynami- ing of the impact that injected current stimuli has on the cal model as a path integral. We utilize Laplace’s method identifiability of parameters in a neuronal model. Param- to approximate the integral and obtain our estimates, and eter estimation results will be shown from 4D-variational we compare predictions of the model forward in time with data assimilation (4D-var) and an Unscented Kalman Fil- data to validate our estimates. ter (UKF). We will test the performance of characteristic currents, including steps, ramps, and chaotic currents. The Daniel Breen ability of these various stimulus protocols to enable state Department of Physics and parameter estimation will be assessed using simulated UC San Diego data from the Morris-Lecar model and a biophysical model DS17 Abstracts 143

of mammalian circadian clock neurons in the suprachias- of available nonlinearities have hitherto been highly lim- matic nucleus. ited. Microstructural geometric nonlinearities, such as those present in granular media, are one approach to cre- Matthew Moye ating effective material nonlinearities. In this presentation, Department of Mathematical Sciences we will discuss recent progress in our exploration of how New Jersey Institute of Technology the microstructural geometry of mechanical metamaterials [email protected] can be tailored to enable variable effective material non- linearities, and present a categorization of several major underlying mechanisms. Our study involves microstructure MS6 design using finite element modeling, and experimental val- Effective Dynamics of Multiple Molecular Motors idation via mechanical testing of additively manufactured metamaterials. This work opens the door for the experi- The transport of cargo attached to multiple motors may mental realization of a wide class of theoretically proposed be modeled as a system of stochastic differential equations. nonlinear dynamical systems. Motors may switch between attached and detached states, each of which having separate equations for determining Nicholas Boechler motor positions. In this talk, we derive effective velocities University of Washington and diffusions for such motor systems. This involves law of Department of Mechanical Engineering large numbers and central limit theorem arguments taken [email protected] from renewal theory and multiscale averaging techniques. Identical motor systems and nonidentical systems, both cooperative and tug of war, will be considered. Results MS7 will be compared with experimental observations and other Impact Dispersion Using 2D and 3D Composite models. Granular Packing

Joe Klobusicky We present a study of efficient dispersion of an impact Geisinger Health Systems onto structured and potentially scalable granular beds. We [email protected] use discrete element method based dynamical simulations of shock wave propagation and dispersion in 2D and 3D arrangements of granular spheres. The spheres are geo- MS6 metrically packed in a nested columnar structure, which Biological Applications of Diffusion in a Randomly leads to the severe attenuation and spreading of the inci- Switching Environment dent energy within the structure. We further show that by incorporating inhomogeneity in material properties, or A number of diverse biological systems involve diffusion by introducing layers of a dissimilar material in the mid- in a randomly switching environment. For example, such dle of the arrangement, impact mitigation can be enhanced processes arise in brain biochemistry, cell signaling, in- significantly. Such an arrangement can therefore be useful sect respiration, and intracellular virus trafficking. Mathe- in the design of effective impact decimation systems. Us- matically, these processes often involve imposing randomly ing a 2D arrangement we first show the basic idea behind switching boundary conditions on either a PDE or SDE. In impact dispersion in such an arrangement. With this un- this talk, we will describe the mathematical tools for ana- derstanding the system is scaled to 3D. The influence of lyzing these systems and show how this analysis can yield the system size and material properties on the wave prop- new biological insight. agation within the packing is also presented.

Sean Lawley Surajit Sen Duke University SUNY Buffalo [email protected] sen@buffalo.edu

MS6 Mukesh Tiwari Ambani Institute of Information and Communication A Non-Markov Model for Swimming Droplets Technology I present a model for analyzing self-propelling droplets mukesh [email protected] that interact through their created concentration gradients. This non-Markov stochastic model shows the existence of T R Krishna Mohan constant velocity solutions and super-diffusive scalings. A Centre for Mathematical Modelling and Computer tunable parameter controls the history of interaction, al- Simulations lowing the system to go from having complete memory to [email protected] behaving like interacting electrostatic potentials.

Katherine Newhall MS7 Dept. of Mathematics Topological Sound and Odd Viscosity in Chiral Ac- University of North Carolina at Chapel Hill tive Materials [email protected] Active materials are composed of interacting particles in- dividually powered by motors. In this talk, we focus on MS7 chiral active materials that violate parity and time rever- Exploring Microstructural Mechanisms for Tailor- sal symmetry. First, we show how to generate topologi- ing Effective Material Nonlinear Response cal sound in fluids of self-propelled particles exhibiting a spontaneous chiral active flow under confinement. These Material nonlinearity has been shown to enable a wide topological sound modes propagate unidirectionally, with- capacity for stress wave tailoring, however, the choices out backscattering, along either sample edges or domain 144 DS17 Abstracts

walls and despite overdamped particle dynamics. Next, nonlinearities. we discuss an exotic transport coefficient characteristic of quantum Hall fluids, called odd viscosity, which controls Eurika Kaiser the hydrodynamics of classical fluids composed of active University of Washington rotors. This odd viscosity couples pressure to vorticity [email protected] leading to transverse flow in Burgers shocks. We envision that such transverse response may be exploited to design Bernd Noack self-assembled hydraulic cranks that convert between linear LIMSI-CNRS and rotational motion in microscopic machines powered by Technical University Braunschweig active rotors fluids. [email protected]

Vincenzo Vitelli Andreas Spohn Associate Professor,Soft Condensed Matter Theory Group Institute PPRIME, ENSMA Instituut-Lorentz, Universiteit Leiden [email protected] [email protected] Robert K. Niven MS7 The University of New South Wales, Australia [email protected] Formation of Rarefaction Waves and Reverse Shocks in Strain-Softening Lattices Louis N. Cattafesta FCAAP, Florida State University, USA We investigate the feasibility of forming shocks and rar- [email protected] efaction waves in nonlinear mechanical metamaterials. As a prototypical system, we assemble two types of mechanical metamaterials: (i) 1D and 2D square lattices made of thin- Marek Morzynski walled elliptical unit cells, and (ii) 3D structures based on Poznan University of Technology volumetric origami cells, e.g., Tachi Miura Polyhedron and [email protected] Kresling origami. These lattices exhibit controllable strain- softening behavior, which can be tuned by modifying their Steven Brunton, Bingni W. Brunton initial geometrical and loading conditions. Preliminary re- University of Washington sults based on computational simulations show their unique [email protected], [email protected] mechanisms of impact mitigation based on the formation of rarefaction waves and reverse shocks. These numerical Nathan Kutz results are complemented by analytical predictions based University of Washington on quasi-continuum approximation of a generalized inviscid Dept of Applied Mathematics Burgers model. [email protected] Jinkyu Yang University of Washington MS8 [email protected] Space-Time Computational Methods for Coherent Sets Hiromi Yasuda, Hryunryung Kim Department of Aeronautics and Astronautics The decomposition of the state space of a dynamical sys- University of Washington tem into metastable sets is important for understanding [email protected], [email protected] its essential macroscopic behavior. The concept is quite well understood for autonomous dynamical systems, and Christopher Chong recently generalizations appeared for non-autonomous sys- Bowdoin College tems: coherent sets. Aiming at a unified theory, in this talk [email protected] we first present connections between the measure-theoretic autonomous and non-autonomous concepts. We will do this by considering the augmented state space. Second, Panayotis Kevrekidis we introduce a data-based method to compute coherent University of Massachusetts sets. It uses diffusion maps for spatio-temporal trajectory [email protected] data, and it is consistent in the infinite-data limit with the widely-used transfer operator approach for coherent sets.

MS8 Peter Koltai Data-Driven Techniques for Modeling, Control and Free University Berlin Sensor Placement [email protected]

We present a cluster-based ROM (CROM) strategy to dis- till nonlinear mechanisms in an unsupervised manner di- MS8 rectly from data. This strategy uses cluster analysis to par- Trajectory-based Computational Analysis of Co- tition snapshot data into a small number of representative herent Structures in Flows states in the state space. The transitions between these states are then dynamically modeled as a Markov process, The notion of coherence in time-dependent dynamical sys- which is related to approximating the Perron-Frobenius op- tems is used to describe mobile sets that do not freely mix erator. CROM has potential applications for the system- with the surrounding regions in phase space. Recently, dif- atic identification of physical mechanisms of complex dy- ferent computational methods have been proposed to iden- namics, for the identification of precursors to desirable and tify coherent behavior in flows directly from Lagrangian undesirable events, and for flow control design exploiting trajectory data, such as obtained from particle tracking DS17 Abstracts 145

algorithms. In this context, spatio-temporal clustering al- Danobat Group gorithms have been proven to be very effective for the ex- [email protected] traction of coherent sets from sparse and possibly incom- plete trajectory data. Inspired by these recent approaches, we consider an unweighted, undirected network, which is MS9 constructed based on relative trajectory positions. Clas- Analytical Results in Nonlinear Dynamics of Turn- sical graph algorithms are then employed to analyze the ing network properties and to extract coherent behavior of the underlying flow. The proposed method is very fast to run, Nonlinear dynamics and bifurcation analysis of regenera- and we demonstrate that is produces useful results in a tive machine tool vibrations in turning are considered. The number of example systems. Furthermore, we point out global dynamics of turning involves switching between two theoretical links to other approaches. vector fields that represent metal cutting and free oscil- lations of the tool. The dynamics during cutting is gov- Kathrin Padberg-Gehle erned by a delay-differential equation, which is typically Institute of Scientific Computing nonlinear through the characteristics of the cutting force. TU Dresden Machine tool vibrations are associated with the loss of sta- [email protected] bility of the equilibrium. The stability is lost via Hopf bifurcation, which is typically subcritical and implies the onset of unstable periodic solutions. When the amplitude MS8 of the periodic oscillations gets so large that the tool loses Numerical Studies of Turbulent Rayleigh-B´enard contact with the workpiece, a nonsmooth fold bifurcation Convection Flows in Large-aspect-ratio Cells takes place that gives rise to a large-amplitude attractive solution. The large-amplitude solution is called chatter: it We present results of three-dimensional direct numerical is a complicated (sometimes even chaotic) motion that in- simulations of Rayleigh-Benard convection in flow domains volves intermittent loss of contact between the tool and the with side lengths that are significantly larger than the workpiece. At the intersections of the stability boundaries, height of the layer. In the turbulent flow regime, time- double Hopf bifurcations take place. Here, the dynamics is averaged mean velocity patterns get organized in roll pat- more complicated: in addition to two unstable periodic so- terns that are similar to the weakly nonlinear regime right lutions and a stable chaotic one, an unstable quasi-periodic above the linear stability threshold of convection. These solution might also coexist with the linearly stable equi- patterns are found to evolve very slowly with respect to librium. In this work, analytical methods are presented to time on the largest horizontal scales. First results of a approximate the arising periodic and the quasi-periodic so- parametric study at different Prandtl and Rayleigh num- lutions, and the region of coexistence is predicted by simple bers are presented in which we determine the typical cor- closed-form formulas. relation scales of the patterns and their contribution to the global transfer of heat by means of Lagrangian methods. Tamas G. Molnar, Zoltan Dombovari Department of Applied Mechanics Joerg Schumacher Budapest University of Technology and Economics Technische Universitat Ilmenau [email protected], [email protected] Institute of Thermodynamics and Fluid Mechanics [email protected] Tamas Insperger Budapest University of Technology and Economics Department of Applied Mechanics MS9 [email protected] Effect of Unstable Quasiperiodic Orbits on Heavy- Duty Milling Operation Gabor Stepan Department of Applied Mechanics We present theoretical and experimental study how unsta- Budapest University of Technology and Economics ble quasiperiodic oscillations can affect regenerative time- [email protected] periodic milling processes. Due to the nonlinearity in the cutting force characteristics, subcritical branch emerges from the Hopf bifurcation point of the periodic solution MS9 correspodning to the stationary cutting process. This Time-Periodic Delay Models of Milling Processes branch was calculated by numerical continuation till the point where the corresponding solution grazes the peri- During machining the regeneration of surface waviness can odic switching surfaces originated from cutting edge fly- cause unstable self-excited vibrations known as chatter. over. Also, results were compared with experiments per- Despite over one century of research, there remain many formed in heavy-duty milling machine equipped with inter- challenges in the avoidance of chatter because it constrains nal inertial drive. Starting with stable stationary cutting, the productivity of machining operations. For the case of perturbation was introduced for several revolution of the milling the rotating cutting tool possesses multiple teeth, milling tool then self-excitation was formed freely. By vary- which each remove material from the workpiece. The cut- ing the driving current of the inertial actuator, the size of ting force is a function of the thickness of this material, the domain of attraction was determined. and consequently the force is also dependant on the rela- tive vibration between the tool and the workpiece. If the Zoltan Dombovari dynamics of the system are linear then this leads to a de- Dynamics and Control, IK4Ideko layed differential equation with linear time periodic coeffi- 20870 Elgoibar, Basque Country, Spain cients due to the tool rotation. The stability of this system [email protected] can be investigated using various techniques, such as Time Finite Element Analysis, Semi-discretisation, or a Fourier Jokin Munoa series expansion of the time-periodic terms. Recent work Ideko Research Alliance IK4 by the authors has focussed on the special case of variable 146 DS17 Abstracts

helix tools. Here, the time delay between successive teeth [email protected] varies across the axial length of the cutter. This leads to a distributed time delay that can be used to enhance the sta- bility of the system. The present contribution will describe MS10 the challenges of modelling the stability of this system. It Revisiting Contact and Disease Transmission will be shown that the Fourier series and Fourier trans- Through the Lens of Fluid Dynamics form provide an elegant means of assessing stability, but that other un-modelled phenomena could be important for The mechanisms governing the transfer of pathogens be- this configuration of milling tool. tween infected and non-infected members of a population are critical in shaping the outcome of an epidemic. This is Neil D. Sims true whether one considers human, animal or plant pop- Department of Mechanical Engineering ulations. Despite major efforts aimed at the mathemat- University of Sheffield ical modeling and mitigation of infectious diseases, the n.sims@sheffield.ac.uk fundamental mechanisms of pathogen spreading for most infectious diseases remain poorly understood. Drawing Luis Urena upon clinical data, fluid experiments and theoretical mod- The University of Sheffield eling I will discuss the dynamics of transmission of various ueluis1@sheffield.ac.uk pathogens through the lens of fundamental fluid fragmen- tation and nonlinear dynamics.

Erdem Ozturk Lydia Bourouiba The University of Sheffield Massachusetts Institute of Technology Advanced Manufacturing Research Centre [email protected] e.ozturk@sheffield.ac.uk

MS10 MS9 Division Patterns and Dynamics of Hematopoietic State-Dependent Delay Effects in Drilling Stem Cells Mathematical modelling has long been employed to under- In this effort, a reduced-order system of a drill string, which stand cellular dynamics and to supplement experimental is a flexible structure used in oil well drilling operations, is knowledge and measurements. Mathematical techniques used to study coupled axial-torsion dynamics of the system. applied in the study of hematopoietic stem cells (HSCs) The system model has a state-dependent delay. The dif- and stem cell biology include both deterministic popula- ferent parameters that are considered include the cutting tion level models and stochastic applications, and are able coefficient, penetration rate, and the rotation speed. The to recreate and help interpret experimental labelling data. effects of the state-dependent delay on the system dynam- Models can advance our understanding of the homeosta- ics are examined by carrying out stability analyses, and sis of hematopoiesis and contribute to our comprehension it is discussed as to why the consideration of this state- of the dynamics of cancers like chronic myeloid leukemia dependent delay is important for drill-string dynamics. (CML), including rare presentations such as periodic CML. In this talk, I will present a delay differential equations Balakumar Balachandran model of HSC dynamics that incorporates the long-term, Dept. of Mechanical Engineering intermediate-term, and short-term HSCs, multipotent pro- University of Maryland genitors, and neutrophils to characterise the relative preva- [email protected] lences of symmetric self-renewal and differentiation, and asymmetric division present in the HSC populations. Xie Zheng Historically, the HSCs were thought to be a relatively Department of Mechanical Engineering dormant and homogeneous population, but new evidence University of Maryland, College Park, MD 20742 points to heterogeneous subpopulations with variable levels [email protected] of mitotic activity. By studying the ratios of each division type, this work sheds light on how HSCs constantly re- generate the blood system, and how deviations from the MS10 homeostatic control of blood cell production can induce pathological presentations like CML. Spatial and Temporal Spread of Infectious Pathogens Morgan Craig Harvard University Infectious pathogens are as mobile as the hosts they are [email protected] infecting. Understanding mobility is therefore important if one wants to understand how pathogens spread in space and time. I will present some models that have been used MS10 to address the mobility of individuals, with a focus on the Normal and Pathological Dynamics of Platelets in use of these models in the context of epidemiology. These Humans models consist typically of large systems of ordinary dif- ferential equations or continuous time Markov chains. I Periodic hematological diseases are dynamic diseases in will discuss some of the mathematical challenges arising which the number of one or more of the red blood cells, when considering such systems and show some real world white blood cells, and platelets oscillate in time. While applications. several periodic hematological diseases such as cyclic neu- tropenia and periodic chronic myelogenous leukemia are Julien Arino now well understood, the pathogenesis of cyclic thrombocy- University of Manitoba, Canada topenia (oscillating platelet numbers) remains unclear. We DS17 Abstracts 147

present a comprehensive mathematical model of platelet, reveals a novel method for gradient sensing and insight into megakaryocyte, and thrombopoietin dynamics in humans the biochemical mechanisms that ensure the establishment and use this model to investigate the etiology of cyclic of a unique polarity site. thrombocytopenia. Carefully estimating model parame- ters from laboratory and clinical data, we then argue that Timothy Elston a subset of parameters are involved in the genesis of cyclic University of North Carolina, Chapel Hill thrombocytopenia based on clinical information. We pro- [email protected] vide model fits to the existing data for both platelet counts and thrombopoietin levels by changing four parameters that have physiological correlates. Our results indicate MS11 that the primary change in cyclic thrombocytopenia is an interference with, or destruction of, the thrombopoietin re- Model for Cell Polarization Based on Coupled ceptor with secondary changes in other processes, including Membrane-Bulk Diffusion immune-mediated destruction of platelets and megakary- ocyte deficiency and failure in platelet production. Division, differentiation and proliferation of living cells rely on mechanisms of symmetry breaking. A key element of Gabriel Langlois these mechanisms is emergence of asymmetric (polar) dis- Brown University tributions of signaling molecules, often in form of molecular gabriel provencher [email protected] clusters. Clustering may be directed by external cues, but can also occur spontaneously. Multiple positive feedback Morgan Craig loops play a crucial role in establishing yeast polarity. We Harvard University discuss a model for stationary patterns and self-sustained [email protected] spatiotemporal oscillations based on passive diffusion of Rho GTPase in the interior coupled to reactions on the Tony R. Humphries membrane of lower dimensionality. The coupling of the McGill University bulk and active membranes arises through both nonlinear Mathematics and Statistics flux boundary conditions for the bulk diffusion field and [email protected] from feedback terms, depending on the local bulk concen- tration, to the dynamics on each membrane. The proposed Michael Mackey model can explain behavior of Cdc42 signaling molecule in McGill University, Canada budding and fission yeast. [email protected] Alexandra Jilkine Joseph M. Mahaffy University of Notre Dame San Diego State University [email protected] Dept of Mathematical Sciences mahaff[email protected] MS11 Jacques Belair Partner Search Strategy and Mechanisms of Polar- D´epartement de math´ematiques et de statistique ization During Fission Yeast Mating Universit´edeMontr´eal [email protected] Cell pairing is central for many processes, including im- mune defense, neuronal connection or sexual reproduction. Thibault Moulin How does a cell precisely orient towards a partner, espe- Laboratoire Chrono-Environnement, cially when faced with multiple choices? During condi- Universit´e de Bourgogne Franche-Comt´e tions of nitrogen starvation, the model eukaryote S. pombe [email protected] (fission yeast) undergoes sexual sporulation. Because fis- sion yeast are non-motile, contact between opposite mating Sean Sinclair types is accomplished by polarizing growth in each mat- Department of Mathematics and Statistics ing type towards the selected mate, a process known as McGill University shmooing. We used a combination of computational mod- [email protected] eling and experiments (performed by collaborators in the group of Sophie Martin, University of Lausanne) to model Liangliang Wang the role of the exploratory Cdc42-GTP zone that appears Department of Statistics and Actuarial Science and disappears along the cortex and stabilizes in response Simon Fraser University to secreted pheromone. We find that efficient pair forma- liangliang [email protected] tion occurs through the combination of local pheromone re- lease, short pheromone decay length, and local pheromone sensing. This result matches experimental observations of MS11 cells lacking the predicted GTPase-activating protein for Mathematical Models for Cell Polarization and Ras, which exhibit stabilized zones at reduced pheromone Gradient Sensing levels. We propose that during mating, Cdc42 responds to a Ras activator-inhibitor dynamical system along the Directed or polarized growth and the detection of chemical cell membrane, undergoing a transition from exploratory gradients are two fundamental cellular processes. Here we to stable localized states in response to opposite mating combine mathematical modeling with various experimen- type pheromone. tal approaches to investigate the molecular mechanisms that underline both processes during the mating response Dimitrios Vavylonis, Bita Khalili of Saccharomyces cerevisiae (budding yeast). Our analysis Lehigh University 148 DS17 Abstracts

[email protected], [email protected] tonomously Driven Electric Vehicles

We present a new algorithm for model predictive control MS11 of non-linear systems with respect to multiple, conflicting A PDE-DDE Model for Cell Polarization in Fission objectives. The idea is to provide a possibility to change Yeast the objective in real-time, e.g. as a reaction to changes in the environment or the system state itself. The algo- We consider a one-dimensional model of cell polarization rithm utilizes elements from various well-established con- in fission yeast consisting of a hybrid partial differential cepts, namely multiobjective optimal control, economic as equationdelay differential equation system. The model de- well as explicit model predictive control and motion plan- scribes bulk diffusion of the signaling molecule Rho GT- ning with motion primitives. In order to realize real-time Pase Cdc42 in the cytoplasm, which is coupled to a pair applicability, we split the computation into an online and of delay differential equations at the ends of the cell via an offline phase and we utilize symmetries in the open-loop boundary conditions. The latter represent the binding of optimal control problem to reduce the number of multiob- Cdc42 to the cell membrane and rerelease into the cyto- jective optimal control problems that need to be solved in plasm via unbinding. The nontrivial nature of the dynam- the offline phase. The results are illustrated using the ex- ics arises from the fact that both the binding and unbind- ample of an electric vehicle where the longitudinal dynam- ing rates at each end are taken to depend on the local ics are controlled with respect to the concurrent objectives membrane concentration of Cdc42. In particular, the asso- arrival time and energy consumption. ciation rate is regulated by positive feedback and the dis- sociation rate is regulated by delayed negative feedback. Sebastian Peitz We use linear stability analysis and numerical simulations University of Paderborn to investigate the onset of limit cycle oscillations at the [email protected] end compartments for a cell of fixed length. We find that the critical time delay for the onset of oscillations via a Kai Sch¨afer Hopf bifurcation increases as the diffusion coefficient de- Paderborn University creases. We then solve the diffusion equation on a grow- [email protected] ing domain under the additional assumption that the to- tal amount of the signaling molecule increases as the cell Sina Ober-Bl¨obaum length increases. We show that the system undergoes a Oxford University transition from asymmetric to symmetric oscillations as the [email protected] cell grows, consistent with experimental findings of new- end-take-off in fission yeast. Julian Eckstein, Ulrich K¨ohler Bin Xu Hella KGaA Hueck & Co. University of Utah [email protected], [email protected] [email protected] Michael Dellnitz University of Paderborn, Germany MS12 [email protected] Transactive System Design and Analysis for Smart Grid Applications MS12 As demand response is becoming increasingly important for Computational Methods for Dynamic Constrained smart grid applications, various control schemes have been Optimization developed to engage responsive loads in demand response programs in order to provide various ancillary services to An important class of problems in optimal control design is the power grid. Among the existing control strategies, defined with constraints that are not static but described transactive coordination and control have attracted consid- by continuous dynamics of a nonlinear system. Such con- erable research attentions. It uses economic or market-like straints are infinite dimensional. To solve a problem with constructs to manage distributed smart grid assets and is such constraints, one needs to apply a discretization scheme amenable to problems where self-interested users are coor- to get a finite dimensional approximation. The dimension- dinated to achieve global control objectives. Transactive ality of such approximate problem can become quite large control framework actually complements the conventional and the problem is often computationally challenging to centralized control framework associated with direct load handle. In this talk we present a computational approach control. It is based on distributed control and has the for dynamic constrained optimization that circumvents the significant advantages of scalability, flexibility and interop- high dimensionality by using a reduced-space optimization erability when applying to the large-scale systems such as approach together with state-of-the-art ODE/DAE inte- power grid. On the other hand, with transactive control grators and adjoint-based optimization gradient evalua- framework implemented at the distribution level, it will be tion. We will demonstrate this approach in applications promising to integrate demand response into the wholesale to power transmission planning and operation, where, for transactive operations to realize the transactive operation example, computing dynamic security constrained optimal framework for the power system. power flow is essential for deciding the appropriate control actions. We believe our approach will enable a number Jianming Lian, Karanjit Kalsi of optimal power flow computations currently run off-line PNNL to run in real time and, hence, provide grid operators far [email protected], [email protected] more flexibility in decision making. Added flexibility will help incorporate in the power grid more renewables such as solar and wind, whose behavior is less predictable than MS12 the behavior of traditional energy resources. We will dis- A Multiobjective MPC Approach for Au- cuss how to manage complexity of these computations and DS17 Abstracts 149

make it easily accessible to domain experts in the industry. Unsteady Flows

Transport barriers are time-varying generalizations of in- Slaven Peles variant manifolds which separate the phase space into dis- Lawrence Livermore Natinoal Laboratory tinct regions between which few, if any, orbits pass. In con- [email protected] trast to the separatrices of autonomous systems, transport barriers are not typically invariant sets which has made Cosmin G. Petra their characterization notoriously difficult. Recently, the Lawrence Livermore National study of Lagrangian coherent structures (LCS) and almost- Center for Advanced Scientific Computing invariant sets have led to an improved understanding of [email protected] transport in these systems. Specifically, these approaches have yielded reliable diagnostic tools such as the finite-time Lyapunov exponent (FTLE), geodesic deviation, as well as MS12 estimates of diffusion and filamentation. In this work we combine numerical diagnostics with a-posteriori analysis Optimal Sensor Placement Using Graph Metrics and validated numerics to obtain computer assisted proofs of existence for some LCS/almost-invariant sets. In par- The problem of assigning limited resources to nodes on a ticular, we introduce a method to compute rigorous enclo- graph arises either directly or indirectly in a wide variety of sures of FTLE and geodesic deviation for codimension-one applications like influence maximization in social networks. surfaces. As a by-product, we also obtain mathematically Such problems are typically formulated as variants of the rigorous bounds on the location as well as interval of exis- k-median problem for which solutions are obtained using tence for transport barriers. This information is useful not approximation algorithms. In this work, we propose an al- only for characterization but also for benchmarking exist- ternative approach for identifying nodes in a graph for as- ing diagnostics. The utility of our methods are illustrated signment of resources (such as sensors and advertisements in examples. etc.) The approach relies on computing the minimum of a coverage metric that is defined in terms of the eigenvectors Shane D. Kepley of the associated graph Laplacian. This solution is shown Florida Atlantic University to serve as an alternative for the solution to the k-median [email protected] problem. We then exploit this approach to optimally place heterogeneous sensors over large non-convex regions. This Jason D. Mireles James optimal sensor placement approach finds solutions that are Rutgers University computationally efficient with high coverage. [email protected] Tuhin Sahai United Technologies MS13 [email protected] Nontrivial Dynamics in the Forced Navier-Stokes Equations: A Computer-Assisted Proof George Mathew iRhythm Technologies In this talk, we introduce a rigorous numerical method george [email protected] to prove existence of periodic orbits in the forced Navier- Stokes (NS) equations. After reformulating the NS equa- tions in their vorticity formulation, we expand the periodic MS13 orbits using Fourier series both in space and in time and Rigorous Integration Forward in Time of Pdes Us- solve for the Fourier coefficients in a Banach algebra of ing Chebyshev Basis geometrically decaying sequences. The proof of existence follows by applying a Newton-Kantorovich type argument We present a rigorous numerical procedure for integration (the radii polynomial approach) close to a numerical ap- forward in time of one-dimensional dissipative PDEs with proximation. We use this approach to prove existence of periodic boundary conditions using the Chebyshev’s series two-dimensional periodic orbits in the NS equations. expansion in time of the solution. Our method is a Newton operator like approach, and we also make use of the radii Jean-Philippe Lessard polynomial method. Central to our approach is a proof Universit´eLaval of stability of a norm for inverses of linear operator with [email protected] respect to the Galerkin approximation dimension. Using this result we are able to apply the contraction principle in Jan Bouwe Van Den Berg infinite dimensional Banach space, for a sufficiently small VU University Amsterdam time-step. [email protected]

Jacek Cyranka Maxime Breden Rutgers University ENS Paris-Saclay & Universit´eLaval [email protected] [email protected]

Jan-Philippe Lessard Lennaert van Veen Universit´eLaval UOIT [email protected] [email protected]

MS13 MS13 Validated Computation of Transport Barriers in Recent Progress on Computational Theorems in 150 DS17 Abstracts

Dynamics such different features with individual reactions of the un- derlying complex biochemical signaling networks, due to In nonlinear analysis we often simulate dynamics of a sys- the large dimension of the system and its inherent non- tem of ODEs on a computer, or calculate a numerical so- linearities. How can we reduce dimensions for the sake lution to a partial differential equation. This gives very of mathematical tractability yet retaining essential bio- detailed, stimulating information. However, it would be physical features? What are the criteria for the choice of even better if we can be sure that what we see on the these features? To address these questions, I will focus on screen genuinely represents a solution of the problem. In metabolic modeling of the calcium-triggering molecule in- particular, rigorous validation of the computations would ositol 1,4,5-trisphosphate (IP3) in astrocytes the predomi- allow such objects to be used as ingredients of (forcing) nant cortical glial cell type. By simple analytic arguments, theorems. The past decade has seen enormous advances in I will illustrate how the interplay between different time the development of computer assisted proofs in dynamics. scales of IP3 production and degradation, could account Attention is now turning to infinite dimensional nonlin- for a large spectrum of healthy and pathological astrocytic ear dynamics generated by PDEs, integral equations, delay calcium signaling, pinpointing to specific molecular mecha- equations, and infinite dimensional maps. In this talk we nisms. The possibility to clearly identify these mechanisms will review recent developments and set the stage for the direct follows from the implemented system reduction pro- other talks in this minisymposium. We will discuss exis- cedure, accounting for different steps of propagation of per- tence, continuation and bifurcations of periodic orbits in turbations, from activated GPCRs on the extracellular side systems of ODEs, delay equations as well as PDEs (includ- of astrocytes, to targets in their interior. ing spatio-temporal periodic patterns). Other examples in- clude the rigorous computation of invariant manifolds and Maurizio De Pitt`a connecting orbits for ODEs, PDEs and delay equations. University of Chicago [email protected] Jan Bouwe Van Den Berg VU University Amsterdam Department of Mathematics [email protected] MS14 Mathematical Investigation of Ion Dynamics in As- trocytes and the Extracellular Space MS14 Role of Neuron-Glia Interaction in Epileptogenesis Astrocytes are glial cells in the brain that each wrap around thousands of synapses. Neurotransmitters released by neu- Neuronal activity is strongly reduced in chronically isolated rons can activate receptors on astrocytes, leading to the re- cortex. In response to prolonged synaptic inactivity astro- lease of IP3, and initiating calcium transients in astrocytes. cytes release tumor necrosis factor alpha (TNFa), which It is believed that these transients allow astrocytes to com- diffuses and binds to its dedicated receptors on the neurons municate with nearby neurons, but the mechanisms are still causing a reduction in the number of postsynaptic GABA being investigated. One proposed pathway is through the and an increase in the number of AMPA and NMDA re- sodium-calcium exchanger activated by the increase in as- ceptors. This process called homeostatic plasticity (HSP) trocyte cytosolic calcium. The exchanger activity affects upregulates depolarizing influences (such as excitatory in- the sodium-potassium pump, enabling astrocytes to regu- trinsic and synaptic conductances) and downregulates hy- late extracellular ion concentrations, and thus the neuronal perpolarizing ones (such as inhibitory conductances). We excitability. We study viability of such communication propose and test a hypothesis that homeostatic plasticity, pathway. Using experimental data collected by our col- which normally maintains a moderate level of activity in laborators, we first study the calcium responses evoked by the cortex, fail to control normal excitability in heteroge- short puffs of ATP. We develop an open-cell, single com- neous networks, where there are subpopulations of neurons partment minimal ODE model that captures the experi- with severely different levels of activity conditions found mentally observed diversity of calcium transients. By vary- in traumatized cortex. Thus, according to our hypothesis, ing the strength of calcium channels, we manipulate the trauma induced deafferentation leads to an acute decrease underlying bifurcation structure of the ODE system, thus of neuronal activity in affected areas, triggering upregu- examining the specific roles of individual calcium fluxes lation of neuronal excitability that in conditions of severe and making testable predictions. Building off of this under- deafferentation may lead to instabilities and development standing, we then introduce sodium and potassium fluxes. of epilepsy. Altering the strengths of these new fluxes, we explore the impacts calcium transients in astrocytes can have on ex- Maxim Bazhenov tracellular ionic concentrations and the firing patterns of Department of Cell Biology and Neuroscience neighboring neurons in healthy and pathological states. University of California, Riverside [email protected] Gregory A. Handy University of Utah Mathematics Department MS14 [email protected] Minimal Modeling of GPCR-Mediated Calcium Signaling Marsa Taheri University of Utah Calcium signaling mediated by Gprotein-coupled receptors Department of Bioengineering (GPCRs) is deployed by many different cells to carry and [email protected] propagate information from the extracellular environment to targets in their interior. In healthy vs. pathological John A. White cellular states, GPCR-mediated intracellular calcium dy- Biomedical Engineering namics may vary largely, showing disease-specific spatial- Boston University temporal features. It is challenging however to associate [email protected] DS17 Abstracts 151

Alla Borisyuk Complex Systems Lab University of Utah Indian Institute of Technology Indore Dept of Mathematics [email protected] [email protected]

MS15 MS14 Inferring Network Topology from Observational The Role of Astrocytic Glutamate Uptake in Neu- Data: Potentials and Pitfalls ronal Ion Homeostasis: A Case Study of Spreading Depolarization We analyse climate dynamics from a complex network approach. This leads to an inverse problem: Is there Simultaneous changes in ion concentrations, glutamate, a backbone-like structure underlying the climate system? and cell volume together with exchange of matter between For this we propose a method to reconstruct and analyze a cell network and vasculature are ubiquitous in numerous complex network from data generated by a spatio-temporal brain pathologies. A complete understanding of patho- dynamical system. We use different classic techniques logical conditions as well as normal brain function, there- for this reconstruction and find various pitfalls. There- fore, hinges on elucidating the pathways involved in these fore, three other procedures are proposed based on recur- mostly interdependent variations. In this paper, we com- rence, on conditional expectations and on a macroscopic bine the Hodgkin–Huxley type spiking dynamics, dynamic approach. These approaches enable us to uncover relations ion concentrations and glutamate homeostasis, neuronal to global circulation patterns in oceans and atmosphere, in and astroglial volume changes, and ion exchange with vas- particular teleconnections. Potentials of these techniques culature into a comprehensive model to elucidate the role are demonstrated. of glutamate uptake in the dynamics of spreading depo- larization (SD) - the electrophysiological event underlying Juergen Kurths numerous pathologies including migraine, ischemic stroke, Humboldt Univ,Germany, Potsdam Institute for Climate aneurysmal subarachnoid hemorrhage, and trauma. Our Impact results demonstrate that glutamate signaling is the intrin- Research, Germany, and Aberdeen University, UK sic mechanism of SD, and that impaired glutamate uptake [email protected] leads to recovery failure of neurons from SD. We confirm predictions from our model experimentally by showing that inhibiting astrocytic glutamate uptake using TFB-TBOA MS15 nearly quadruples the duration of SD in cortical slices from Experimental Observation of Spiral Wave Chimera juvenile rats. The model equations are either derived from in a Very Large Network of Chemical Oscillators a combination of first physical principles of electroneutral- ity, osmosis, and conservation of particles and known phys- I will present a versatile experimental setup based on op- iological facts. Accordingly, we claim that our model is tically coupled catalytic micro-particles, that allows for broadly applicable to other brain pathologies and normal the experimental study of synchronization patterns in brain function. very large networks of relaxation oscillators under well- controlled laboratory conditions. In particular I will show Ghanim Ullah, Niklas Hubel our experimental observation of the spiral wave chimera, University of South Florida predicted by Kuramoto in 2003. This pattern features a [email protected], [email protected] wave rotating around a spatially extended core that con- sists of phase-randomized oscillators. We study its ex- Jokubas Ziburkus istence depending on coupling parameters and observe a University of Houston transition to incoherence via core growth and splitting. [email protected] The spiral wave chimera is likely to play a role in cardiac and cortical cell ensembles, as well as in cilia carpets.

MS15 Jan Totz Institut f¨ur Theoretische Physik Eigenvector Localization in Complex Networks Technische Universit¨at Berlin [email protected] Analysis of eigenvector localization properties are impor- tant due to its various applications including networks cen- trality, spectral partitioning, and disease spreading phe- Kenneth Showalter nomena in complex networks. We evolve an initially ran- West Virginia University dom network with an edge rewiring technique using inverse Department of Chemistry participation ratio as a fitness function to obtain a net- [email protected] work, which has the most localized principal eigenvector, and analyze the structural properties of the optimized net- Harald Engel works. It turns out that in the optimized network there Institut f¨ur Theoretische Physik, Technische Universit¨at exists a set of edges that are crucial for the localization, Ber rewiring only one of them leads to a complete delocaliza- 10623 Berlin, Germany tion of the principal eigenvector. We analytically derive [email protected] the condition for changes in the IPR values for the edge rewiring. The work has potential applications in designing technological networks such as computer virus propagation MS15 network, transport network, and systems requiring vibra- Partial Cascades on One-Dimensional Geographic tion confinement such as spring-mass-damper systems, and Networks piezoelectric networks. We examine the question of cascades on networks where Sarika Jalan the connections between nodes are heavily dependent on 152 DS17 Abstracts

some underlying spatial geography. Our focus is on the in our study was strongly dependent upon a spatial pos- Centola-Macy model of cascade dynamics, where nodes will itive feedback term. Simulation data support a role for activate if they have a sufficient number (not a sufficient cytoskeletal remodeling in this spatial feedback, constitut- proportion) of active neighbors. We consider cases where ing a cellular memory regarding directions. Finally, we the connections are short enough that the cascade can be show that additional feedback links providing information viewed as a wave front propagation and there is no activa- about the gradient to multiple levels in the pathway can tion of new clusters. Of particular interest is the possibility help the cell to refine initial inaccuracy in the polarization of the termination of the cascade and the probability dis- direction. tribution of final cascade sizes. We model the number of agents that activate on each time step as a Markov chain Elijah Roberts and estimate the extinction probability as a function of Department of Biophysics time. We derive a model to predict the final cascade size Johns Hopkins University distribution for one-dimensional networks with uniformly [email protected] distributed nodes.

Yosef M. Treitman MS16 Rensselaer Polytechnic Institute Making Rare Events Happen: Prediction and Con- [email protected] trol of Extinction and Switching in Heterogeneous Networks

MS16 We consider epidemic extinction as rare events in finite Noise-Induced Rare Events in Population Dynam- networks with broad variation in local connectivity. Given ics: the Role of Spatial Degrees of Freedom random networks with a given degree distribution, we are able to predict the most probable, or optimal, paths to We study the influence of the network topology on the extinction in various configurations, including truncated statistics of interest including rare events of interacting power-laws. We find that paths for heterogeneous networks particle systems on complex graphs. As an example, we follow a limiting form in which infection first decreases in investigate the metastability and fixation properties of a low-degree nodes, which triggers a rapid extinction in high- set of evolutionary processes. In the framework of evolu- degree nodes, and finishes with a residual low-degree ex- tionary game theory, where the fitness and selection are tinction. The usefulness of our approach is further demon- frequency dependent and vary with the population compo- strated through optimal control strategies that leverage the sition, we analyze the dynamics of snowdrift games (char- dependence of finite-size fluctuations on network topology. acterized by a metastable coexistence state) on scale-free Interestingly, we find that the optimal control is a mix of networks. Using an effective diffusion theory in the weak treating both high and low-degree nodes based on theoreti- selection limit, we demonstrate how the scale-free structure cal predictions, in contrast to methods that ignore dynam- affects the systems metastable state and leads to anoma- ical fluctuations. This research is supported by the Office lous fixation compared to the non-spatial well-mixed case. of Naval Research. In particular, we analytically and numerically show that Ira B. Schwartz the probability and mean time to fixation are character- Naval Research Laboratory ized by stretched-exponential behaviors with exponents de- [email protected] pending on the networks degree distribution.

Michael Assaf Jason Hindes Racah Institute of Physics US Naval Research Laboratory Hebrew University of Jerusalem US Naval Research Laboratory [email protected] [email protected]

Brandon Lindley MS16 Ralph Wagner Associates How Do Cells Extract Information About the Di- [email protected] rection of Signaling Molecule Gradients to Make Decisions? Leah Shaw College of William and Mary Nearly all eukaryotic cells organize macromolecules by po- Dept. of Applied Science sitioning them in specific locations relative to an external [email protected] signal. This organizing process is known as polarization and is often directed along a gradient of an external sig- naling molecule. Like many cellular processes, polariza- MS16 tion decisions are made using a relatively small number of Epidemic Extinction in Adaptive Networks with molecules and are thus susceptible to internal and external Avoidance Rewiring fluctuations during signal transduction. How cells manage to transform a noisy external signal into accurate direc- During epidemic spread on a network, individual nodes tional sensing is an open question in cellular biology. Here, may adapt their connections to reduce their chance of in- I present results showing that spatial structure in the sig- fection. A commonly modeled form of adaptation is avoid- naling molecule distribution plays a key role in extracting ance rewiring, where a noninfected node rewires a link away directional information. Our simulation data show that from an infected node, instead connecting to another non- an architecture combining bistability with spatial positive infected node. In a susceptible-infected-susceptible (SIS) feedback permits the cell to both accurately detect and in- disease model, small rewiring results in a single stable ternally amplify an external gradient. We observe strong steady state, but larger rewiring leads to a backward bifur- polarization in all individual cells, but in a distribution of cation and bistability between an endemic state and a dis- directions centered on the gradient. Polarization accuracy ease free state. We compare stochastic extinction in both DS17 Abstracts 153

the small and larger rewiring cases. Applying large devia- part of the talk, we offer a taxonomy of the current lit- tion theory on an adaptive network, we predict extinction erature by focusing on non-coherent detection methods. times and find the most probable path to extinction. The Finally, several concluding remarks are discussed. analysis is compared with stochastic simulations of the net- work. Georges Kaddoum D´epartement de g´enie ´electrique Leah Shaw Ecole´ de technologie sup´erieure College of William and Mary [email protected] [email protected]

Jason Hindes MS17 US Naval Research Laboratory Chaotic OFDM System for Secure Multi-User US Naval Research Laboratory Communications [email protected] Chaotic modulation schemes, Orthogonal Frequency Divi- Ira B. Schwartz sion Multiplexing(OFDM) and Interference Alignment(IA) Naval Research Laboratory are combined in order to obtain an inversely proportional [email protected] relationship between throughput and detectability of trans- mit signals in multi user environment. A mixture of chaotic modulation schemes is used to generate chaotically modu- MS17 lated symbols for each sub-carrier of the OFDM transmit- Chaotic Sequences for Synchronization and Com- ter. This technique can eliminate the physical layer secu- munication rity concerns in conventional OFDM systems. Through- put degradation due to interference can be minimized us- A major impediment to coherent chaotic communications ing IA. In particularly, IA can be applied independently is the difficulty of synchronizing the receiver to the trans- inside disjoint subsets of subcarriers [J. Reitterer and M. mitter. In this work, a set of dictionary sequences is cre- Rupp, Interference alignment in UMTS Long Term Evolu- ated from a chaotic map. The dictionary sequences may tion, 19th European Signal Processing Conference, 2011]. be concatenated in a way that minimizes the discontinu- At the receiver side, interference suppression is performed ity between the end of 1 sequence and the beginning of on disjoint subsets of sub-carriers. Then, a mixture of cor- the next sequence. The receiver has a copy of these dictio- relators and mean value functions are implemented to de- nary sequences, so it uses a correlation detector followed by modulate chaotic signals [H. Leung, S. Shanmugam, N. a Viterbi decoder to estimate the transmitted sequences, Xie, S. Wang, ”An ergodic approach for chaotic signal es- allowing the receiver to be synchronized to the transmit- timation at low SNR with application to ultra-wide-band ter. Once synchronization is achieved, information can be communication,” IEEE Trans. on Information on Signal transmitted by multiplying the transmitted signal by + or Processing, 2006]. The statistical random tests, time do- 1 to encode a binary 1 or 0. The binary values are detected main randomness tests and frequency domain randomness in the receiver by cross correlation. The bit rate can be in- tests are used to confirm the improvements in physical layer creased by using the multiple components of the chaotic security. The improvements in throughput is investigated map to create multiple orthogonal signals. The encoding to ensure the inversely proportional relationship between technique is equivalent of binary phase shift keying, and detectability and throughput. gives the same bit error rate. Henry Leung, Chatura Seneviratne Thomas L. Carroll University of Calgary Naval Reseach Laboratory Department of Electrical and Computer Engineering [email protected] [email protected], [email protected]

MS17 MS17 Chaos Theory in Communications: Applications Wireless Communication with Chaos and Challenges The constraints of a wireless physical media, such as multi- Since the early 1990s, a large number of chaos-based com- path propagation and complex ambient noises, prevent in- munication systems have been proposed exploiting the formation from being communicated at low bit error rate. properties of chaotic waveforms. The motivation lies in Surprisingly, it has only recently been shown that, from the significant advantages provided by this class of non- a theoretical perspective, chaotic signals are optimal for linear signals. For this aim, many communication schemes coherent communication. It maximises the receiver signal- and applications have been specially designed for chaos- to-noise performance, consequently minimizing the bit er- based communication systems where energy, data rate, and ror rate. This talk demonstrates numerically and experi- synchronization awareness are considered in most designs. mentally that chaotic systems can in fact be used to cre- Recently, the major focus, however, has been given to the ate a reliable and efficient wireless communication system. non-coherent chaos-based systems to benefit from the ad- We also show that the chaotic signals provide a easy way vantages of chaotic signals and non coherent detection and to decrease the inter-symbol interference caused by multi- to avoid the use of chaotic synchronization, which suffers path. We propose an impulsive control method to generate from weak performance in the presence of additive noise. chaotic wave signals that encode arbitrary binary informa- This talk presents a comprehensive survey of the widely tion signals, and a sub-optimal threshold to detect infor- used wireless radio frequency chaos-based communication mation bits. The experimental validation is conducted by systems. First, it outlines the challenges of chaos imple- inputting the signals generated by an electronic transmit- mentations and synchronization methods, followed by com- ting circuit to an electronic circuit that emulates a wire- prehensive literature review and analysis of chaos-based less channel, where the signals travel along three differ- coherent techniques and their applications. In the second ent paths. The output signal is decoded by an electronic 154 DS17 Abstracts

receiver, after passing through a match filter. The sim- bilistic inference models of decision making, and derive an ulation results also are given to show that the proposed ideal observer model for inferring the present state of the sub-optimal threshold can achieve very competitive per- environment along with its rate of change. Moment closure formance. then allows us to obtain a low-dimensional system that per- forms comparable inference. These computations can be Hai-Peng Ren implemented by a neural network model whose connections Xi’an Univeristy of Technology are updated according to an activity-dependent plasticity PRChina rule. Our work therefore builds a bridge between statisti- [email protected] cal decision making in volatile environments and stochastic nonlinear dynamics of neural systems. Chao Bai, Jun-Liang Yao Xi’an University of Technology Zachary P. Kilpatrick [email protected], [email protected] Department of Applied Mathematics 526 UCB, University of Colorado, Boulder, CO, 80309 Murilo Baptista USA University of Aberdeen [email protected] [email protected] Adrian Radillo Celso Grebogi University of Houston King’s College [email protected] University of Aberdeen [email protected] Kresimir Josic University of Houston Department of Mathematics MS18 [email protected] Models of Value-Sensitive Decision Making Alan Veliz-Cuba Decision making is crucially important at all levels of bi- Department of Mathematics ological complexity. Two-alternative choice tasks, for ex- University of Dayton ample, are widely studied in terms of accumulation of rel- [email protected] ative evidence. Related models are known to implement a speed-accuracy tradeoff [Bogacz et al., Psychol. Rev. 113: 700 (2006)]. However, based on findings in house-hunting MS18 honeybees [Pais et al., PLOS ONE 8: e73216 (2013)], and Collective Behavior and Individual Variability: recent results obtained from behavioral data [Teodorescu From Personalities to Heterospecific Interactions et al., Psychon. Bull. Rev. 23: 22 (2016)], we follow an approach that describes value-based decisions, which Interactions between groups and sub-groups - be they should be related to a speed-value tradeoff rather than a strains or species have been reported in many cases speed-accuracy tradeoff [Pirrone et al., Front. Neurosci. across many taxae. We here focus on interactions that 8: 1 (2014)]. Here, we compare the performance of dif- are either quantitatively or qualitatively different reflecting ferent nonlinear neuro-models of decision making that dif- inter-individual variability within a group or heterospecific fer in the way the inhibition mechanism is implemented, communication. We first propose a simple model of ag- i.e. feed-forward, mutual and interneuronal inhibition. As- gregation based on earlier experiments accounting for the signing quality values to available options we are able to presence of personalities inside a group and discuss the study the dynamics of the decision-maker depending on differences in the subsequent collective behaviours as com- the quality of options and the balance between positive pared to a group composed by clones. We subsequently and negative feedback. Considering two options, which consider the case of interactions between two Physarum are of equal quality to the decision-maker, we apply the polycephalum individuals of the same or different strain interneuronal inhibition model to a hypothetical animal and determine their role in their decision-making patterns. making foraging decisions. As a result, we find that in an Finally, a generic model accounting for both conspecific ongoing decision-making process oscillatory behavior may and heterospecific interactions is outlined predicting differ- improve the performance of the animal. Our results mo- ent collective behaviours without any change of individuals tivate an integrated functional and mechanistic study of algorithm as some key generic parameters such as the car- animal decision-making. rying capacity, the number of individuals involved and the strength of inter-attraction between sub-groups are varied. Thomas Bose, Andreagiovanni Reina, James A R A key result is the possibility for sub-groups to segregate Marshall between patches and for transition between different pat- Department of Computer Science terns, even in absence of active agonistic behaviour. University of Sheffield, Regent Court, Sheffield, S1 4DP, UK Stamatios C. Nicolis t.bose@sheffield.ac.uk, a.reina@sheffield.ac.uk, Unit of Social Ecology james.marshall@sheffield.ac.uk Universit´e Libre de Bruxelles, 1050 Bruxelles, Belgium [email protected] MS18 Evidence Accumulation in Dynamic Environments MS18 Excitability and Feedback in Regulation of Forag- To make decisions a constantly changing world, organ- ing Harvester Ants isms must account for environmental volatility and appro- priately discount old information when making decisions Harvester ant colonies regulate the rate at which foragers based on such accumulated evidence. We introduce proba- leave the nest based on brief antennal interactions between DS17 Abstracts 155

incoming food-bearing foragers and potential foragers in and of disorder. the nest entrance chamber. Despite such limited interac- tions, ant colonies exhibit steady-state foraging rates that Jean-Fran¸cois Mercier are robust to perturbation and adaptive to changes in tem- Applied Mathematics Department perature and humidity. To examine the underlying feed- ENSTA ParisTech back mechanisms of foraging regulation, we have adapted [email protected] a low-dimensional nonlinear model of neuronal excitability using the analogy between a spiking neuron and a forag- Bruno Lombard ing ant. Our model closes the loop around the dynamics LMA inside the nest, which maps incoming foragers to outgoing [email protected] foragers, with the dynamics outside the nest, which maps outgoing foragers to incoming foragers. Inside the nest we use a leaky integrator to model accumulation of evidence MS19 from interactions and excitability dynamics to model the Symmetry-Induced Dynamic Localization in Lat- decision of potential foragers to go out and forage. Out- tice Structures side the nest, we use a time delay distribution to model foraging. Closing the loop provides a mechanism for sta- The existence of internal symmetry in lattice structures bility of the steady-state incoming and outgoing rates. To may give rise to stationary compact solutions, even in the explain adaptation of foraging rates to changes in temper- absence of disorder and nonlinearity. These compact solu- ature and humidity, we propose another feedback mecha- tions are related to the existence of flat dispersion curves nism that manages volatility in the excitability dynamics. (bands). Nonlinearity can be a destabilizing factor for such With a small number of parameters, our model qualita- compactons. This can be illustrated even by a simple one- tively captures a range of behaviors observed in harvester site model, in which the compacton corresponds to a sin- ants, and presents opportunities for generalization to other gle hidden antisymmetric mode. This antisymmetric mode contexts. can lose its stability through parametric resonance, when accounting for nonlinear interactions. The phenomenon is Renato Pagliara examined for a chain of linearly connected massless boxes Mechanical and Aerospace Engineering containing two masses each, coupled to the box by cubic Princeton University, Princeton, NJ USA internal interaction forces. Linear stability of the compact [email protected] mode is performed rigorously for a single element by Hills method, revealing a finite width instability tongue in the parameter plane. Numerical integration for a closed chain Deborah M. Gordon of several boxes confirms the emergence of either a sta- Stanford University ble static compacton solution or dynamic delocalization, [email protected] depending on the parameters, in correspondence with the single element stability map. In addition, a chain with in- Naomi E. Leonard ternal configurational symmetry and vibro-impact on-site Princeton University potentials is investigated. Rigorous stability analysis is [email protected] performed for an asymptotically long chain using the no- tion of the saltation matrix and existence of finite stability zones of a compacton solution in the nonlinear chain is MS19 revealed.

A Model for Nonlinear Acoustic Waves in a Non- Nathan Perchikov uniform Lattice of Helmholtz Resonators Technion - Israel Institute of Technology [email protected] We are interested in the dynamics of nonlinear solitary acoustic waves in lattices. The main feature of these waves Oleg Gendelman is that they propagate without change of shape and with a Technion Israel Institute of Technology velocity depending of their amplitude. To study the propa- [email protected] gation of high amplitude acoustic pulses in a 1D waveguide connected to a lattice of Helmholtz resonators, an homog- enized model has been proposed by Sugimoto (J. Fluid. MS19 Mech., 244 (1992)). This model takes into account both Strongly Nonlinear Acoustic Metamaterials with the nonlinear wave propagation and various mechanisms Passive Self-Tuning and Wave Redirection Prop- of dissipation. We have already developed a numerical erties modeling of this model and we have successfully compared simulations with experimental data. In Sugimoto’s model, We present analytical and numerical results concerning all the resonators are the same. The drawback of this passive nonlinear targeted energy transfer TET leading approach is that the reflection of an incident wave by a to wave redirection in networks of weakly coupled granu- defect cannot be considered. To remedy this limitation, lar media. In particular, we consider two weakly coupled we propose an extension of the model, predicting two-way uncompressed granular chains of semi-infinite extent, com- propagation across variable resonators. Thanks to a dis- posed of perfectly elastic spherical beads with Hertzian crete description of the resonators, the new model takes interactions, mounted on linear elastic foundations. One into account two important features: resonators of differ- of the chains is regarded as the ”excited” chain, and the ent strengths and back-scattering effects. Moreover, we other is designated as the ”absorbing” chain. Both chains reformulate the viscothermal losses, which leads to a for- are initially at rest and a stratification of the coupling be- mulation suitable for an energy balance. Comparisons with tween the two chains is introduced. Under both impulsive experimental data show that a closer agreement is obtained and harmonic excitations of the excited chain we study the than with the original Sugimoto’s model. Numerical exper- nonlinear dynamical mechanism for TET in this granular iments are also proposed to highlight the effect of defects network, and identify it as a transient resonance capture. 156 DS17 Abstracts

This is manifested as a macro-scale analog of the Landau- properties of the Koopman operator are often computed Zener tunneling quantum effect in space. The TET mech- using a class of algorithms known as Dynamic Mode De- anism provides an efficient way for wave redirection in the composition (DMD), however, there has been no general network, whereby breathers originally initiated in the ex- theorems on convergence of these algorithms. In this talk, cited chain are eventually redirected to the absorbing chain we present the proof of convergence for some general classes only. Numerical simulations fully validate the theoretical of dynamical systems and discuss some examples from com- analysis and results. putational fluid dynamics.

Mohammed A. Hasan Hassan Arbabi University of California, San Diego University of California, Santa Barbara [email protected] [email protected]

Yuli Starosvetsky Igor Mezic Technion, Israel Institute of Technology UCSB [email protected] [email protected]

Leonid Manevitch Semenov Institute of Chemical Physics MS20 Russian Academy of Sciences The Computation of Invariant Manifolds for Par- [email protected] tial Differential Equations by Set Oriented Numer- ics

Alexander Vakakis In this talk we present a novel numerical framework for the University of Illinois computation of finite dimensional invariant manifolds for [email protected] infinite dimensional dynamical systems. With this frame- work we extend classical set oriented numerical schemes (for the computation of invariant manifolds in finite di- MS19 mensions) to the infinite dimensional context. The under- Traveling Waves in Nonlinear Metamaterials: lying idea is to utilize appropriate embedding techniques Granular Chains and Beyond for the globalization of these manifolds in a certain finite dimensional space. Finally, we illustrate our approach by In this talk we will present some recent progress motivated the approximation of invariant manifolds in the context of by granular crystals and related applications. We will ex- partial differential equations. amine traveling waves in prototypical such chains and sub- sequently generalize consideration to FPU type lattices of Michael Dellnitz different types. We will revisit and establish a criterion for University of Paderborn, Germany the stability of traveling waves and analyze their spectra. [email protected] We will connect such traveling waves to discrete breathers and connect the criterion to one for the stability of discrete Adrian Ziessler breathers. We will also explore two complementary aspects Paderborn University for examining the spectra of traveling waves, namely that [email protected] of a steady state of a co-traveling frame which leads to an advance-delay equation, and that of a periodic orbit modulo shifts that gives rise to a suitable Floquet multi- MS20 plier problem. In addition to the theoretical analysis and Fast Computation of Coherent Sets connections, a number of computational examples will be presented and wherever possible discussion of relevant ex- Finite-time coherent sets are regions in the domain of some periments will be highlighted. flow that minimally mix with the rest of the domain and roughly keep their geometric shape over some finite pe- Haitao Xu riod of time, while the flow around them is more turbu- University of Minnesota lent. Prominent examples in fluid flows are ocean eddies [email protected] or atmospheric vortices. Techniques for the detection and computation of coherent sets are available which are based Panayotis Kevrekidis on spectral properties of appropriately defined transfer and University of Massachusetts Laplacian operators. We develop efficient numerical meth- [email protected] ods for the discretization of the associated eigenproblems, enabling a fast and accurate extraction of finite-time coher- ent sets, even for scattered and possibly sparse trajectory MS20 data. This is joint work with A. Denner (TUM), G. Froy- Computation of the Koopman Spectrum in Com- land (UNSW) and D. Karrasch (TUM), cf. the talk by G. plex Flows Froyland in ”Koopman Operator Techniques in Dynamical Systems: Theory - Part II of II”. Koopman operator theory is a powerful framework for study of high-dimensional systems including fluid flows. In Oliver Junge particular, the eigenvalues and eigenfunctions of the Koop- Center for Mathematics man operator, if existent, play a central a role in study of Technische Universit¨at M¨unchen, Germany such systems. For example, Koopman modes, which are [email protected] projection of observables onto Koopman eigenfunctions, describe the periodically evolving spatial structures in the flow, and they have become a standard tool for analysis MS20 and low-dimensional modeling of fluid flows. The spectral Quantifying the Influence of Lateral Boundaries on DS17 Abstracts 157

Turbulent Transport electronic circuits in a chaotic regime, we study the loss of inter-layer synchronization as a function of the betweenness Turbulent flows almost by definition mix efficiently: any centrality of the removed links. initial patch of marked fluid will rapidly smear out and be intermingled with other fluid by the action of the flow. Irene Sendi˜na-Nadal But this turbulent transport is also well known to be influ- Rey Juan Carlos University, enced, at least in finite time, by the presence of coherent Mostoles, Madrid, Spain structures that may either locally enhance or suppress mix- [email protected] ing. Such structures may be pinned to features at bound- aries, potentially inducing long-time effects on the mixing Inmaculada Leyva structure of the flow. We study such effects in a quasi-two- University of Rey Juan Carlos dimensional laboratory turbulent flow in the presence of Madrid, Spain canonical lateral boundary features, and use transfer op- [email protected] erators to quantity the effects of these boundaries on the mixing in the flow. Ricardo Sevilla-Escoboza Centro Universitario de los Lagos Nicholas T. Ouellette Universidad de Guadalajara Yale University [email protected] Department of Mechanical Engineering and Materials Science [email protected] Ricardo Gutierrez Department of Chemical Physics The Weizmann Institute of Science MS21 [email protected] A Multiplex View of PageRank Through Biplex Markov Chains Javier Buldu Universidad Polit´ecnica de Madrid, Spain & PageRank is the basic ingredient of the (probably) most Universidad Rey Juan Carlos, Madrid, Spain famous web searcher (Google), but it also has many appli- [email protected] cations to different real-life problems, ranging from biolog- ical systems to cibersecurity. In this talk, a multiplex view Stefano Boccaletti of the classic PageRank is presented including some ana- CNR-Istituto dei Sistemi Complessi lytical relations between this new approach and the usual [email protected] centrality measures. In addition to this, an analysis of the control of the PageRank of a node is presented in terms of personalization vectors and some sharp analytical results MS21 about the limits of this control are included. Analysis of Chinese Airline Network As a Multi- layer Network Miguel Romance Centre for Biomedical Technology (CTB) We encapsulate the Chinese Airline Network (CAN) into Technical University of Madrid, Pozuelo de Alarc´on, multi-layer infrastructures via the ’k-core decomposition’ Madrid method. The network is divided into three layers: Core [email protected] layer, containing airports of provincial capital cities, is densely connected and sustains most flight flow; Bridge layer, consisting of airports in Tier 2 and Tier 3 cities, MS21 mainly connects two other layers; and Periphery layer, Inter-Layer Synchronization in Nonidentical Mul- comprising airports of remote areas, sustains little flight tilayer Networks flow. Moreover, it is unveiled that CAN stays the most robust when low-degree nodes or high flight flow links are Inter-layer synchronization is a dynamical state occurring removed, which is similar to the Worldwide Airline Net- in multi-layer networks composed of identical nodes. The work (WAN), albeit less redundant. state corresponds to have all layers synchronized, with nodes in each layer which do not necessarily evolve in uni- Zhen Wang son. So far, the study of such a solution has been restricted Qingdao University, Qingdao, China to the case in which all layers had an identical connectiv- [email protected] ity structure. When layers are not identical, the inter- layer synchronous state is no longer a stable solution of the system. Nevertheless, when layers differ in just a few MS21 links, an approximate treatment is still feasible, and allows Generalizing the Master Stability Function to Mul- one to gather information on whether and how the system tilayer Networks may wander around an inter-layer synchronous configura- tion. In this Minisyposium, we will report the details of The structure of many real-world systems is best captured an approximate analytical treatment for a two-layer multi- by networks consisting of several interaction layers. Un- plex, which results in the introduction of an extra inertial derstanding how a multi-layered structure of connections term accounting for structural differences. Numerical val- affects the synchronization properties of dynamical systems idation of the predictions highlights the usefulness of our evolving on top of it is a highly relevant endeavour in math- approach, especially for small or moderate topological dif- ematics and physics, and has potential applications to sev- ferences in the intra-layer coupling. Moreover, we identify a eral socially relevant topics, such as power grids engineering non-trivial relationship between the betweenness centrality and neural dynamics. We present a general framework to of the missing links and the intra-layer coupling strength. assess the stability of the synchronized state in networks Finally, by the use of two multiplexed identical layers of with multiple interaction layers, deriving a necessary con- 158 DS17 Abstracts

dition that generalizes the Master Stability Function ap- will end by discussing challenges to the field and require- proach. ments for the brave folk that wish to embark upon it.

Charo I. del Genio Angelean O. Hendrix School of Life Sciences GlaxoSmithKline University of Warwick, U.K. [email protected] [email protected]

MS22 MS22 Modeling Lung Airway Liquid Dynamics for Hy- Dimer Dynamics and Degenerate Transversally In- perosmotic Treatment Design in Cystic Fibrosis tersecting Manifolds Airway disease in Cystic Fibrosis (CF) is characterized We consider a pharmacological model of dimerisation, i.e., by lung infection, inflammation, and impaired mucocil- a receptor binding two ligand (drug) molecules. This model iary clearance (MCC) arising from depletion of the air- is an extension of the well studied target mediated drug way surface liquid (ASL) at the organ-scale. Dysfunction disposition model (TMDD) in which the receptor binds in the CF transmembrane conductance regulator protein to one ligand molecule. It is assumed that the binding (CFTR) has been reported to cause cell-scale dysregulation is the fastest process. This gives a separation of time in ion and liquid transport alone and via other transport- scales, which allows us to use geometric singular perturba- related proteins. We have used in vitro measurements of tion theory to analyse these models. In both models, the ion transport and liquid absorption dynamics in cells with slow manifold consists of two components, which intersect and without CF to develop an ordinary differential equa- transversely in the origin. The dimerisation model leads to tion model of liquid and solute transport in cultured human a degenerate intersection. To analyse such intersection, we bronchial epithelia (HBE). Our cell-scale model accurately consider a general two parameter slow-fast system in which characterizes transcellular liquid transport in HBE cultures the critical set consists of a one dimensional manifold and and recapitulates experimental findings of liquid and solute a two dimensional manifold, intersecting at the origin. Us- hyperabsorption. From cell-scale predictions we conclude ing geometric desingularisation, we determine the fate of that reduced paracellular integrity is the predominant fac- the incoming one dimensional manifold and show that for a tor leading to increased liquid and DTPA absorption in subset of the parameter set there is a jump away from the CF. At the lung-scale, we have developed a physiologically intersection at the origin and away from the critical set. motivated pharmacokinetic model of the action of hyper- For the remaining parameter set, there is an exchange of tonic saline (HS) as an inhaled therapy to improve MCC stability between the attracting components of the critical and reduce absorption of DTPA, and thus airway liquid, set and the direction of the continuation can be expressed in CF. This multi-scale understanding of fluid trafficking in terms of the parameters. The parameters for the dimeri- in individual patients facilitates the design and optimiza- sation model fit into the latter category and we will give the tion of treatment schedules and doses in order to maintain approximation of the dynamics in the dimerisation model. consistent airway hydration.

Robert S. Parker, Matthew Markovetz Gianne Derks, Philip J. Aston Department of Chemical and Petroleum Engineering University of Surrey University of Pittsburgh Department of Mathematics [email protected], [email protected] [email protected], [email protected] Timothy Corcoran Christine Gavin Department of Pulmonary, Allergy, and Critical Care University of Surrey University of Pittsburgh [email protected] [email protected]

MS22 MS22 Beyond Dose Predictions: How Mathematical Modelling the Impact of Physicochemical Drug Pharmacology Supports Drug Development from Properties on Drug Reservoirs in the Skin Target Identification to Clinical Outcomes Treatment of chronically ill patients, such as those suffer- As mathematicians we are often looking for interesting ing from bipolar disorder or drug addiction, often involves AND impactful projects. Drug development presents a long-term drug prescription of potentially harmful drugs plethora of opportunities for the right person. Beyond es- (if they reach toxic concentrations in the blood). To main- tablished methods used for PK/PD and dose selection. We tain drug concentrations within the therapeutic window will review the pipeline for those unfamiliar with drug de- requires accurate and accessible monitoring which is very velopment and discuss industry wide needs for support in challenging. One mechanism that has been proposed is to target identification, lead optimization, interspecies trans- monitor the drugs non-invasively through the skin. Whilst lation and experimental design. Along this pipeline there in principle this is an obvious approach to take, it quickly are many open questions for mathematicians interested in becomes complicated by the fact that drugs that have cir- graph theory, ODEs, PDEs, optimization, control theory, culated for long periods in the body build up reservoirs even chaos theory. We will also be discussing how to iden- or archives in the outer skin layers. We have developed a tify these problems and engage with industry to increase model structure that explores the initial build up of the the impact of your work and what to expect once you start drug reservoir in the skin and its effect on subsequent drug on them. A recent case study, utilizing mathematical phar- extraction across the skin. The model system involves sys- macology for dose selection of a luminally restricted com- tems of coupled PDEs, the parameters of which depend on pound, will used to demonstrate the broader impact using the physicochemical properties of the drug. This talk will: this approach can have outside of the original problem. We motivate the choice of model formulation; justify the pa- DS17 Abstracts 159

rameter ranges in terms of properties including lipophilic- by multiple motors is the attachment and detachment rates ity, molecular weight and binding; and explore how physic- of molecular motors from a microtubule. These rates are ochemical properties affect the model outcomes. It will usually taken as given constants, but the attachment rate conclude by discussing implications of the model outcomes is clearly affected by the local spatial environment and the and, in response to that, future work. detachment rate is affected by the force experienced by the motor through its tethering to the cargo. I will report Jane White on some ongoing efforts to characterize these attachment University of Bath and detachment processes through more spatially explicit England models. Particular attention will be given to relating the [email protected] reattachment of a motor to a microtubule to a first passage time problem for the detached motor. Jennifer Jones Department of Mathematical Sciences Peter R. Kramer [email protected] Rensselaer Polytechnic Institute Department of Mathematical Sciences Begona Delgado-Charro [email protected] Department of Pharmacy and Pharmacology [email protected] Abhishek Choudhary Rensselaer Polytechnic Institute [email protected] MS23 Formation of Pseudo-cleavage Furrow during Cell MS23 Polarization in C. Elegans TheParadoxof Codependence Among Antagonistic Intracellular polarization, where a cell specifies a spatial Motors in Intracellular Transport axis by segregation of specific factors, is a fundamental biological process. In the early embryo of the nematode Transport in neurons is intrinsically bidirectional, with worm C. elegans, polarization is often accompanied by each movement modality carried out by molecular motors deformations of the cortex. It has been suggested that in either the kinesin (anterograde) or the dynein (retro- the eggshell surrounding the early embryo plays a role in grade) families. Because all motors are present at a given polarization although its function is not understood. In time there must be competition and/or cooperation among this work we develop a mathematical model which couples motors that simultaneously bind a single vesicle to nearby reaction-diffusion equations to a phase field model of the microtubules. The prevailing tug-of-war model captures cell cortex and incorporates the eggshell to monitor cor- this dynamic, but fails to account for a recently recognized tical invaginations during the early development of the C. phenomenon: that in many situations, disabling one fam- elegans embryo. We investigate the potential rigidity effect ily of motors somehow inhibits the performance of motors of the geometric constraint imposed by the presence and that are working in the opposite direction. In this talk we size of the eggshell on polarization dynamics. Our model will survey a few proposed mechanisms that may account suggests that the geometric constraint of the eggshell is es- for this behavior and will look at recent work that focuses sential for proper polarization and the size of the eggshell on a potential role played by the helper protein dynactin. also affects the dynamics of the polarization. Therefore we conclude that geometric constraint on a cell might affect Scott McKinley the dynamics of a biochemical processes. Tulane University [email protected] Betul Aras Ohio State University MS23 [email protected] Combining Computational Fluid Dynamics and Yongcheng Zhou Electron Tomography to Study the Mechanics of Department of Mathematics Kinetochore Microtubules Colorado State University The accurate segregation of chromosomes, and subsequent [email protected] cell division, in Eukaryotic cells is achieved by the inter- actions of an assembly of microtubules (MTs) and motor- Adriana Dawes proteins, known as the mitotic spindle. We use a com- Department of Mathematics/Department of Molecular bination of our computational platform for simulating cy- Genetics toskeletal assemblies and our structural data from high- Ohio State University resolution electron tomography of the mitotic spindle, to [email protected] study the kinetics and mechanics of MTs in the spindle, and their interactions with chromosomes during chromo- Ching-Shan Chou some segregation in the first cell division in C.elegans em- Department of Mathematics bryo. We focus on kinetochore MTs, or KMTs, which have Ohio State University one end attached to a chromosome. KMTs are thought [email protected] to be a key mechanical component in chromosome seg- regation. Using exploratory simulations of MT growth, bending, hydrodynamic interactions, and attachment to MS23 chromosomes, we propose a mechanical model for KMT- Spatial Modeling of Motor Attachment and De- chromosome interactions that reproduces observed KMT tachment length and shape distributions from electron tomography. We find that including detailed hydrodynamic interactions One ingredient of several models for the transport of cargo between KMTs is essential for agreement with the experi- 160 DS17 Abstracts

mental observations. this must be exploited to make constrained controller de- sign feasible and tractable. We introduce a new ”system Ehssan Nazockdast level” (SL) approach involving three complementary SL Courant Institute of Mathematical Sciences elements. System Level Parameterizations (SLPs) gener- New York University alize state space and Youla parameterizations of all sta- [email protected] bilizing controllers and the responses they achieve, and combine with System Level Constraints (SLCs) to param- Sebastian F¨urthauer eterize the largest known class of constrained stabilizing Flatiron Institute, Center for Computational Biology controllers that admit a convex characterization, gener- [email protected] alizing quadratic invariance (QI). We further provide a catalog of useful SLCs, most importantly including spar- Stefanie Redemann, Thomas M¨uller-Reichert sity, delay, and locality constraints on both communica- Experimental Centre, Medical Faculty Carl Gustav Carus, tion and computing internal to the controller, and external Technische Universit¨at Dresden, 01307 Dresden, Germany system performance. The resulting System Level Synthe- [email protected], sis (SLS) problems that arise define the broadest known [email protected] class of constrained optimal control problems that can be solved using convex programming. An example illustrates how this system level approach can systematically explore Michael Shelley tradeoffs in controller performance, robustness, and syn- Group Leader in Biophysical Modeling Group, thesis/implementation complexity. Center for Computational Biology, Simons Foundation [email protected] Nikolai Matni California Institute of Technology [email protected] MS24 Optimal Control Re-Examined Using the Geomet- Yuh-Shyang Wang, John Doyle ric Methods of Dynamical Systems Caltech Department of Control and Dynamical Systems In this talk, we look at optimal control from the perspective [email protected], [email protected] of dynamical systems, and demonstrate how this geometric understanding may help us in the optimal design of con- trollers. We start with analyzing the stable and unstable MS24 manifolds of the Hamilton equation arising from the Pon- tryagin Maximum Principle (PMP) in optimal control. We Automata Theory and Dynamic Programming demonstrate through numerical simulations how the tra- jectories of the Hamilton boundary value problem (BVP) We investigate the synthesis of optimal controllers for behave with different time horizon, control constraints, and continuous-time and continuous-state systems under be- end-time state conditions. Due to difficulties in solving the havioral specifications. The specification is expressed as a BVP in applications, optimal controller designs are often deterministic, finite automaton (the specification automa- carried out using model predictive control (MPC). We an- ton) with transition costs, and the optimal system behav- alyze the closed-loop systems arising from MPC and from ior is captured by a cost function that is integrated over PMP through the example of an optimal adaptive cruise time. We construct a dynamic programming problem over controller, where we project the trajectories of the Hamil- the product of the underlying continuous-time, continuous- ton equations into the phase plane and compare them with state system and the discrete specification automaton. To the trajectories arising from MPC. With a geometrical un- solve this dynamic program, we propose controller synthe- derstanding of the optimal control, we design an optimal sis algorithms based on Approximate Dynamic Program- connected cruise controller using MPC, where the equipped ming (ADP) for both linear and nonlinear systems under vehicle exploits motion information receives from multiple temporal logic constraints. We argue that ADP allows vehicles ahead via vehicle-to-vehicle communication. We treating the synthesis problem directly, without forming consider state and control constraints of the vehicle and expensive discrete abstractions. We show that, for linear use state predictors to compensate for the communication systems under co-safe temporal logic constraints, the ADP delay in the controller. solution reduces to a single semidefinite program. Further- more, we will also show how to extend these techniques Jin Ge to perform safe data-driven controller learning, and learn- University of Michigan, Ann Arbor ing by expert demonstrations—thereby incorporating data [email protected] and side information in a provably safe way. Ivan Papusha Gabor Orosz University of Texas University of Michigan, Ann Arbor [email protected] Department of Mechanical Engineering [email protected] Jie Fu Worcester Polytechnic Institute MS24 [email protected] A System-Level Approach to Controller Synthesis Min Wen Biological and advanced cyberphysical control systems of- University of Pennsylvania ten have limited, sparse, uncertain, and distributed com- [email protected] munication and computing in addition to sensing and ac- tuation. Fortunately, the corresponding plants and per- Ufuk Topcu formance requirements are also sparse and structured, and University of Texas DS17 Abstracts 161

[email protected] values of α.

Richard Murray Jonathan C. Jaquette Control and Dynamical Systems Rutgers University California Institute of Technology [email protected] [email protected] Jean-Philippe Lessard Universit´eLaval MS24 [email protected] Learning Model Predictive Control for Iterative Konstantin Mischaikow Tasks Department of Mathematics Rutgers, The State University of New Jersey Learning Model Predictive Control (LMPC) is a control [email protected] technique for system performing iterative tasks. In LMPC, the data collected from each task execution are exploited to improve the system performance. A safe set and a terminal Jan Bouwe Van Den Berg cost function are used to guarantee recursive feasibility and VU University Amsterdam improving performance for the closed-loop system. In this Department of Mathematics talk, we discuss the LMPC design, how to construct the [email protected] safe set and the terminal cost from data. We show that the proposed LMPC converges to a local optimal trajectory. MS25 The effectiveness of the LMPC will be demonstrated by experimental results on a self-driving race car. Rigorous Continuation of Periodic Orbits and Val- idation of Hopf Bifurcations Ugo Rosolia, Francesco Borrelli Periodic solutions to ODEs are hard to determine by direct University of California, Berkeley calculation but are relatively easy to compute numerically. [email protected], [email protected] By means of a computer, it is also possible to continue a branch of periodic solutions in case of parameter depen- dency of the vector field. In this talk, the mathematically MS25 rigorous validation of such ”numerical” branches of peri- Algorithms for Processing the Labeled Conley In- odic solutions will be presented. The method relies on dex the application of radii polynomials to polynomial vector fields of any order. An interesting feature of the presented Conley index theory has proven to be a powerful tool for method is the generality of the approach with respect to automating the rigorous exploration of dynamical systems. dimensionality: the presented bounds do not suffer greatly When searching for highly complicated dynamics, however, in higher dimensions. Furthermore, ongoing research on the Conley index may also become highly complicated, and rigorous validation of Hopf bifurcations will be presented, difficult to interpret or use in rigorous arguments. I will since Hopf bifurcations are a common source of periodic present an automated approach to processing Conley in- solutions. The first goal of this research is to validate, in dex information for discrete-time dynamical systems, joint the same general setting of polynomial vector field of any work with Sarah Day and Rodrigo Trevi˜no. This approach order and dimensionality, the Hopf bifurcation point. The produces a topologically semi-conjugate symbolic dynam- second goal is to develop techniques for the validated con- ical system whose topological entropy serves as a lower tinuation of branches of cycles starting at a validated Hopf bound for that of the system under study. I will describe bifurcation point. several applications and modifications of this approach, culminating in recent work with Sarah Day which uses sofic Elena Queirolo shifts to in some sense capture as much information from Vrije Universiteit Amsterdam the index as possible. [email protected]

Rafael Frongillo MS25 University of Colorado, Boulder [email protected] Validated Saddle-node Bifurcations and Applica- tions to Lattice Dynamical Systems

The use of rigorous verification methods is a powerful tool MS25 which permits progress in the analysis of dynamical pro- Advances on Wright’s Conjecture: Counting and cesses that is not possible using purely analytical tech- Discounting Periodic Orbits in a Delay Differential niques. In this talk we describe a set of tools for branch val- Equation idation, which allows for the rigorous verification of branch behavior, bifurcation, and solution index on branches gen- Despite the deceptively simple appearance of Wright’s erated through a saddle-node bifurcation. While the pre- equation: y(t)=−αy(t − 1)[1 + y(t)], a complete un- sented methodology can be applied in a variety of settings, derstanding of its global dynamics remains out of grasp. we illustrate the use of these tools in the context of mate- A long-standing conjecture claims that Wright’s equation rials science. In particular, lattice models have been pro- does not have any periodic orbits when α<π/2andthere posed as a more realistic reflection of the behavior of ma- is a unique slowly oscillating periodic orbit when α>π/2. terials than traditional continuum models, since they can In this talk I will present recent progress on this conjecture. account for phenomena such as pinning. However, in the In particular I will discuss applications of rigorous numer- context of bifurcation theory, questions about lattice dy- ics to study the Hopf bifurcation at α = π/2, as well as namical systems are significantly harder to answer than for the asymptotic stability of periodic orbits for mesoscopic a continuum model. We show that computer-assisted proof 162 DS17 Abstracts

techniques can be used to answer some of these questions, Edriss S. Titi and apply these tools to the discrete Allen-Cahn equation. Texas A&M University This provides results on the existence of branches of mosaic Weizmann Institute of Science solutions and their robustness as it relates to grain size. We [email protected], [email protected] also demonstrate that there are situations in which classi- cal continuation methods can fail to identify the correct branching behavior. MS26 Synchronization of Chaotic Dynamical Systems Us- Evelyn Sander ing Time-Averaged Partial Observations of the George Mason University Phase Space [email protected] We study the synchronization of chaotic systems when the Thomas Wanner coupling between them contains both time averages and George Mason University stochastic noise. Our model dynamics are given by the Department of Mathematical Sciences Lorenz equations which are a system of three ordinary dif- [email protected] ferential equations in the variables X, Y and Z.Ourtheo- retical results show that coupling two copies of the Lorenz equations using a feedback control which consists of time MS26 averages of the X variable leads to exact synchronization Data Assimilation Algorithms for Geophysical provided the time-averaging window is known and suffi- Models and Charney’s Conjecture ciently small. In the presence of noise the convergence is to within a factor of the variance of the noise. We also We propose data assimilation algorithms for the B´enard consider the case when the time-averaging window is not convection problem and the 3D Planetary Geostrophic known and show that it is possible to tune the feedback model of ocean, that employ coarse spatial mesh obser- control to recover the size of the time-averaging window. vations of the temperature alone. We show that the al- Further numerical computations show that synchronization gorithms converge to the reference solution for both the is more accurate and occurs under much less stringent con- B´enard problem in porous media and the 3D Planetary ditions than our theory requires. Geostrophic model cases. On the other hand, recent nu- merical results indicate that the algorithm doesn’t converge Jordan Blocher for the B´enard model. This justifies an earlier conjecture University of Nevada Reno of Charney which states that temperature history of the [email protected] atmosphere, for certain simple atmospheric models, deter- mines all other state variables. Vincent R. Martinez Tulane University Aseel Farhat [email protected] Department of Mathematics University of Virginia Eric Olson [email protected] UNR [email protected] Evelyn Lunasin United States Naval Accademy [email protected] MS26 Determining the Global Dynamics of the 2D Edriss S. Titi Navier-Stokes Equations by 1D ODE Texas A&M University Weizmann Institute of Science One of the main characteristics of infinite-dimensional dis- [email protected], [email protected] sipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long- time dynamics is determined by finitely many parameters MS26 – finite number of determining modes, nodes, volume el- Numerical Approximation of a Data Assimilation ements and other determining interpolants. In this talk Algorithm by a Post-processing Galerkin Method I will show how to explore this finite-dimensional feature of the long-time behavior of infinite-dimensional dissipa- We consider a data assimilation algorithm for recover- tive systems to design finite-dimensional feedback control ing the exact value of a reference solution of the two- for stabilizing their solutions. Notably, it is observed that dimensional Navier-Stokes equations, by using continuous this very same approach can be implemented for designing in time and coarse spatial observations. The algorithm is data assimilation algorithms of weather prediction based given by an approximate model which incorporates the ob- on discrete measurements. In addition, I will also show servations through a feedback control (nudging) term. We that the long-time dynamics of the Navier-Stokes equations obtain an analytical uniform in time estimate of the er- can be imbedded in an infinite-dimensional dynamical sys- ror committed when numerically solving this approximate tem that is induced by an ordinary differential equations, model by using a post-processing technique for the spectral named determining form, which is governed by a globally Galerkin method, inspired by the theory of approximate in- Lipschitz vector field. Remarkably, as a result of this ma- ertial manifolds. This is a joint work with C. Foias and E. chinery I will eventually show that the global dynamics S. Titi. of the Navier-Stokes equations is be determining by only one parameter that is governed by an ODE. The Navier- Cecilia F. Mondaini, Ciprian Foias Stokes equations are used as an illustrative example, and Texas A&M University all the above mentioned results equally hold to other dissi- [email protected], fi[email protected] pative evolution PDEs, in particular to various dissipative DS17 Abstracts 163

reaction-diffusion systems and geophysical models. bines correlations among edges with distance-dependent connection probability. This model is an extension of Edriss S. Titi the SONET model [Zhao et al, 2011] that adds spatial Texas A&M University structure by allowing heterogeneous first order connectivity Weizmann Institute of Science statistics. The second order connectivity statistics, i.e., the [email protected], [email protected] covariances among the edges that determine the frequency of convergent, divergent, chain, and reciprocal connections, Ciprian Foias remain homogeneous across the network. We use this Texas A&M University model to generate ring models with period spatial struc- fi[email protected] ture that is combined with the SONET microstructure. We analyze these networks to predict how the SONET param- Michael S. Jolly eters as well as the global parameters predict synchrony Indiana University across the network. Department of Mathematics [email protected] Duane Nykamp School of Mathematics Dan Lithio University of Minnesota Indiana University [email protected] [email protected] Samantha Fuller University of Minnesota MS27 [email protected] Independent Noise Synchronizing Population Rhythms of Networks of Pulse-Coupled Oscillators MS27 The coherent dynamics of coupled oscillators have been Granger Causality in Sparse Neuronal Network studied extensively. In the neuronal networks of the brain such coherence is observed to generate a rich variety of Granger causality (GC) is a mathematical tool to reveal rhythms, among them the well-studied γ-rhythm. It has linear dependencies between random variables. It has been been suggested that coherence across different brain areas widely used in neuroscience to examine the signals to show may play an important role in the communication between the effective connectivity between neurons in a network. these areas. Motivated by the observation of the coexis- Recent computational research has shown that GC can tence of different γ-rhythms in the olfactory system and of successfully reconstruct neuronal network provided either the coherence of γ-rhythms across different brain areas, we voltage trace or spike train of all the neurons. But it is gen- consider the interaction between networks of pulse-coupled erally unrealistic experimentally. In this talk, we will show inhibitory neurons, with each network supporting its own that for a sparse neuronal network, only partial recordings γ-rhythm. We focus on the effect of noise on the synchro- are needed to recover the local network structure, and the nization of coupled networks. Surprisingly, we find that possible cause of error is also analyzed. different γ-rhythms, each corresponding to the coherent activity of neurons in one of the coupled networks, can be- Yangyang Xiao come synchronized by noise, even if that noise is completely NewYorkUniversityAbuDhabi uncorrelated between different neurons. By reducing the [email protected] coherent dynamics of the two networks to an iterated map we show that key for the enhanced synchronization is the noise-induced phase heterogeneity, which allows the faster network to suppress the spiking of a fraction of the neu- MS27 rons in the slower network. The resulting synchroniza- A Probability Polling State of Neuronal Systems tion can increase the learning speed of neurons that read Underlying Maximum Entropy Coding Principle the coupled networks via synapses exhibiting spike-timing- dependent plasticity. Supported by NSF-CMMI 1435358 How to extract information from exponentially growing recorded neuronal data is a great scientific challenge. In recent experiments, it has been found that the second or- John Meng,XizeXu der maximum entropy model, by using only firing rates Northwestern University and second order correlations of neurons as constraints, [email protected], can well capture the observed distribution of neuronal fir- [email protected] ing patterns in many neuronal networks, thus, conferring its great advantage in that the degree of complexity in the Hermann Riecke analysis of neuronal activity data reduces drastically from Applied Mathematics O(2n)toO(n2), where n is the number of neurons under Northwestern University consideration. In this talk, we address the question of what [email protected] kind of dynamical states of neuronal networks allows the network to possess a coding scheme dictated by the Maxi- mum Entropy Principle (MEP). For asynchronous neuronal MS27 networks, when considering the probability increment of a Distance-Dependent Edge-Correlations and Syn- neuron spiking induced by other neurons, we found a prob- chrony in Neuronal Networks ability polling (p-polling) state that underlies the success of the second order maximum entropy model. We show To examine how network microstructure and global struc- that this p-polling state can arise in vitro and in vivo. Our ture network patterns can influence synchronization in a theoretical analysis of the p-polling state and its relation- neuronal network, we developed a network model that com- ship to MEP provides a new perspective to the information 164 DS17 Abstracts

coding of neuronal network dynamics in the brain. Center for Applied Mathematics Cornell University Douglas Zhou [email protected] Shanghai Jiao Tong University [email protected] MS28 Zhiqin John Xu Interplay of Coupling and Common Noise at the Abu Dhabi and Courant Institute Transition to Synchrony in Oscillator Populations New York University [email protected] There are two ways to synchronize oscillators: by coupling and by common forcing, which can be pure noise. By virtue David Cai of the Ott-Antonsen ansatz for sine-coupled phase oscil- Courant Institute for Mathematical Sciences, NYU lators, we obtain analytically tractable equations for the Shanghai Jiao-Tong University case where both coupling and common noise are present. [email protected] While noise always tends to synchronize the oscillators, the repulsive coupling can act against synchrony, and we focus on this nontrivial situation. For identical oscillators, the MS28 fully synchronous state remains stable for small repulsive coupling; moreover it is an absorbing state which always Connecting Hyperbolic Geometry to Kuramoto wins over the asynchronous regime. For oscillators with a Oscillator Systems; New Classes of Gradient Phase distribution of natural frequencies, we report on a counter- Models and Completely Integrable Dynamics intuitive effect of dispersion (instead of usual convergence) Kuramoto oscillator networks have the special property of the oscillators frequencies at synchrony; the latter effect that their time evolution is constrained to lie on 3D or- disappears if noise vanishes. bits of the M¨obius group acting on the N-fold torus T N − Arkady Pikovsky which explains the N 3 constants of motion discovered Department of Physics and Astronomy by Watanabe and Strogatz. The dynamics for phase mod- N−1 University of Potsdam, Germany els can be further reduced to 2D invariant sets in T [email protected] which have a natural geometry equivalent to the unit disk D with hyperbolic metric. We show that the classic Ku- Denis S. Goldobin ramoto model with order parameter Z1 (the first moment of the oscillator configuration) is a gradient flow in this Institute for Physics, University of Potsdam metric with a unique fixed point on each generic 2D invari- Potsdam, Germany ant set, corresponding to the hyperbolic barycenter of an [email protected] oscillator configuration. This gradient property makes the dynamics especially easy to analyze. We exhibit several Michael Rosenblum new families of Kuramoto oscillator models which reduce Potsdam University to gradient flows in this metric; some of these have a richer Department of Physics and Astronomy fixed point structure including non-hyperbolic fixed points [email protected] associated with fixed point bifurcations. Anastasyia Pimenova Jan Engelbrecht Institute of Continuous Media Mechanics, Ural Brahch of Department of Physics, Boston College RAS [email protected] Perm, Russia [email protected] MS28 Frequency Spiral MS28 Dynamics of Weakly Inhomogeneous Oscillator We study the dynamics of coupled phase oscillators on a Populations: Perturbation Theory on Top of two-dimensional Kuramoto lattice with periodic boundary WatanabeStrogatz Integrability conditions. For coupling strengths just below the transi- tion to global phase-locking, we find localized spatiotem- As has been shown by Watanabe and Strogatz (WS) (1993, poral patterns that we call frequency spirals. These pat- Phys. Rev. Lett. 70, 2391), a population of identical phase terns cannot be seen under time averaging; they become oscillators, sine-coupled to a common field, is a partially visible only when we examine the spatial variation of the integrable system: for any ensemble size its dynamics re- oscillators’instantaneous frequencies, where they manifest duce to equations for three collective variables. Here we themselves as two-armed rotating spirals. In the more fa- develop a pertur- bation approach for weakly nonidentical miliar phase representation, they appear as wobbly peri- ensembles. We calculate corrections to the WS dynamics odic patterns surrounding a phase vortex.Unlike the sta- for two types of perturbations: those due to a distribution tionary phase vortices seen in magnetic spin systems, or of natural frequencies and of forcing terms, and those due the rotating spiral waves seen in reaction-diffusion sys- to small white noise. We demonstrate that in both cases, tems, frequency spirals librate: the phases of the oscilla- the complex mean field for which the dynamical equations tors surrounding the central vortex move forward and then are written is close to the Kuramoto order parameter, up backward, executing a periodic motion with zero winding to the leading order in the perturbation. This supports the number. We construct the simplest frequency spiral and validity of the dynamical reduction suggested by Ott and characterize its properties using analytical and numerical Antonsen (2008, Chaos 18, 037113) for weakly inhomoge- methods. Simulations show that frequency spirals in large neous populations. lattices behave much like this simple prototype. Vladimir Vlasov Bertrand Ottino-Loffler Istituto Italiano di Tecnologia DS17 Abstracts 165

[email protected] tion equations fail. As a side product we verify local ex- istence and uniqueness of the stochastic Swift-Hohenberg Michael Rosenblum and the complex Ginzburg Landau equation on the whole Potsdam University real line in weighted spaces that allow for growth at infin- Department of Physics and Astronomy ity of solutions. Moreover we use energy estimates to show [email protected] that solutions of the Ginzburg-Landau equation are Hlder- continuous, which just gives enough regularity to proceed Arkady Pikovsky with the error estimates. Department of Physics and Astronomy University of Potsdam, Germany Dirk Bl¨omker [email protected] Universitat Augsburg Germany [email protected] MS29 The Motion of a Droplet for the Stochastic Mass- Guido Schneider conserving Allen-Cahn Equation University of Stuttgart Department of Mathematics We study the stochastic mass-conserving Allen-Cahn equa- [email protected] tion posed on a smoothly bounded domain of R2 with ad- ditive, spatially smooth, space-time noise. This equation Luigi A. Bianchi describes the stochastic motion of a small almost semicircu- TU Berlin lar droplet attached to domain’s boundary and moving to- [email protected] wards a point of locally maximum curvature. We apply Itˆo calculus to derive the stochastic dynamics of the center of the droplet by utilizing the approximately invariant man- MS29 ifold introduced by earlier by Alikakos, Chen and Fusco for the deterministic problem. In the stochastic case de- Stochastic Dynamics: Data-Driven Information pending on the scaling, the motion is driven by the change Extraction via Effective Reduction in the curvature of the boundary and the stochastic forc- ing. Moreover, under the assumption of a sufficiently small Dynamical systems arising in engineering and science of- noise strength, we establish stochastic stability of a neigh- ten evolve under random fluctuations and multiple scales. borhood of the manifold of boundary droplet states in the The random fluctuations may be Gaussian or non-Gaussian L2-andH 1-norms, which means that with overwhelming described by Brownian motion or α-stable Levy motion, probability the solution stays close to the manifold for very respectively. Stochastic dynamical systems with α-stable long time-scales. Levy motions have attracted a lot of attention recently. The non-Gaussianity index α is a significant indicator for Peter W. Bates various dynamical behaviors. The speaker will present re- Michigan State University cent work on data assimilation and extraction of dynam- Department of Mathematics ical behaviors (e.g., transitions, mean residence time, es- [email protected] cape probability), via effective approximations, such as slow manifold, homogenization, and averaging, for stochas- Dimitra Antonopoulou tic dynamical systems with non-Gaussian Levy noise. University of Chester [email protected] Jinqiao Duan Illinois Institute of Technology Dirk Bl¨omker Department of Applied Mathematics Universitat Augsburg [email protected] Germany [email protected] MS29 Georgia Karali Couple Periodic-patches of a Stochastic Microscale University or Crete to Predict Emergent Macroscale Dynamics [email protected] Molecular simulations, while in principle deterministic on the microscale, are chaotic and so effectively stochastic on MS29 meso/macro-scales. Many other microscale simulations are Stochastic Modulation Equations on Unbounded directly stochastic. Furthermore, as for many microscale Domains systems, molecular simulations are much the easiest to code with periodic boundary conditions in space. We seek to The reduction of complicated dynamics on large domains distribute many small periodic-patches of such stochastic via modulation equations is a well known tool to reduce microscale simulators over a macroscale grid. Then the re- the complexity of the model close to a change of stability. search aim is find a coupling between the small patches in While for deterministic models this is a well established order to effectively and cheaply simulate the macroscale dy- theory, there are hardly any results for stochastic models namics. A stochastic slow manifold analysis of the system on unbounded domains. In the talk we consider the approx- of stochastic periodic-patches indicates that a proportional imation via modulation equations for nonlinear SPDEs on control applied within action regions of each patch can unbounded domains with additive space time white noise. be effective. The analysis suggest Lagrange interpolation As a first example we study the stochastic Swift-Hohenberg of core-patch averages provides a useful control. A pilot equation on the whole real line, where due to the weak reg- implementation of the coupling to a small triply-periodic ularity of solutions the all standard methods for modula- patch of atomistic simulation indicates that the scheme can 166 DS17 Abstracts

usefully predict the emergent macroscale. tive synchronization using the Lorenz chaotic oscillator is shown. The potential of this synchronization approach is il- Anthony J. Roberts lustrated for the bistatic radar synchronization. The setup University of Adelaide includes a Lorenz oscillator at the transmitter and another [email protected] at the receiver. Due to the system and space propaga- tion losses, the response oscillator must accept the scaled version of the transmitted chaotic waveform as the driver MS30 variable. Since the scaling factor is not known a priori, Nonlinear Dynamics in Optimal Communication we propose the generalized projective synchronization. A Waveforms circuit implementation of this approach is presented where the transmitted waveform is replicated at the receiver. The Standard methods of communication theory are used to proposed synchronization is found to be very robust for derive the optimal communications waveform correspond- changes in the control parameters of the oscillators. The ing to a simple infinite impulse response (IIR) matched application of this approach is illustrated for the bistatic filter. Surprisingly, we find that the resulting waveform radar system where synchronization is necessary to obtain is chaotic. Specifically we presume a simple linear RLC the range-Doppler information of the target. The short matched filter and derive a fixed basis function that max- time cross-correlation between the transmitted and the imize the receiver signal-to-noise performance. A commu- synchronized waveforms is very high and the sidelobes are nication waveform is constructed by linear superposition of below -15 Db. This indicates the advantages of chaotic syn- the fixed basis functions with binary weight at each sym- chronization compared to other traditional synchronization bol time. It is shown that the resulting waveform has an techniques utilized for bistatic radars. embedded shift, implying the waveform is chaotic. Further- more, the optimal waveforms are naturally generated by a Chandra S. Pappu piecewise linear manifold oscillator that is easily realized in Union College a hybrid analog-digital electronic circuit. We extrapolate [email protected] from this result and argue that the optimal communication waveform for any stable IIR filter is similarly chaotic. If Benjamin C. Flores true, this conjecture implies nonlinear dynamics and chaos University of Texas at El Paso are essential to a full understanding of modern communi- Electrical and Computer Engineering cation theory. bfl[email protected] Ned J. Corron, Jonathan Blakely U.S. Army AMRDEC [email protected], MS30 [email protected] A Demonstration Wireless Exact Solvable Chaotic Communication System

MS30 A wireless communication system based on an exact solv- Recent Discoveries in Solvable Chaotic Systems able chaotic equation has been demonstrated. The system consists of a data input section, a chaos oscillator con- Exact solutions are rare in the study of chaotic dynam- troller, an exact solvable chaotic oscillator, an AM wireless ics, but some examples do exist. Recently, analytic chaotic communication system, a matched filter and a data out- solutions have been found for a class of hybrid dynamical put section. ST microcontrollers are used to input data systems, which has allowed for new methods of exploitation (data input section) to and output data (data output sec- of chaos. The key characteristic of these chaotic solutions tion) from the communication system. The exact solvable is an analytic expression for the waveform that can be writ- chaotic oscillator has a fundamental frequency of 18 kHz. ten as an infinite sum of copies of a single basis function, It produces a baseband chaotic signal using a single tran- enabling the use of well established, conventional commu- sistor sinusoidal oscillator circuit with a signum function nication theory, and signal processing. Here we describe based nonlinearity. The oscillator is controlled into two a solvable 2nd order chaotic differential equation with a distinct orbits, representing 1s and 0s, using proportional circuit implementation that can be used for acoustic de- feedback control. A standard AM transmitter up converts tection and ranging. Specifically, we highlight the benefits the chaos modulated signal onto a 2.3 GHz carrier for wire- of using solvable chaos in application such as matched fil- less transmission to a receiver that down converts it back tering and compressibility for efficient waveform storage. to baseband. A matched filter specifically designed for the Furthermore, we detail more recent discoveries in manip- exact solvable chaotic oscillator is then used to recover the ulating hybrid systems that bring rise to new phenomena. information from the received signal. In particular, we describe how solvability is preserved even in the presence of complex clock patterns, the dynamics of Keaton Rhea, Andrew Muscha, Remington Harrison, hybrid systems with a multi-level discrete state, as well as Frank Werner, Robert Dean a generalization of the discrete state rule which allows for Auburn University a period doubling route to chaos. [email protected], [email protected], [email protected], Marko S. Milosaviljevic [email protected], [email protected] US Army RDECOM [email protected] Darren Boyd NASA-MSFC MS30 [email protected] Circuit Implementation of Generalized Projective Synchronization for Bistatic Radar Applications MS31 An electronic implementation of the generalized projec- Nonlinear Dynamics of Vehicle Networks with DS17 Abstracts 167

Long Range Vehicle-to-vehicle Communication [email protected]

The development and deployment of connected and au- MS31 tomated vehicles is currently capturing the interest of academia and industry. These vehicles must be able to A Delay-Based Controller Design Using Lambert interact in a complex environment of vehicular traffic con- W function in a Multi-Agent Consensus Dynamics sisting of vehicles of different levels of autonomy including A class of linear time invariant consensus dynamics is con- conventional human-driven vehicles. Such an environment sidered in this study, where agents’ decision making is af- is full of nonlinearities, time delays and uncertainties. We fected by a time delay. Here we focus on achieving fast use bifurcation analysis to characterize large scale patterns consensus by means of pole placement. Specifically, we pro- in traffic such as stop-and-go jams, and then use this insight pose a delay-based controller for each agent, whereby this to design control algorithms for connected and autonomous controller structure enables a decomposition of the system vehicles. We demonstrate that by inserting these vehicles characteristic equation, which is then effectively analyzed into the flow of conventional vehicles allows us to change with established tools based on Lambert W function. We the large scale traffic dynamics even for low penetrations show that the delay-based controller can easily be tuned of connected and autonomous vehicles. to place system rightmost roots to achieve fast consensus for a given delay in the system, and present case studies Sergei S. Avedisov where the delay is time-varying and how the delay-based University of Michigan, Ann Arbor controller performs on the consensus dynamics. [email protected] Adrian Ramirez Northeastern University Gabor Orosz Boston, MA University of Michigan, Ann Arbor [email protected] Department of Mechanical Engineering [email protected] Min Hyong Koh Northeastern University [email protected] MS31

Emergent Task Differentiation on Network Filters Rifat Sipahi Department of Mechanical and Industrial Engineering Northeastern University In this talk, we analyze the emergence of task differen- [email protected] tiation in a model complex system, characterized by an absence of hierarchical control, yet able to exhibit coor- dinated behavior and collective function. The analysis fo- MS31 cuses on linear network filters, i.e., networks of coupled lin- Stability Charts for Continua Subjected to Delayed ear oscillators with a differentiated steady-state response Feedback to exogenous harmonic excitation. It demonstrates how an optimal allocation of excitation sensitivities across the The lecture provides a summary of engineering problems network nodes in a condition of resonance may be con- where the underlying mechanical model is a controlled con- structed either using global information about the network tinuum and the corresponding mathematical model is a topology and spectral properties or, alternatively, through partial differential equation with delayed boundary con- the iterated dynamics of a nonlinear, nonsmooth learning ditions. The examples include the stabilization of rockets paradigm that only relies on local information within the with delayed follower forces, material forming processes like network. Explicit conditions on the topology and desired boring or deep drilling, and electroacoustics. The detailed resonant mode shape are derived to guarantee local asymp- derivation of the stability chart of the electroacoustic prob- totic stability of fixed points of the learning dynamics. The lem is presented. The characteristic roots of the linear analysis also demonstrates the possibly semi-stable nature system do not depend continuously on the delay param- of the fixed point with all zero excitation sensitivities, a eters. The physical interpretation of these mathematical condition of system collapse that can be reached from an results require special considerations. The possible bifur- open set of initial conditions, but that is unstable under cations are also checked along the stability limits, and the the learning dynamics. Theoretical and numerical results vibration frequencies of the possible primary and secondary also show the existence of periodic responses, as well as of Hopf and double Hopf bifurcations are calculated. connecting dynamics between fixed points, resulting in re- current metastable behavior and noise-induced transitions Li Zhang along cycles of such connections. Nanjing University of Aeronautics and Astronautics [email protected] Mehdi Saghafi Gabor Stepan University of Illinois at Urbana-Champaign Department of Applied Mechanics mehdi.saghafi@gmail.com Budapest University of Technology and Economics [email protected] Harry Dankowicz University of Illinois at Urbana-Champaign Department of Mechanical Science and Engineering MS32 [email protected] Dynamics Near Cubic Homoclinic Tangencies in Symplectic Maps Whitney Tabor University of Connecticut We study the orbit behavior near cubic homoclinic tangen- 168 DS17 Abstracts

cies in two-dimensional symplectic maps. We distinguish historic behaviour occurs persistently for initial conditions two types of cubic homoclinic tangencies, and each type in a set with positive Lebesgue measure. The family ap- gives different first return maps derived to diverse conser- pears in the unfolding of a degenerate differential equation vative cubic H´enon maps with quite different bifurcation whose flow has an asymptotically stable heteroclinic cycle diagrams. In this way, we establish the structure of bifurca- involving two-dimensional connections of non-trivial peri- tions of periodic orbits in two parameter general unfoldings odic solutions. We show that the degenerate problem also generalizing to the conservative case the results previously has historic behaviour, since for an open set of initial con- obtained for the dissipative case. ditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the Marina Gonchenko centres of gravity of the periodic solutions and their Flo- University of Barcelona quet multipliers. This is a joint work with I. Labouriau [email protected] (University of Porto).

Sergey Gonchenko Alexandre A. Rodrigues N.I.Lobachevsky˜ Nizhny Novgorod University University of Porto [email protected] [email protected]

Ivan Ovsyannikov University of Bremen, Germany MS32 [email protected] Splitting of Separatrices at a Periodically Forced Hamiltonian-Hopf Bifurcation MS32 Mixed Dynamics in Planar Reversible Diffeomor- Abstract Not Available At Time Of Publication. phisms Arturo Viero Existence of open regions of structural instability was a University of Barcelona crucial discovery for Bifurcation Theory. In this sense, the [email protected] pioneering works of Newhouse [1970, 1979] were seminal: there exist open regions in the space of dynamical systems where systems with homoclinic tangencies are dense (in r MS33 the C -topology, r ≥ 2). A particular case where such do- mains exist in any neighbourhood around a planar diffeo- Noise-Induced Effects on Spontaneous and Spiral morphism having an homoclinic tangency. This tangency is Wave Activity of the Biopacemaker typically quadratic and constitutes a codimension-1 bifur- cation. From the 70s there have been important contribu- Spontaneous activity is the core of a normal heartbeat. The tions to extend the work of Newhouse: Shilnikov, Gavrilov, limitations in the natural hearts pacemaker can necessitate Palis, Viana, Duarte, Tedeschini- Lalli, Yorke, Gonchenko, the implantation of an electronic pacemaker. The biolog- Turaev, Romero, and many others. One of the main dif- ical pacemaker approach has been proposed as a possible ficulties studying such regions in the space of dy- namical alternative with the creation of a new multicellular pace- systems is its extreme richness. This means that to provide making substrate. Interaction between cell types is central a complete description of their dynamics becomes a very to the function of this new approach. The present project difficult target. From the works above it was suggested as focuses on studying the effect of current noise on sponta- one of the most important properties in these Newhouse neous activity (focal and reentrant dynamics) of multicel- regions the coexistence of infinitely many periodic orbits lular mixture of two different cell population, pacemaker of different types of stability (attracting, repelling, saddle and quiescent cardiac cells. Simulations were carried out and elliptic). This leads to the concept of the so-called in a 2D spatial cardiac model. Currents were represented Mixed dynamics. The aim of this work is to present some by the Luo-Rudy I ionic model and a bias current that was results and techniques ”`a la Shilnikov” proving mixed dy- introduced to induce spontaneous activity in pacemaker namics for reversible diffeos. It is based on papers with A. cells. Sensitivity to the current noise was simulated as a Delshams, S., V. and M. Gonchenko and O. Sten’kin. function of spontaneous activity of pacemaker cells for dif- ferent stochastically distributed pacemaker cells patterns. J. Tomas Lazaro The presence of noise lowered the minimum density of pace- Universitat Politecnica de Catalunya maker cells needed for focal spontaneous activity. The ef- [email protected] fects of noise on spatial-temporal dynamics are more im- portant with weak pacemaker cells. Sensitivity to reentrant MS32 activity highlights the synergetic interaction between pace- maker action potential morphology and the spatial pattern On Taken’s Last Problem: Times Averages for Het- distribution. The project will provide a better understand- eroclinic Attractors ing of noise interaction with initiation of spontaneous ac- In this talk, I will discuss some properties of a robust fam- tivity within the concept of biological pacemaker. ily of smooth ordinary differential equations exhibiting tan- gencies for a dense subset of parameters. We use this to Alireza Aghighi find dense subsets of parameter values such that the set Institute of Biomedical Engineering of solutions with historic behaviour contains an open set. Universit´edeMontr´eal This provides an affirmative answer to Taken’s Last Prob- [email protected] lem (F. Takens (2008) Nonlinearity, 21(3) T33–T36). A limited solution with historic behaviour is one for which Philippe Comtois the time averages do not converge as time goes to infin- Institute of Biomedical Engineering, ity. Takens’ problem asks for dynamical systems where Universite de Montreal DS17 Abstracts 169

[email protected] caused by fiber rotation.

Oliver Steinbock MS33 Department of Chemistry and Biochemistry Semianalytical Approach to Criteria for Ignition of Florida State University Excitation Waves [email protected]

We consider the problem of ignition of propagating waves Zhihui Zhang in one-dimensional bistable or excitable systems by an in- Florida State University stantaneous spatially extended stimulus. Earlier we pro- [email protected] posed a method (Idris and Biktashev, PRL, vol 101, 2008, 244101) for analytical description of the threshold con- ditions based on an approximation of the (center-)stable MS33 manifold of a certain critical solution. Here we general- Fractional-Order Voltage Dynamics Suppresses Al- ize this method to address a wider class of excitable sys- ternans and Promotes Spontaneous Activity in a tems, such as multicomponent reaction-diffusion systems Minimal Cardiomyocyte Model and systems with non-self-adjoint linearized operators, in- cluding systems with moving critical fronts and pulses. We Electrical activity in cardiomyocytes is typically modeled also explore an extension of this method from a linear to using an ideal parallel resistor-capacitor circuit. However, a quadratic approximation of the (center-)stable manifold, studies have suggested that the passive properties of cell resulting in some cases in a significant increase in accuracy. membranes may be more appropriately modeled with a The applicability of the approach is demonstrated on five non-ideal capacitor, in which the current-voltage relation- test problems ranging from archetypal examples such as ship is given by a fractional-order derivative. Fractional- the Zeldovich–Frank-Kamenetsky equation to near realis- order membrane potential dynamics introduces capacitive tic examples such as the Beeler-Reuter model of cardiac memory effects, i.e., dynamics are influenced by the prior excitation. While the method is analytical in nature, it is membrane potential history. We recently showed that recognised that essential ingredients of the theory can be fractional-order dynamics alter ionic currents and spik- calculated explicitly only in exceptional cases, so we also ing rates in neurons. Here, we investigate fractional- describe methods suitable for calculating these ingredients order dynamics in a cardiomyocyte model using the 3- numerically. variable Fenton-Karma model. Simulations, performed for fractional-orders between 0.5 and 1 and variable cycle Burhan Bezekci lengths, showed that the action potential duration (APD) University of Exeter, Exeter, UK was shortened as the fractional-order decreased, for all cy- [email protected] cle lengths. At short cycle lengths at which APD alternans was present in the first-order model, alternans was sup- Ibrahim Idris pressed. Finally, for small fractional-order, spontaneous Bayero University, Kano, Nigeria action potentials followed pacing. Short-term memory ef- [email protected] fects were represented by a hypothetical memory current, which was primarily outward for fractional-order closer to 1, shortening APD, and primarily inward for fractional- Radostin Simitev order closer to 0.5, generating spontaneous action poten- University of Glasgow, Glasgow, UK tials. Collectively, our results suggest the capacitive mem- [email protected] ory may play a role in both alternans formation and pace- making. Vadim N. Biktashev College of Engineering, Mathematics and Physical Tien Comlekoglu Sciences Virginia Commonwealth University University of Exeter [email protected] [email protected] Seth Weinberg Department of Biomedical Engineering MS33 Virginia Commonwealth University Local Heterogeneities in Cardiac Systems Suppress [email protected] Turbulence by Generating Multi-Armed Rotors

Ventricular fibrillation is an extremely dangerous cardiac MS34 arrhythmia that is linked to rotating waves of electric Neural Synchronization During Neocortical activity and chaotically moving vortex lines. These fil- Seizures aments can pin to insulating, cylindrical heterogeneities which swiftly become the new rotation backbone of the The question of the role of synchronization during seizures local wave field. For thin cylinders, the stabilized rota- has been intensely debated both by the clinical and non- tion is sufficiently fast to repel the free segments of the linear dynamics community. We have investigated this in turbulent filament tangle and annihilate them at the sys- focal seizures in the rat neocortex. The seizures are in- tem boundaries. The resulting global wave pattern is pe- duced by 4-aminopyridine, which blocks potassium chan- riodic and highly ordered. Our cardiac simulations show nels. Seizure recording is obtained using a local field po- that also thicker cylinders can establish analogous forms of tential recording as well as imaging with a voltage-sensitive tachycardia. This process occurs through the spontaneous dye in order to record voltage changes throughout the af- formation of pinned multi-armed vortices. The observed fected area. We find a significant increase in synchroniza- number of wave arms N depends on the cylinder radius tion during the seizure events, quantified using the stochas- and is associated to stability windows that for N =2,3 tic phase synchronization index. In agreement with the ex- partially overlap. We also discuss the effects of anisotropy perimental observations, the elimination of potassium con- 170 DS17 Abstracts

ductance in an array of simulated coupled neurons results mechanisms for partially synchronized states. In oscilla- in seizure-like mean field oscillations. More specifically, in tions close to Hopf bifurcation and nearly identical natural the computational model, seizure initiation occurs as a bi- frequencies, we describe the emergence of chimera states. furcation when the potassium conductance is varied as a The experiments point out the importance of low level of control parameter. Our results suggest that experimen- heterogeneities (e.g., surface conditions) and optimal level tally observed changes in field potential amplitude and fre- of coupling strength and time-scale as necessary compo- quency during the course of a seizure may be explained by nents for the realization of the chimera state. For sys- noise-induced transitions among multistable states. tems with large heterogeneities, a remnant chimera state is identified where the pattern is strongly affected by the Daisuke Takeshita presence of frequency clusters. For experimental conditions University of Helsinki where chimera states are not possible, we analyze the spa- daisuke.takeshita@helsinki.fi tially organized partially synchronized states as a function of underlying heterogeneities and network topologies. As Sonya Bahar prototype system, we consider three oscillators with super- Center for Neurodynamics and Dept. of Physics and imposed local and global coupling topologies. An approx- Astronomy imate formula is derived for the mixed local/global cou- University of Missouri at St. Louis pling topology for the critical coupling strength at which [email protected] full synchrony is achieved. The formula is verified with macrocells, in which the coupling topology is systemati- cally varied from local to global coupling. MS34 Engineering Reaction-Diffusion Networks with Istvan Z. Kiss Properties of Neural Tissue Department of Chemistry Saint Louis University We present an experimental system of networks of coupled [email protected] non-linear chemical reactors, which we theoretically model within a reaction-diffusion framework. The networks con- sist of patterned arrays of diffusively coupled nanoliter- MS34 scale reactors containing the Belousov-Zhabotinsky (BZ) Order and Information Flow in an Array of Elec- reaction. Microfluidic fabrication techniques are developed tronic Neurons that provide the ability to vary the network topology, the reactor coupling strength and offer the freedom to choose A small array of FitzHugh-Nagumo neurons was con- whether an arbitrary reactor is inhibitory or excitatory structed. The nine neurons, connected in a ring with near- coupled to its neighbor. This versatile experimental and est neighbor coupling, exhibited either unsynchronized fir- theoretical framework can be used to create a wide vari- ing or synchronized behavior, depending on the coupling ety of chemical networks. Here we design, construct and strength. However a new state, a chimera, appeared when characterize chemical networks that achieve the complex- a single long-distance link (small world coupling)wasin- ity of central pattern generators, which are found in the troduced. autonomic nervous system of a variety of organisms. We It was possible to calculate the time-dependent behavior of envision that this artificial nervous system can serve as the the order parameter of the system. Of greater interest for controller of a synthetic musculature comprised of chemo- understanding computation in such a system, we were also mechanical gels coupled to the BZ layer in order to create able to determine, via symbolic dynamics, the information soft robots capable of autonomous activity. flow in the system as a function of time, both with and without the small world link, and to relate it to the order Seth Fraden parameter. This approach should be generally applicable Brandeis University to more complex arrays of electronic and biological neurons Department of Physics as well. [email protected] Mark Spano Thomas Litschel School of Biological and Health Systems Engineering Brandeis University Arizona State University Department of Biochemistry [email protected] [email protected] Easwara Arumugam MichaelM.Norton Electrical Geodesics, Inc. Brandeis University [email protected] Department of Physics [email protected] MS35 Using Bistability, Oscillations and Slow-Fast Dy- MS34 namics to Engineer Population Density Control Partially Synchronized States in Small Networks and Stable Coexistence of Competitive Bacterial of Electrochemical Oscillators: Effect of Hetero- Strains geneities and Network Topology In order to safely and reliably translate more than a decade When electrochemical reactions take place on electrode ar- of research on engineered gene regulatory networks to med- rays, a network can form through the potential drop among ical or industrial settings, it is necessary to develop strate- the elements. The goal of the presentation is to show how gies to systematically control bacterial population dynam- synchronization and network theories can be applied to de- ics. These strategies may be employed to limit growth as a scribe the partially synchronized patterns of the oscillatory safety precaution, to release intracellularly produced chem- chemical reaction system. We consider two fundamental ical compounds via lysis, or to co-culture different bacte- DS17 Abstracts 171

rial strains for multi-step industrial fermentation processes. ronmental stressors. Gating of bacterial persistence has re- Here, we use a reduced deterministic model to identify the cently been linked to toxin-antitoxin (TA) modules, which design principles and theoretical limitations of a recently are operons with an evolutionarily conserved motif that published self-lysis circuit [Din et al., Nature 563, 2016] includes a toxin that halts cell growth and a correspond- that is based on diffusive signaling within the population ing antitoxin that neutralizes the toxin. While many such and activates upon reaching a quorum threshold. We find modules have been identified and studied in a wide range that the conditions for the onset of population size oscil- of organisms, little consideration of the interactions be- lations derived from the model are consistent with experi- tween multiple modules within a single host has been made. mentally observed behavior in microfluidics. Robust oscil- Moreover, the multitude of different antitoxin species are lations are driven by the bistability arising from a core pos- degraded by a relatively small number of proteolytic path- itive feedback loop and their period can be experimentally ways, strongly suggesting competition between antitoxins tuned and analytically predicted in certain limits. Since for degradation machinery, i.e. queueing coupling. Here the circuit makes population growth self-limiting, we also we present a theoretical understanding of the dynamics use the model to explore the possibility of co-culturing two of multiple TA modules that are coupled through either strains that use different quorum sensing systems. Depend- proteolytic queueing, a toxic effect on cell growth rate, or ing on the individual parameters and cross-talk between both. We conclude that indirect queueing coordination be- the quorum-sensing systems, we find distinct multi-strain tween multiple TA modules may be central to controlling dynamics that could be useful for a variety of applications. bacterial persistence.

William H. Mather Philip Bittihn Virginia Tech Physics BioCircuits Institute [email protected] University of California, San Diego [email protected] MS35 Dynamic Ligand Discrimination in the Notch Sig- MS35 naling Pathway Cooperation in Synthetic Bacterial Communi- ties: Optimality, Fragility and Interaction Non- A mysterious aspect of Notch signaling, and many other Additivity developmental signaling pathways, is the use of distinct ligands in different developmental contexts despite their Cooperation is one of the major social interactions that similar abilities to interact with receptors. With Notch in contribute to the spatiotemporal organization of bacterial particular, because the pathway provides no known mech- communities, such as biofilms. Understanding its func- anism by which signal-receiving cells might discern the tional traits is important for characterizing, predicting and identity of an activating ligand, it has remained unclear eventually engineering the dynamics of complex communi- whether, and how, cells could adopt distinct gene expres- ties. Although bacterial cooperation has attracted consid- sion programs or cell fates in response to signaling by dif- erable interest over the decades, what remains unsolved ferent ligands. Using quantitative single-cell imaging in is how its key molecular traits contribute to the struc- cell-culture, we discovered that the Notch pathway uses tural characteristics of a community. Using engineered dynamics to discriminate signaling by the closely related Lactococcus lactis consortia as model systems, we have ligands Dll1 and Dll4 through a single receptor, Notch1. investigated how three key aspects of cooperationoptimal- Dll1 activates Notch1 in discrete, frequency-modulated ity, fragility, and competition non-additivitycontribute to pulses that preferentially regulate the Notch target gene community organization. Our studies show that (10 opti- Hes1. By contrast, Dll4 activates Notch1 in a sustained, mality of cooperation is determined by the stoichiometric amplitude-modulated manner that up-regulates different ratio of molecules engaged in cooperation, (2) that fragility direct targets such as Hey1/L.Mathematical modeling of of cooperation increases with the decrease of cell popula- the Hes/Hey circuit, including auto- and cross-repression tions, and (3) that cooperation can revoke the additivity of between factors, reveals how Notch signaling dynamics competition in communities consisting of multiple species. can be decoded through an incoherent feed-forward ar- Furthermore, we extended our exploration from well-mixed chitecture to activate distinct target genes. Taken to- liquid settings to solid agar plates, under which hetero- gether,these data show that the Notch pathway uses dy- geneous spatial structures of communities emerged. This namics to effectively establish multiple channels of com- study yields new insight into bacterial cooperation, provid- munication through a single receptor. ing broad implications for understanding and engineering complex community structures including biofilms. Sandy Nandagopal California Institute of Technology Ting Lu [email protected] University of Illinois [email protected] MS36 Traveling Waves in Fluids: Bifurcations with Re- MS35 spect to the Wave Speed Excitable Toxin-Antitoxin Modules Coordinated Through Intracellular Bottlenecks In this talk, I will present some new results on traveling waves in fluids. Two examples will be featured promi- Chronic infections and pathogenic biofilms present a seri- nently: electromigration dispersion waves in a capillary ous threat to the health of humans by decreasing life ex- and acoustic waves in thermoviscous perfect gases. In the pectancy and quality. The resilience of these microbial former case, a reduction to Darboux’s equation is obtained. communities has been attributed to the spontaneous for- In the latter case, an “elementary” nonlinear ODE (specif- mation of persister cells, which constitute a small fraction ically, a special case of Abel’s equation) is derived for invis- of the population capable of surviving a wide range of envi- cid but thermally conducting gases, while the case of vis- 172 DS17 Abstracts

cous but non-thermally conducting gases leads to a more new class of bifurcation, which is discussed in more detail complicated nonlinear ODE. In all contexts, exact (but im- in Part II. plicit) solutions are constructed for traveling waves (specif- ically, “kinks”) connecting distinct equilibria of their re- Aminur Rahman, Denis Blackmore spective ODE. In both the case of electromigration disper- New Jersey Institute of Technology sion waves and acoustic waves in viscous but non-thermally [email protected], [email protected] conducting gases, the governing ODE is shown to exhibit a transcritical bifurcation with respect to the dimension- less wave speed at the value of unity. I will conclude the MS36 talk by highlighting the generic nature of such transcritical The Annihilation of Periodic Points and Barriers bifurcations at unit wave speed in fluid mechanics. to Transport by Discontinuous ‘Slip’ Deformations in a Fluid Flow Ivan C. Christov Purdue University Fluid mixing and fluid particle transport has primarily [email protected] been considered in smooth flows, driven by ‘stretching and folding’ motions. However, there exist a broad range of flows that admit discontinuous ‘slip’ deformations; includ- MS36 ing valved flows, granular flows and shear banding mate- Course-Grained Models for Interacting Flapping rials. In these systems mixing and particle transport can Swimmers also be generated by ‘cutting and shuffling’ motions. We demonstrate the interplay between these discontinuous de- We present the results of a theoretical investigation into formations and bifurcations of periodic points in a model the dynamics of interacting flapping swimmers. Our study fluid flow with valves. In particular, we show that dis- is motivated by ongoing experiments in the NYU Applied continuous deformations destroy some Lagrangian struc- Math Lab, in which freely-translating, heaving airfoils in- tures associated with bifurcations, including barriers to teract hydrodynamically to choose their relative positions fluid transport, while leaving other structures intact. and velocities. We develop a discrete dynamical system in which flapping swimmers shed point vortices during each Lachlan Smith flapping cycle, which in turn exert forces on the swimmers. Dept. of Chemical & Biological Engineering We present a framework for finding exact solutions to the Northwestern University evolution equations and for assessing their stability, which [email protected] is nontrivial owing to the temporal nonlocality in the gov- erning equations. Our analysis gives physical insight into Murray Rudman the preference for certain observed ”schooling states.” Nu- Monash University merical simulation of the governing equations reveals that [email protected] the swimmers may undergo a complex nonlinear dynamics while maintaining a bound state. The model may be ex- Daniel Lester tended to arrays of flapping swimmers, and configurations Dept. of Civil, Environmental & Chemical Engineering in which the swimmers’ flapping frequencies are incommen- RMIT surate. Generally, our results indicate how hydrodynamics [email protected] may mediate schooling and flocking behavior in biological contexts. Guy Metcalfe School of Mathematical Sciences Anand Oza,LeifRistroph Monash University Courant Institute, NYU [email protected] [email protected], [email protected]

MichaelJ.Shelley MS37 New York University Realization of Nonlinear Resonances in Multiple Courant Inst of Math Sciences Modes of a Microcantilever Polymer System [email protected] Advances in micro/nano-scale fabrication techniques have led the extensive development of microelectromechanical MS36 systems (MEMS). Among them, resonator-based MEMS Bifurcations in Walking Droplet Dynamics designs often employ beam-type mechanical structures in micro/nanometer scales, designed to exhibit mechanical Bouncing droplets on a vibrating fluid bath can exhibit motion usually at their high resonant frequencies with high behavior analogous to wave-particle duality, such as be- Q factors. Intentional implementation of geometric nonlin- ing propelled by the waves they generate. These droplets earity has suggested ways to break through the limitations seem to walk across the bath, and thus are dubbed walk- of linear MEMS by utilizing beneficial nonlinear charac- ers. Walkers can exhibit exotic dynamical behavior which teristics, not attainable in a linear setting. Integration of give strong indications of chaos, but many of the interest- nonlinear couplings to an otherwise purely linear micro- ing dynamical properties have yet to be proven. In a re- cantilever is a promising way to intentionally realize ge- cent work by Gilet (PRE 2014) a discrete dynamical model ometric nonlinearity. Here, we demonstrate a nonlinear is derived and studied numerically. We prove the exis- microresonator consisting of a silicon microcantilever and tence of Neimark-Sacker bifurcations for a variety of eigen- a polymer attachment exhibiting strong nonlinear harden- mode shapes of the waves and parameter regimes from this ing behavior not only in the first flexural mode but also in model. Then we reproduce numerical experiments done by higher modes. In this design, we intentionally implement a Gilet and produce new numerical experiments and apply drastic and reversed change in the axial vs. bending stiff- our theorem to the test functions used for that model in ness between the Si and polymer components by varying addition to new test functions. We also show evidence of a the geometric can material properties. By doing so, res- DS17 Abstracts 173

onant oscillations induce large axial stretching within the multitude of the resonances. The domains of the NNMs polymer component, which effectively introduces geomet- instability are found analytically in the framework of the ric stiffness and damping nonlinearity. The efficacy of the developed approach. It was shown that the nonlinear reso- design and the mechanism of geometric nonlinearity are nant interaction of the long wavelength NNMs leads to the confirmed through a comprehensive experimental, analyt- intensive energy exchange or energy localization in some ical, and numerical analysis on the nonlinear dynamics of domain of the lattice, which are described by the LPTs. this system. The threshold corresponding to LPT instability (the tran- sition to energy localization) is also estimated analytically. Keivan Asadi The numerical simulation of the large-amplitude dynamics Dept. of Mechanical and Aerospace Engineering, The of the finite model of the Sine-lattice confirms the con- Ohio Sta clusions of the asymptotic analysis with a good accuracy. [email protected] The problem of chaotic regimes in the short discrete Sine- lattices is discussed. Snehan Peshin, Junghoon Yeom Dept. of Mechanical Engineering, Michigan State Leonid Manevitch University, Semenov Institute of Chemical Physics [email protected], [email protected] Russian Academy of Sciences [email protected] Hanna Cho Ohio State University Valeriy Smirnov [email protected] N.N. Semenov Institute of Chemical Physics, RAS [email protected]

MS37 Energy Exchange in Strongly Nonlinear Systems: MS37 Canonical Formalism Resonance Energy Transfer Between Charged Par- ticles and Electromagnetic Waves Over recent years, a lot of progress has been achieved in understanding of the relationship between localization and In the present talk we review recent studies of multi-scale transport of energy in essentially nonlinear oscillatory sys- plasma systems where the nonlinear resonant interaction tems. In this paper we are going to demonstrate that the between charged particles and electromagnetic waves plays structure of the resonance manifold can be conveniently de- an important role; and discuss particular examples of the scribed in terms of canonical action-angle variables. Such acceleration of charged particles by scattering on reso- formalism has important theoretical advantages: all res- nances and capture into resonance. We focus on a sys- onance manifolds may be described at the same level of tem, where both scattering and capture allow particles to complexity, appearance of additional conservation laws on investigate different domains of the phase space. We com- these manifolds is easily proven both in autonomous and pute the effects of a single scattering and a single capture non-autonomous settings. The harmonic balance - based on adiabatic invariants, and show how these effects can be complexification approach, used in many previous studies combined into a Fokker-Plank-type equation for the prob- on the subject, is shown to be a particular case of the ability distribution function for an adiabatic invariant. We canonical formalism. Moreover, application of the canonic compute the rates of mixing and energy transport from the averaging allows treatment of much broader variety of dy- wave to the particles. namical models. As an example, energy exchanges in sys- tems of coupled trigonometrical and vibro-impact oscilla- Dmitri Vainchtein tors are considered. Other examples address dynamics of Drexel Plasma Institute forced vibro-impact oscillator and of Hamiltonian chain of [email protected] such coupled oscillators. Anton Artemyev Oleg Gendelman Institute of Geophysics and Planetary Physics, UCLA Technion Israel Institute of Technology [email protected] [email protected] Fan Wu Themistoklis Sapsis Central South University, China Massachusetts Institute of Techonology [email protected] [email protected]

MS38 MS37 Large Deviations in Stochastic Hybrid Networks High-amplitude Stationary Dynamics, Energy Transfer and Localization in Sine-lattice We construct a path-integral representation of solutions to a stochastic hybrid system or piecewise deterministic The strongly nonlinear dynamics of the discrete model of Markov process, consisting of one or more continuous vari- the harmonically coupled pendula (Sine-lattice) is consid- ables evolving according to a piecewise-deterministic dy- ered in a wide interval of oscillation amplitudes. The semi- namics. The differential equations for the continuous vari- inverse procedure is used for the derivation of the main ables are coupled to a set of discrete variables that sat- asymptotic approximation, that allows to estimate the non- isfy a continuous-time Markov process, which means that linear normal modes‘ (NNMs) spectrum as well as to ana- the differential equations are only valid between jumps in lyze the non-stationary resonance dynamics in terms of the the discrete variables. Examples of stochastic hybrid sys- limiting phase trajectories (LPTs). It was shown that the tems arise in biophysical models of stochastic ion channels, spectrum of the NNMs at the large amplitudes is specified motor-driven intracellular transport, gene networks, and by the non-monotonic dispersion curves that leads to the stochastic neural networks. We use the path-integral rep- 174 DS17 Abstracts

resentation to derive a large deviation action principle for [email protected] a stochastic hybrid system, which can be used to solve first passage time problems. We also derive a Feynman-Kac for- mula for the residence time of a stochastic hybrid system. MS38 Designing Noisy Networks

Paul C. Bressloff A heteroclinic network is a finite set of dynamical states University of Utah connected by trajectories in phase space. Heteroclinic net- Department of Mathematics works appear robustly in a range of applications and have bressloff@math.utah.edu been used as a way to model a number of types of biologi- cal and cognitive processes. Excitable networks are a close relation of heteroclinic networks for which finite amplitude MS38 perturbations are required to transition from one state to Can Finite Size Effects in Networks Be Repre- another. I will present methods for realizing an arbitrary sented by Stochastic Mean-Field Dynamics? directed graph as a heteroclinic or excitable network in phase space. These networks attractors display a high sen- The complex dynamics of phase oscillator networks a re- sitivity to low amplitude noise both in terms of the regions ceived great attention in recent years. The thermodynamic of phase space visited and in terms of the sequence of the limit with infinitely many nodes often allows for analytical transitions around the network. In some circumstances, treatment with the approach of Ott and Antonsen as most the noise may even cause long-time correlations, or mem- popular example [Ott, Antonsen, Chaos 18, 037113 (2008)]. ory, in the sequence of transitions between states. I will In this talk the focus is on finite-size networks for which the give some results that quantify the effect of noise on these analytic treatment is limited. For a fully connected, homo- attractors; specifically, how to characterize residence times geneous network we start off with the Ott and Antonsen at states and switching rates between states. I will also manifold that covers its mean-field or order parameter dy- discuss several new applications of this work to modelling namics. By reducing network size stepwise we investigate neural processes. to what extent the original low-dimensional order param- eter dynamics stays in tact while erratic behavior emerges Claire M. Postlethwaite due to the random realization of the oscillators natural University of Auckland frequencies. For different network sizes we analyze numer- [email protected] ically obtained time series of the order parameter using the Kramers-Moyal expansion. This expansion into corre- Peter Ashwin sponding conditional cumulants [Friedrich, Peinke. Phys- University of Exeter ical Review Letters 78(5):863 (1997); Gradiek, Siegert, [email protected] Friedrich, Grabec. Physical Review E 62(3A):3146 (2000)] yields so-called drift and diffusion coefficients that repre- sent the deterministic and stochastic components of the MS39 dynamics, respectively. While the first largely resembles Data Assimilation for Geophysical Fluids: Back the dynamics in the thermodynamic limit presuming suf- and Forth Nudging (BFN) and Observers ficiently large networks, the latter does support the idea that finite-size effects give rise to diffusion-like stochastic The Back and Forth Nudging (BFN) algorithm is a proto- components in the mean-field dynamics. type of a new class of data assimilation methods, although the standard nudging algorithm is known for a couple of Andreas Daffertshofer decades. It consists in adding a feedback term in the model MOVE Research Institute Amsterdam equations, measuring the difference between the observa- VU University Amsterdam tions and the corresponding space states. The idea is to a.daff[email protected] apply the standard nudging algorithm to the backward (in time) nonlinear model in order to stabilize it. The BFN algorithm is an iterative sequence of forward and backward MS38 resolutions, all of them being performed with an additional Noise-Induced Optimal Dynamics in Self- nudging feedback term in the model equations. We also Organizing Adaptive Networks present the Diffusive Back and Forth Nudging (DBFN) algorithm, which is a natural extension of the BFN to Recent studies have shown that adaptive networks driven some particular diffusive models. These nudging-based al- by simple local rules can organize into critical global steady gorithms can be extended to more complex observers, with states, providing another framework for self-organized crit- the aim of correcting non-observed variables, and improv- icality (SOC). In this talk, based upon the paper [C. ing the convergence of the algorithm and estimation of the Kuehn, Time-scale and noise optimality in self-organized model state, but also with the aim of identifying model critical adaptive networks, Phys. Rev. E 85, 026103, 2012], parameters. I shall focus on the important convergence to criticality and show that noise and time-scale optimality are reached at Didier Auroux finite values. This is in sharp contrast to the previously be- University of Nice, France lieved optimal zero noise and infinite time-scale separation [email protected] case. Furthermore, a noise-induced breakdown of SOC is observed. The noise-optimality effect can be understood particularly well using a simple conceptual model, which MS39 reveals a new version of stochastic resonance, which we Forecast Sensitivity to Observations and Observa- call steady-state stochastic resonance. tion Impact for Data Assimilation Systems

Christian Kuehn A posteriori, it is possible to evaluate the accuracy of fore- Technical University of Munich cast skill in the data assimilation system. Using an adjoint DS17 Abstracts 175

technique that can be either explicit or ensemble based, daniel [email protected] forecast skill can be traced back to the observations used in the past analysis, leading to quantitative evaluation of observation impact and sensitivity to forecast skill. Thus MS40 Forecast Sensitivity to Observations (FSO) is a diagnos- The Role of Localization and Center-Surround tic tool that complements traditional data denial of the Structure in Compressive Sensory Signal Process- observing system experiments. In this talk, we will evalu- ing ate FSO of the operational numerical weather prediction. By careful examination of the innovation, i.e., difference For successful preservation of sensory information across in model background and observations, we will assess the pathways with widely varying numbers of neurons, it is effect of observation and model biases on FSO. necessary for neuronal network architecture to facilitate efficient signal processing. The center-surround receptive Kayo Ide field structure, ubiquitous in the visual system, is hypoth- University of Maryland, College Park esized to be evolutionarily advantageous in image process- [email protected] ing tasks. We address the potential functional benefits and shortcomings of localization and center-surround receptive fields in the context of an integrate-and-fire neuronal net- MS39 work model. Based on the sparsity of natural scenes, we Multilevel Monte Carlo Methods for Data Assimi- derive a compressive-sensing framework for input image lation reconstruction utilizing evoked neuronal firing rates. We investigate how the accuracy of input encoding depends For half a century computational scientists have been nu- on the network architecture, and demonstrate that center- merically simulating complex systems. Uncertainty is re- surround antagonism facilitates marked improvements in cently becoming a requisite consideration in complex ap- natural scene processing beyond uniformly-random excita- plications which have been classically treated determinis- tory connectivity. However, for specific classes of images, tically. This has led to an increasing interest in recent we show that the center-surround structure may underlie years in uncertainty quantification (UQ). Another recent common optical illusions. In the context of signal process- trend is the explosion of available data. Bayesian infer- ing, we expect this work may suggest new sampling proto- ence provides a principled and well-defined approach to cols useful for extending conventional compressive sensing the integration of data into an a priori known distribution. theory. The posterior distribution, however, is known only point- Victor Barranca wise (possibly with an intractable likelihood) and up to a Swarthmore College normalizing constant. Monte Carlo methods have been de- [email protected] signed to sample such distributions, such as Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) samplers. Recently, the multilevel Monte Carlo (MLMC) MS40 framework has been extended to some of these cases, so that approximation error can be optimally balanced with Modeling Within and Across Area Neuronal Vari- statistical sampling error, and ultimately the Bayesian in- ability in the Visual System verse problem can be solved for the same asymptotic cost Shared variability among neurons (noise correlations) have as solving the deterministic forward problem. This talk been commonly observed in multiple cortical areas (Cohen will concern the recent development of MLMC data assim- and Kohn, 2011). Moreover, noise correlations are mod- ilation methods, which combine dynamical systems with ulated by cognitive factors, such as task engagement and data in an online fashion. Examples are ML particle filters attention (Cohen and Maunsell, 2009; Doiron et al. 2016). and ensemble Kalman filters. While there is much discussion about the consequences of noise correlations on neuronal coding, there is a general Kody Law lack of understanding of the circuit mechanisms that gener- Oak Ridge National Laboratory ate and modulate shared variability in the brain. Recently, [email protected] simultaneous microelectrode array recordings from V1 and MT in behaving monkeys (Ruff and Cohen, 2016) suggest that attention not only decreases correlations within a cor- MS39 tical area (MT), but also increases correlations between The Bayesian Formulation and Well-posedness of cortical areas (V1 and MT). The differential modulation Fractional Elliptic Inverse Problems of between-areas and within-area noise correlations impose further constraints on circuit mechanisms for the gener- We study the inverse problem of recovering the order and ation and propagation of noise correlations. We develop the diffusion coefficient of an elliptic fractional partial dif- a spiking neuron network with spatiotemporal dynamics ferential equation from a finite number of noisy observa- that internally generates shared variability matching the tions of the solution. We work in a Bayesian framework low dimensional structure widely reported across cortex. and show conditions under which the posterior distribution This variability results from macroscopic chaos in popu- is given by a change of measure from the prior. Moreover, lation rates, which correlates neurons from the same re- we show well-posedness of the inverse problem, in the sense current network while decoupling them from feedforward that small perturbations of the observed solution lead to inputs. Attention is modeled as depolarizing the inhibitory small Hellinger perturbations of the associated posterior neuron population, which reduces the internally generated measures. We thus provide a mathematical foundation to shared variability and allows the network to better track the Bayesian learning of the order – and other inputs – of input signal. fractional models. Chengcheng Huang Daniel Sanz-Alonso New York University University of Warwick [email protected] 176 DS17 Abstracts

Douglas Ruff, Marlene Cohen, Brent Doiron nary Differential Equations University of Pittsburgh douglas.ruff@gmail.com, [email protected], Recent work on reservoir computing has brought to the [email protected] forefront a new set questions about the typical behavior of large sets of ordinary differential equations with fixed struc- ture and random variations within that structure. Exam- MS40 ples include situations in which large blocks of linear con- Spatial Variation of Spike Initiation Threshold in stant coefficient subsystems that are coupled through non- Large Active Dendritic Arbors linearities of a fixed form. On one hand, such a structure can be used as a computationally efficient alternative to The integration of synaptic inputs in a neuron can be non- the kernel methods appearing in support vector machines. linear not just at the axon, but also locally in the dendrites In another context, such models give rise to chaotic behav- if they possess active voltage-gated ion channels. Such non- ior whose underlying periodic solutions can be tuned to linearities can lead to, for example, compartmentalized re- closely match a given pattern. In this talk we will present sponses to inputs, with branches acting as individual non- some specific examples illustrating these points, as well as linear units in which dendritic spikes occur. In a straight parts of what may become a general theory. cable, above- and below-threshold conditions are known to be separated by an unstable threshold or critical sur- Roger Brockett face. Here we demonstrate methods that can be used to Division of Engineering and Applied Sciences find the critical synaptic conductance leading to a thresh- Harvard University old solution, and apply them to large, branched dendritic [email protected] arbors with voltage-gated conductances. The methods al- low efficient description as to how the critical threshold is Daniel J. Gauthier affected by factors such as branching geometry, constric- Duke University tions or varicosities, and spatial variation of conductance [email protected] densities.

William Kath MS41 Department of Applied Mathematics, Northwestern University Reservoir Computing Using Autonomous Boolean Department of Neurobiology, Northwestern University Networks Realized on a Field-Programmable Gate [email protected] Array In its typical configuration, a reservoir computer requires a MS40 randomly connected network with 100s to 1,000s of nodes, which is challenging to realize experimentally. One ap- Distinct Roles of Interneurons in Hippocampal proach is to use an autonomous time-delay Boolean net- Network Oscillations work for the reservoir, which can be realized on a com- The functional role of interneurons in the generation of mercial device known as a field-programmable gate array hippocampal oscillations remains to be elucidated. For in- (FPGA). This platform allows for typical node time scales terneurons recorded in mice hippocampal CA1 area, by us- of under a nanosecond, opening up the possibility of very ing time-delayed mutual information, we characterize two high speed operation of the reservoir computer, along with subsets of interneurons whose firing activities share high flexible design rules for optimizing the reservoir dynam- mutual information with theta-band (4-12 Hz) and ripple- ics. We will describe the performance characteristics of band (100-250 Hz) LFP signals, respectively. Information FPGA-based autonomous time-delays Boolean networks flow direction further suggests their unique contribution to for various tasks, such as forecasting a chaotic time se- theta and ripple oscillations. Finally, we discuss the limi- ries, classifying written-digit images, and identifying simple tation of Granger Causality analysis in this case as well as spatial-temporal patterns. We gratefully acknowledge the in general neuronal data analysis. financial support of the U.S. Army Research Office grant #W911NF-12-1-0099. Songting Li Courant Institute of Mathematical Sciences Daniel J. Gauthier New York University Duke University [email protected] [email protected]

Douglas Zhou Daniel Canaday, Aaron Griffith Shanghai Jiao Tong University Ohio State University, Columbus, OH, USA [email protected] [email protected], griffi[email protected]

David Cai Nicholas Haynes Courant Institute for Mathematical Sciences, NYU Department of Physics Shanghai Jiao-Tong University Duke University [email protected] [email protected]

Jiamin Xu, Longnian Lin Otti D’Huys East China Normal University Aston University, Birmingham, UK jimmy [email protected], [email protected] [email protected]

MS41 MS41 Qualitative Properties of Large Systems of Ordi- Nonlinear Dynamics of Reservoir Computing and DS17 Abstracts 177

Prediction are manipulated by powerful techniques of molecular biol- ogy in hundreds of laboratories. Atoms (and thus charges) Reservoir computing algorithms use a general-purpose, can be substituted a few at a time. The location of ev- neural-network dynamical system to perform machine ery atom is known in favorable cases. Ion channels are learning tasks. This talk will focus on the tasks of predict- one of the few living systems of great importance ing a chaotic time series whose dynamics are unknown, and whose natural biological function can be well de- of inferring unobserved state variables of a chaotic system scribed by a tractable set of dynamical equations. from observed variables. In each case, the observed time Ion channels are studied by Poisson Drift Diffusion equa- series is input to the reservoir system, creating a coupled tions familiar in plasma and semiconductor physics called drive-response system. I will describe specific algorithms, Poisson Nernst Planck or PNP in biology. My collabora- and discuss how their success is related to the concept of tors and I have (almost) derived relevant dynamics from generalized synchronization. stochastic differential equations. Dimensional reduction yields tractable inverse, variational, and forward problems. Brian R. Hunt Dynamical models fit data with only a handful of parame- University of Maryland ters that do not change even when concentration boundary [email protected] conditions change value by 107.

MS41 Bob Eisenberg Department of Molecular Biophysics and Physiology Predicting Spatial-Temporal Dynamics Using Rush University Reservoir Computers [email protected] We consider the problem of predicting the future time evo- lution of a spatiotemporally chaotic process purely from previous measurements of time series from that process MS42 (without knowledge of a model producing the time series). The Steric Pnp-Ok Model Examples will be presented showing that reservoir comput- ing is an extremely useful approach to this general problem. The Poisson-Nernst-Planck (PNP) theory is one of the Issues considered will include design choices for reservoir most widely used analytical methods to describe electroki- parameters, parallel implementation schemes for favorable netic phenomena for electrolytes. The model, however, scaling to very large systems, and whether, when the pre- considers isolated charges and thus is valid only for dilute diction time is exceeded, the reservoir output continues to ion concentrations. The key importance of concentrated reproduce the ergodic dynamical properties of the original electrolytes in applications has led to the development of process (i.e., does the reservoir give the process ’climate’). the steric PNP model which accounts for ion-ion steric in- [Supported by ARO.] teractions. This model has been successfully applied to Edward Ott the study of transport in ion channels. In this study, I University of Maryland will consider the steric Poisson-Nernst-Planck (PNP) the- Inst. for Research in Electronics and Applied Physics ory and in particular show that in certain cases, the model [email protected] gives rise to singular solutions. I will present a high-order model in which solutions are smooth, and will show that the model gives rise to multiple solutions. Jaideep Pathak UMD [email protected] Nir Gavish Department of Mathematics Technion ITT Zhixin Lu [email protected] University of Maryland [email protected] MS42 Brian Hunt UMD Energetic Variational Approaches in Transport of [email protected] Ionic Particles

Michelle Girvan Transport of charged ions is central to almost all biological University of Maryland activities. Flows of ions produce signaling in the nervous [email protected] system, initiation of contraction in muscle, coordinating the pumping of the heart and regulating the flow of water through kidney and intestine. Ion concentrations inside MS42 cells are controlled by ion channel proteins through the Dynamics of Ions Control Most Biological Systems lipid membranes. In this talk, continuum models are de- rived from the energetic variational approach which include Biology occurs in ionic solutions that are both physical the coupling between the electrostatic forces, the hydrody- and biological plasmas. Dynamics of ions determines the namics, diffusion and crowding (due to the finite size ef- properties of plasmas. Ion channels are proteins with a hole fects), in particular, temperature fields and various bound- down their middle that conduct spherical charges like Na+, ary conditions. The models provide basic understanding K+, Ca2+, and Cl−, with diameter about 0.2 nm through of some important properties of proteins, such as the ion a narrow tunnel of fixed charge (doping) with diameter selectivity and sensor mechanism. about 0.6 nm. Ion channels control dynamics of electric charge and current flow across membranes. Channels are Chun Liu as significant in biology as transistors are in computers: Department of Mathematics, Penn State University most of biology is controlled by channels. Ion channels University Park, PA 16802 178 DS17 Abstracts

[email protected] [email protected]

MS42 MS43 The Data Mining of Sloppiness: Parameter Reduc- Individual Flux Study Via Steady-State Poisson- tion for Complex Dynamical Systems Nernst-Planck Systems: Effects from Boundary Conditions In the past decades, extensive effort has been expended on reducing the number of variables in large, complex dy- We provide a detailed study for ionic flow through ion namic models to obtain effective models of smaller dimen- channels for the case with three ion species, two posi- sion. Nevertheless, the concomitant reduction in the num- tively charged having the same valence and one negatively ber of parameters dictating system response is a relatively charged, and with zero permanent charge. Our focus is recent endeavor. This area is classically covered by sensi- on the effects of boundary conditions on the ionic flow. tivity analysis, which is nevertheless plagued by the fact Beyond the existence of solutions of the model problem, that it is local and linear in nature. Sloppiness and ac- we are able to obtain explicit approximations of individual tive subspaces are two promising current directions, but a fluxes and the I-V relations, from which effects of bound- complete understanding and an effective algorithmic imple- ary conditions on ionic flows are examined in a great detail. mentation of parameter reduction is still largely lacking. In Critical potentials are identified and their roles in charac- this talk, we will explore the link between nonlinear data terizing these effects are studied. Compared to ionic mix- mining and parameter reduction by identifying and ana- tures with two ion species, a number of new features for lyzing sets of effective parameters and possible parameter- mixtures of three ion species arise. Numerical simulations izations. Combining equation–free principles with machine are performed, and numerical results are consistent with learning algorithms, we propose a data–based approach to our analytical ones. resolve “sloppy’ parameter directions in both singularly– and regularly–perturbed kinetic models. Using that frame- work, we fully analyze the paradigmatic Michaeli–Menten– Mingji Zhang New Mexico Institute of Mining & Technology Henri mechanism, connect our results to a singular per- [email protected] turbation analysis and identify the number and nature of effective parameters in that system.

Antonios Zagaris MS43 Wageningen UR Central Veterinary Institute Sparse Sensor Placement for Multiscale Phenom- antonios[email protected] ena Ioannis Kevrekidis Multiscale dynamics pose challenges in determining modal Dept. of Chemical Engineering decompositions with physical meaning which can render Princeton University sensor placement particularly difficult. Localized features [email protected] in space or time may play a crucial role for the phenomenon of interest but may be insufficiently resolved due to their William Holiday low energy contribution. We determine near-optimal spa- Princeton University tial sensors in state space using empirical interpolation [email protected] methods (EIM, discrete EIM) on low-rank dynamic mode decomposition (DMD) modes. Such empirical interpolative methods are commonly employed with proper orthogonal MS43 decompositions in model reduction. We extend this sensor placement approach to multiscale physics problems using Kernel Analog Forecasting and Its Applications multi-Resolution DMD or mrDMD [Kutz et al., 2015], a recursive application of DMD in the time-frequency do- Analog forecasting is a nonparametric technique intro- main that separates flow features occurring at different duced by Lorenz in 1969 which predicts the evolution of timescales. The discovered sensors achieve accurate flow observables of dynamical systems by following the evolu- state reconstruction in representative multiscale examples tion of samples in a historical record of observations of including global ocean temperature data with an energetic the system which most closely resemble the observations El Ni˜no mode. Interestingly, this method places sensors at forecast initialization. We discuss a family of forecast- near coastlines without imposing additional constraints, ing methods which improve traditional analog forecast- which is beneficial from an engineering perspective. ing by combining ideas from kernel methods for machine learning and state-space reconstruction for dynamical sys- tems. A key ingredient of our approach is to replace single- Krithika Manohar analog forecasting with weighted ensembles of analogs con- University of Washington structed using local similarity kernels. The kernels used Dept of Applied Mathematics here employ a number of dynamics-dependent features de- [email protected] signed to improve forecast skill, including Takens’ delay- coordinate maps (to recover information in the initial data Eurika Kaiser, Steven Brunton lost through partial observations) and a directional depen- University of Washington dence on the dynamical vector field generating the data. [email protected], [email protected] We illustrate these techniques in atmosphere ocean science applications. Nathan Kutz University of Washington Zhizhen Zhao Dept of Applied Mathematics University of Illinois DS17 Abstracts 179

[email protected] terns?

In many semi-arid ecosystems, vegetation forms spatial MS44 patterns in an apparent response to water scarcity. Using Introducing Topographical Influences in the hypotheses for vegetation-water interaction, water trans- Extended-Klausmeier Vegetation Model port, and vegetation spread, PDE models can gener- ate comparable patterns via a self-organizing mechanism. Today, an increasing amount of vegetation patterns in These models are often analyzed assuming a uniformly semi-arid climates face the dangers of desertification. To sloped landscape with homogeneous water availability. Us- understand the driving mechanics behind this effect many ing satellite imagery and remote sensing of topography, we ecosystem models have been created over the last years can examine predictions of the models, including a pre- - both very simplistic ones and very realistic ones. The dicted relationship between slope and pattern wavelength. current consensus indicates the importance of water avail- This relationship appears to be weak, perhaps because to- ability for the survivability (and movement) of vegetation. pography has a strong influence on water availability and In turn the water availability is greatly influenced by the transport over long length scales not included in the mod- topography. Often, simple models ignore the topographi- els. Identifying how topography impacts vegetation pat- cal details whereas advanced models can only be studied terns may lead to a new generation of models for this sys- by means of numerical PDE simulations. In this talk, we tem. present one way to add topographical effects in a simpli- fied ecosystem model (extended-Klausmeier) that results in Sarah Iams a heterogeneous PDE that we can still study analytically Harvard University for some cases. Specifically we study the topographical ef- Cambridge, MA fects on localized vegetation pulses, i.e. one-dimensional [email protected] patches. This leads to new observations, including pulses that move downhill. Andrew J. Bernoff Harvey Mudd College Robbin Bastiaansen Department of Mathematics Mathematical Institute [email protected] Leiden University [email protected] Karna V. Gowda Northwestern University Arjen Doelman [email protected] Mathematisch Instituut [email protected] Chad M. Topaz Dept. of Mathematics, Statistics, and Computer Science Martina Chirilus-Bruckner Macalester College University of Leiden [email protected] Mathematical Institute [email protected] Mary Silber University of Chicago Dept. of Engineering Sciences and Applied Mathematics MS44 [email protected] Topographic Controls on Vegetation Patterns

Regular and irregular spatial patterns of vegetation have MS44 been observed in arid and semi-arid landscapes worldwide. Topography and Water Transport in Semi-Arid Such patterns are the result of both endogenous and exoge- Ecosystems nous effects, the former pertaining to the intrinsic proper- ties of the self-organization process, the latter reflecting Abstract Not Available At Time Of Publication. spatial heterogeneities in soil properties and topographic features. In particular, topographic heterogeneities of the Amilcare Porporato landscape can affect water runoff, infiltration, solar radi- Duke University ation and erosion processes, ultimately changing the spa- [email protected] tial redistribution of resources and thus the pattern forma- tion mechanism. While these landform-water-vegetation interactions have been largely recognized, they are often MS45 neglected in many vegetation pattern formation models, The Geometry of Blenders in a Three-dimensional which have mainly focused on flat terrains or uniform slope H´enon-like Family profiles. Here we analyze the effects of accounting for land- scape topographic features in models describing vegetation Blenders are a geometric tool to construct complicated patchiness in water-limited systems. The influence of vari- dynamics in diffeomorphisms of dimension at least three ations in local exogenous features such as topographic slope and vector fields of dimension at least four. They ad- and aspect on pattern formation will be investigated. mit invariant manifolds that behave like geometric objects which have dimensions higher than expected from the man- Sara Bonetti, Amilcare Porporato ifolds themselves. We consider an explicit family of three- Duke University dimensional H´enon-like maps that exhibit blenders in a [email protected], [email protected] specific regime in parameter space. Using advanced numer- ical techniques we compute stable and unstable manifolds in this system, enabling us to show one of the first numer- MS44 ical pictures of the geometry of blenders. We furthermore How Does Topography Impact Vegetation Pat- present numerical evidence suggesting that the regime of 180 DS17 Abstracts

existence of the blenders extends to a larger region in pa- either coincide or have opposite signs). rameter space. Ivan Ovsyannikov Stefanie Hittmeyer, Bernd Krauskopf, Hinke M. Osinga University of Bremen, Germany University of Auckland [email protected] Department of Mathematics [email protected], [email protected], [email protected] MS45 Complex Dynamics of Robust Pulse Generators in Katsutoshi Shinohara Reaction-Diffusion Systems Hitotsubashi University Graduate School of Commerce and Management Heterogeneity is one of the most important and ubiquitous [email protected] types of external perturbations. We study a spontaneous pulse-generating mechanism in an excitable medium with jump-type heterogeneity. Such a pulse generator (PG) has MS45 attracted considerable interest due to the computational Bifurcations of Degenerate Singular Cycles potential of pulse waves in physiological signal processing. Exploring the global bifurcation structure of PGs as peri- We investigate the dynamics near a so-called singular cy- odic solutions, we find firstly the conditions under which cle – a heteroclinic cycle involving a hyperbolic equilibrium they emerge, i.e., the onset of PGs, secondly a candidate point E and a hyperbolic periodic solution P , such that the for the organizing center producing a variety of PGs. We connection from E to P is of codimension one and the con- devise numerical frameworks to trace the long-term behav- nection from P to E occurs at a quadratic tangency (also of ior of PGs as periodic solutions, and we detect the asso- codimension one). We study such a cycle as the organizing ciated terminal homoclinic orbits that are homoclinic to a center of a two-parameter bifurcation scenario. Depend- special type of heterogeneity-induced ordered pattern with ing on properties of the transition maps, we find different a hyperbolic saddle. [Yadome, M., Nishiura, Y. and Ter- types of shift dynamics near the cycle. Breaking one or amoto, T., SIAM Journal on Applied Dynamical Systems both of the connections we further explore the bifurcation 13 (2014), 1168, 34 pages.] diagrams previously begun by other authors. In particular, we identify multipulse homoclinic solutions to E and P as Takashi Teramoto well as multipulse heteroclinic tangencies from P to E,and Asahikawa Medical University bifurcating periodic solutions. [email protected]

Alexander Lohse University of Hamburg MS46 [email protected] Rescuing Pulsatile Insulin Secretion by Wiggling Glucose

Alexandre A. Rodrigues Pancreatic islets manage elevations in the blood glucose University of Porto level by secreting insulin into the bloodstream in a pul- [email protected] satile or oscillatory manner. In this presentation, we show that although islet oscillations are lost by fixing the glu- MS45 cose stimulus at a high level, they may be rescued by sub- sequently converting the glucose stimulus to a sinusoidal Birth of Discrete Lorenz Attractors in Homoclinic wave. We predict that the ability of this wave to rescue and Heteroclinic Tangencies oscillations should depend on the amplitude and frequency A presence of non-transversal homoclinic or heteroclinic of the wave, using insights from an analysis of the dynam- orbits (tangencies) in a dynamical system is regarded as ical system. This prediction is confirmed in experiments a universal criterion of existence of a complex dynam- that employ a specially-designed microfluidics device. Our ics. However, it does not immediately lead to the emer- results suggest a mechanism whereby oscillatory blood glu- gence of genuine strange attractors (those preserving their cose levels recruit non-oscillating islets to enhance pulsatile strangeness under small perturbations) such as the Lorenz insulin output from the pancreas. They also provide sup- attractors. A list of three-dimensional diffeomorphisms port for the main hypothesis of the Dual Oscillator Model, with quadratic tangencies will be presented in which dis- that a glycolytic oscillator endogenous to islet β-cells drives crete Lorenz attractors are born in bifurcations. For some pulsatile insulin secretion. of them a stronger result was proved: in any neighbour- Richard Bertram, Joseph Mckenna hood there exist residual sets in which systems possess a Department of Mathematics countable number of coexisting discrete Lorenz attractors. Florida State University To get Lorenz attractors one needs to have the effective [email protected], [email protected] dimension of the problem to be at least three. This means that there should be no global contraction/expansion and no global center manifolds. To fulfill the first condition the Raghuram Dumpa Jacobian at the saddle should be close to 1 in the homo- Florida State University clinic case, and in the heteroclinic case it is enough to have [email protected] the contracting-expanding configuration, when the Jaco- bian at one saddle is less than one and greater than one Nikita Mukhitov at another saddle. The following conditions prevent the Department of Chemistry and Biochemistry appearance of center manifolds: 1. At least one of the Florida State University saddle fixed point is a saddlefocus; 2. All the fixed points [email protected] are saddles but one of the homoclinic/heteroclinic orbits is non-simple; 3. A saddle is resonant (two stable eigenvalues Michael Roper DS17 Abstracts 181

Florida State University Department of Mathematics [email protected] [email protected]

MS46 MS46 Centre Manifold Deformation in Impaired Ultra- Information Transfer in Gonadotropin-Releasing dian Oscillations of Glucose Regulation Hormone (gnrh) Sensing and Signalling

The accurate regulation and metabolisation of glucose is Biochemical cell signalling networks are noisy communica- essential for releasing the energy essential for the proper tion channels; transmitting biological information and al- functioning of cells and organs. This regulation operates lowing cells to respond appropriately to external cues. In in a cyclic way, as a result of a feedback loop in which the context of reproductive endocrinology, cell signalling phasic insulin release has a crucial role. In this contribu- networks allow pituitary gonadotropes to sense pulses of tion, we evaluate the effect of various physiological param- Gonadotropin-releasing hormone (GnRH) and regulate the eters on the production of the ultradian oscillatory regime production of hormones (luteinizing hormone, LH; and in a nonlinear system comprising two delays. The system, follicle-stimulation hormone, FSH) which control reproduc- based on the model introduced in [Li, Kuang, Mason, Mod- tion. Here, we develop and study a simplified mathemati- eling the glucoseinsulin regulatory system and ultradian cal model of GnRH signalling in gonadotropes. Using the insulin secretory oscillations with two explicit time delays, model we explore the systems sensitivity to GnRH dynam- Jour Theor Biol,2006], takes into account both instanta- ics; showing that the system is robust to changes in pulse neous and delayed insulin responses which, together with width and concentration but sensitive to changes in pulse the hepatic glucose production, have an effect in generating frequency. The model also reveals the interplay between the ultradian regime. It is analysed to show the effect of fast (minute) and slow (minute-hour) negative feedback reduced insulin production or sensitivity on the oscillations loops with maximal information transfer achieved at inter- and several strategies for reinstating its accuracy are de- mediate feedback levels. Consistent with this, experiments scribed through Hopf bifurcation techniques. An analytical revealed that reducing or increasing negative feedback re- description is performed by approximating the centre man- duced information transfer in the system. ifold through the perturbation of periodic solutions. The resulting expressions provide a rational link between all the Margaritis Voliotis physiological parameters and the amplitude and frequency University of Exeter of the periodic solutions. This allows to quantify the con- College of Engineering, Mathematics and Physical tribution of the insulin sensitivity to the oscillatory regime Sciences and hence provide a basis for inverse problems aiming at [email protected] measuring diabetic parameters. Krasimira Tsaneva-Atanasova Benoit Huard University of Exeter Northumbria University [email protected] Department of Mathematics, Physics & Electrical Engineering Craig McArdle [email protected] Laboratories for Integrative Neuroscience and Endocrinology, Maia Angelova, Adam Bridgewater University of Bristol Department of Mathematics and Information Sciences [email protected] Northumbria University [email protected], Amitesh Pratap, Kathryn Garner [email protected] University of Bristol amitesh.pratap@ bristol.ac.uk, [email protected] MS46 An Integrated System Towards Artificial Pancreas MS47 and its Numerical Trials Nonlinear Energy Transfers and Intermittent En- ergy Dissipation in Forced Shear Flows Millions of Americans suffer type 1 diabetes mellitus (T1DM) that is caused by the lack of insulin producing It is believed that the intermittent fluctuations in turbulent pancreatic beta-cells. Exogenous insulin or its analogues shear flows are triggered by the energy transfer to the mean must be daily administered to utilize and lower the chronic flow via nonlinear inertial interactions. However, because high glucose level, ideally, though an artificial pancreas, an of the vast range of active spatial and temporal scales, iden- integrated system consisting of an insulin pump, a glucose tifying the responsible interactions is not straightforward. monitoring system, and closed loop control (CLC) algo- We show that the responsible modes can be formulated as rithms. An effective CLC algorithm is still lacking in han- the (initially unknown) solutions of an appropriate con- dling the delayed effects of insulin in delivery mechanisms, strained variational problem. The variational problem can GMS and the hepatic glucose production (HGP). The tim- be solved at a low computational cost, and the solution is ing discrepancies and dose inaccuracies often cause un- the nontrivial mode with instantaneously maximal trans- desired glucose fluctuations including both hyperglycemia fer of energy to the mean flow. We demonstrate the ap- and dangerous hypoglycemia. Our ultimate goal is to for- plication of this variational method on a direct numerical mulate an integrated system, consisting of several dynam- simulation of the two-dimensional Kolmogorov flow. ical system models, with the aims to develop and validate effective CLC algorithms for artificial pancreas. Mohammad Farazmand School of Physics Jiaxu Li Georgia Institute of Technology University of Louisville [email protected] 182 DS17 Abstracts

Themistoklis Sapsis [email protected] Massachusetts Institute of Techonology [email protected] MS47 Recurrent Flow Analysis in Kolmogorov Flows MS47 Flows over periodic domains which are body forced in Statistics of Spatially-Periodic Turbulence Driven one direction by a steady, trigonometric function of by Steady Forces large, monochromatic wavelength are typically called ‘Kol- mogorov flows’ after Kolmogorov’s (1959) suggestion of a We conduct direct numerical simulations of turbulence 2D version as a simple situation in which to study linear in- sustained by steady spatially-periodic forces under peri- stability. Due to the absence of physical (non-slip) bound- odic boundary conditions in three directions. One of the aries, the flow can be efficiently simulated using spectral most remarkable features of this system is that turbulence methods. These simulations indicate that the flow under- statistics (e.g. the energy dissipation rate) evolve quasi- goes a sequence of bifurcations as the forcing amplitude periodically in time even at high Reynolds numbers. The increases which quickly give way to complex dynamics. In periodicity implies a quasi-periodic sustaining process of this talk I’ll discuss work carried out to probe these com- the turbulence and is explained in term of turbulent energy plex dynamics using a recurrent flow analysis. The fruits cascade events. Another interesting feature is the follow- of this analysis can be used to a posteriori predict cer- ing: when the forcing is sinusoidal with the wave number tain statistics of the dynamics using periodic orbit theory equal to unity, i.e. when the flow is the three-dimensional (Chandler & Kerswell, J. Fluid Mech. 722, 554, 2013, Lu- Kolmogorov flow, the steady laminar solution is linearly cas & Kerswell, Phys. Fluids, 27, 045106, 2015) and un- stable and the onset of turbulence is due to a sub-critical cover key dynamical processes of the flows (Lucas & Ker- transition. We verify that the probability of the survival swell, arxiv:1611.04829). of the transient turbulence is an exponential function of the time in a low-Reynolds-number range. This feature Dan Lucas was previously demonstrated by Linkmann and Morozov DAMTP (2015) with another forcing. It seems that such a behav- Cambridge University ior is universal not only in the wall-bounded turbulence [email protected] but also in turbulence without walls. We also found edge states between the steady flow and the transient turbu- Gary Chandler, Rich Kerswell lence, which might give useful information to understand Department of Mathematics the onset of turbulence in this simple system. University of Bristol, U.K. [email protected], [email protected] Susumu Goto Osaka University [email protected] MS48 Can Inducible Resistance in Plants Cause Her- Lennaert van Veen bivore Aggregations? Spatial Patterns in an In- UOIT ducible Plant/Herbivore Model [email protected] Many theories regarding the evolution of inducible resis- tance have an implicit spatial component, but most rele- MS47 vant population dynamic studies ignore spatial dynamics. We examined a spatially explicit model of plant inducible Localized Turbulence in 2D Kolmogorov Flows resistance and herbivore population dynamics to explore how these influence spatial patterning. Both transient and Spatially-localized turbulent states are mainly observed in persistent spatial patterns developed in all models exam- subcritical wall turbulence such as pipe flow and Couette ined, where patterns manifested as wave-like aggregations flow. We, however, have found localized turbulent states of herbivores and variation in induction levels. Patterns can be isolated also in two-dimensional Kolmogorov flow arose when herbivores moved away from highly induced which is governed by the Navier-Stokes equations with a plants, there was a lag between damage and deployment steady sinusoidal forcing in a doubly-periodic box. We of induced resistance, and the relationship between herbi- will explain the roll of the flow rate in isolating a local- vore density and strength of the induction response had a ized state and controlling their motions and interactions. sigmoid shape. These mechanisms influenced pattern for- Random and/or coherent movements of the center of a lo- mation regardless of the assumed functional relationship calized turbulent state are studied from the dynamical sys- between resistance and herbivore recruitment and mortal- tems point of view. For example, for moderate values of ity. In models where induction affected herbivore mortality, Reynolds number and the flow rate, a localized turbulent large-scale herbivore population cycles driven by the mor- state moves with a constant speed on average and changes tality response often co-occurred with smaller scale spatial its own direction randomly and intermittently. These in- patterns driven by herbivore movement. When the mor- termittent changes of the direction can be understood as tality effect dominated, spatial pattern formation was com- transfer processes among several chaotic attracting sets. pletely replaced by spatially synchronized herbivore popu- lation cycles. Our results present a new type of ecological Yoshiki Hiruta pattern formation driven by induced trait variation, con- Gruduate School of Science sumer behavior, and time delays that has broad implica- Kyoto University tions for the community and evolutionary ecology of plant [email protected] defenses.

Sadayoshi Toh Kurt Anderson Kyoto University University of California Riverside DS17 Abstracts 183

Department of Biology by orders of magnitude, can be detrimental to natural and [email protected] agricultural systems, but the mechanisms driving these outbreaks is unclear. Proposed bottom-up mechanisms Brian Inouye focus on plant responses to herbivory, such as (1) resis- Florida State University tance, where plants produce defensive chemicals that re- Biology Department duce herbivore survival, and (2) compensatory regrowth, [email protected] where plants replace consumed biomass and increase the food available to the herbivore. To quantify their influ- Nora Underwood ence on population dynamics, we analyze discrete-time Florida State University plant-herbivore models that incorporate these two plant [email protected] responses. When plants return to carrying capacity each year, such as in agricultural systems, resistance does not produce outbreaks, but compensatory regrowth produces MS48 outbreaks only when plants overcompensate for herbivory Mathematical Models of RNA Interference in by producing more biomass than was consumed. In sys- Plants tems with both plant responses, compensatory regrowth can dampen outbreaks at high levels of induced resistance In this talk I will discuss a model of plant immune response and produce outbreaks at low levels of induced resistance. with a special emphasis on the role of time delays repre- When plant population sizes can vary across years, such senting maturation time of proliferating tissue and the ac- as in natural systems, plant responses can also cause her- tivation delay of RNA interference. Detailed bifurcation bivore outbreaks that consist of periods of low herbivore analysis of this model will demonstrate how stability and density followed by high density peaks. Effectively using dynamical behaviour is affected by the system parameters plant responses in natural and agricultural systems requires and the time delays [G. Neofytou, Y.N. Kyrychko, K.B. understanding these interactions between plants and herbi- Blyuss, Time-delayed model of immune response in plants, vores to mitigate as opposed to exacerbate pest problems. J. Theor Biol. 389, 28-39 (2016)]. I will also show how RNA interference can mediate interactions between two viruses in the same plant host, allowing for both cross- Christopher Stieha protection and cross-enhancement between viruses [G. Ne- Department of Biology ofytou, Y.N. Kyrychko, K.B. Blyuss, Mathematical model Case Western Reserve University of plant-virus interactions mediated by RNA interference, [email protected] J. Theor Biol. 403, 129-142 (2016).] Brian Lerch Konstantin Blyuss Case Western Reserve University Department of Mathematics [email protected] University of Sussex [email protected] Katja Poveda Cornell University MS48 [email protected] Exploring Fitness Effects of Plant Defenses Against Insect Herbivores Karen Abbott Case Western Reserve University When attacked by insect herbivores, plants emit volatile [email protected] chemicals. These chemicals are known to induce local de- fenses, prime neighboring plants for defense, and attract predators and parasitoids to combat the herbivores but MS49 are coupled with fitness costs. We examine the interac- Analysis of New Walking Droplet Bifurcations tions between the model plant goldenrod and one of its insect herbivores, Trirhabda virgata, in order to explore how plant defense strategies influence both mean fitness Gillet developed a planar, 2-parameter, smooth discrete of a field of goldenrod as well as spatial variation in plant dynamical system model for walking droplet motion of the fitness within the field. form 2 → 2 Karen M. Cumings, Peter R. Kramer fC,μ : R R , Rensselaer Polytechnic Institute Department of Mathematical Sciences where 0

[email protected] Paul Park Northwestern University [email protected] MS49 Topological Bifurcations of Vorticity Zafir Zaman, Mengqi Yu Northwestern University Vortices are the most important coherent structures in fluid Evanston, IL dynamics. A simple way to keep track of the dynamics of zafi[email protected], vortices in 2D is to follow the motion of the critical points [email protected] of the vorticity as they evolve in time. Vortex creation and destruction can be described as bifurcations of these points Paul Umbanhowar with time as the primary bifurcation parameter. We will Northwestern University discuss general equations of motion of the critical points Evanston, IL 60208-3109 obtained from the Navier-Stokes equations, and also con- [email protected] sider some specific flows: The creation of the Karman vor- tex street behind a circular cylinder after the Hopf bifur- Julio M. Ottino cation, and vortex merging between between two initially Northwestern University Gaussian vortices in the core growth model. [email protected] Morten Brons Tech University of Denmark MS49 Department of Mathematics [email protected] Bifurcations in a Soft Billiard Billiard dymaical systems have played a foundational role Morten Andersen, Jesper Schmidt Hansen in statistical mechanics and dynamical systems theory due Department of Science and Environment to their clean definition, rich dynamics and direct applica- Roskilde University tion to many problems in physics. In traditional billiard [email protected], [email protected] models a point particle moves force-free along a geodesic line and specularly reflects from a hard boundary. The Matthias Heil new wrinkle here is a soft boundary. If the boundary is School of Mathematics soft, then the particle will exchange energy with its envi- The University of Manchester ronment on a scale that fits the definition of a small system, [email protected] which puts this soft mechanical nonequilibrium system into the same thermodynamic category as molecular motors and large molecules such as DNA or proteins. Experiments on MS49 a soft triangular billiard can measure the probability dis- Surprisingly Persistent Structures Predicted by tribution function of particle position, from which one can Piecewise Isometries calculate the information entropy as a function of time, energy input (equivalent to temperature) and softness of While chaotic advection has long been studied, mixing the boundary. The approach to equilibrium is very slow, by cutting and shuffling, a different mixing mechanism, exhibiting a stretched exponential form. On raising the is relatively unexplored and imperfectly understood. Un- energy input a soft triangular billiard bifurcates from a like chaotic advection, cutting and shuffling maps do not stationary resting state to bouncing particle motion. How- stretch, possess no positive Lyapunov exponents, and ex- ever, as the enery input decreases, entropy production is hibit no chaotic behavior in the usual sense. Yet they dominated by large and rare fluctuations and the time to can mix quite effectively. Cutting and shuffling can be attain equilibrium diverges. described in terms of the mathematical formalism of Piece- wise Isometries (PWIs) in which an object is divided into Guy Metcalfe a finite number of pieces which are rearranged back into School of Mathematical Sciences the object’s original shape. The trajectories of the cuts Monash University in the limit of infinite iterations forms the exceptional set, [email protected] akin to a stroboscopic map. The complement of the excep- tional set comprises non-mixing regions, similar to regular islands of typical chaotic systems. We apply PWIs to a MS50 hemispherical shell as a zeroth order model of mixing a Nonlinear Energy Transfers in Fluid-Structure In- spherical tumbler half-filled with particles and rotated al- teraction of a Cylinder with An Internal Attach- ◦ ternately by ≤ 90 about two axes. Non-mixing regions in ment the PWI correspond to surprisingly persistent non-mixing elliptic regions and global barriers to mixing occur, both We consider two-dimensional flow past a linearly-sprung similar to structures in experiments (X-ray visualization of cylinder allowed to undergo rectilinear motion normal to a granular flow) and simulations using a continuum model. the mean flow, with an attached “nonlinear energy sink’ This merging of PWIs, dynamical systems, and physical (NES) consisting of a mass allowed to rotate about the applications suggests a novel paradigm for mixing that it cylinder axis, and whose rotational motion is linearly is especially well suited to the study of granular materials. damped by a viscous damper. At Re = 100, the NES- Funded by NSF Grant CMMI-1435065. equipped cylinder undergoes repetitive cycles of slowly de- caying oscillations punctuated by chaotic bursts. During Richard M. Lueptow each slowly decaying cycle, the dynamics is regular and, Northwestern University for large enough values of a mass ratio ε, can lead to sig- Department of Mechanical Engineering nificant vortex elongation with partial stabilization of the [email protected] wake. As ε approaches zero, no such vortex elongation DS17 Abstracts 185

is observed and the wake is similar to that for an NES- [email protected], [email protected] less cylinder. Proper orthogonal decomposition (POD) of the flow field shows that the NES has a drastic effect on the underlying flow structures, imparting significant redis- MS50 tribution of energy among POD modes. We introduce a Tailoring Dispersion Characteristics in Next- quantitative signed measure of the work done by the fluid Generation Metastructures on the cylinder and find that vortex elongation is associ- ated with a sign change of that measure, indicating that a This talk will review our efforts on metastructures that reversal of the direction of energy transfer, with the cylin- employ tailored dispersion characteristics for applications der “leaking energy back’ to the flow, is responsible for ranging from structure-borne wave focusing for enhanced wake elongation. We relate these findings to the mecha- energy harvesting to broadband vibration attenuation us- nism of transient resonance capture into a slow invariant ing locally resonant metamaterials. The two particular ap- manifold of the fluid–structure interaction dynamics. proaches for wave focusing are the use of elastic wave lens and mirror concepts. In the context of wave focusing by dispersion tailoring, we will discuss our computational and Antoine Blanchard experimental results on a Gradient-Index Phononic Crys- University of Illinois - Urbana-Champaign tal Lens (GRIN-PCL) design for dramatically enhanced [email protected] plane wave energy harvesting. The proposed GRIN-PCL is formed by an array of blind holes with different diame- Lawrence Bergman ters on an aluminum plate where the orientation and size of University of Illinois - Urbana - Champaign the blind holes are tailored to obtain a hyperbolic secant [email protected] distribution of refractive index. The design is guided by finite-element simulations of the lowest asymmetric Lamb Alexander Vakakis wave mode dispersion to achieve the desired refractive in- University of Illinois dex profile. We will also summarize our recent efforts [email protected] on elastic wave focusing using structurally-embedded mir- rors through finite-element simulations and experimental validations. Finally, low-frequency dispersion tailoring by means of locally resonant metamaterials will be discussed MS50 for broadband vibration attenuation through bandgap for- mation. Locally-resonant dispersion and bandgap char- Extreme Control of Impulse Transmission by acteristics (especially with changing mass ratio) will also Cylindrical Phononic Crystals be bridged to lens and mirror concepts to enable low- frequency wave focusing.

Here we present a highly tunable periodic device that can Alper Erturk offer two extremes of elastic wave propagation - nearly Georgia Institute of Technology complete transmission and strong attenuation under im- [email protected] pulse excitation. The device is a one-dimensional chain that consists of identical cylinders interacting as per non- linear Hertzs contact law. Under no initial static compres- MS50 sion, the system becomes an essentially non-linear system. Nonlinear Mechanisms of the 2D Energy Redirec- We maintain two different contact angles, which periodi- tion and Absorption in Mechanical Systems Sub- cally vary along the chain, and analyze the resulting dimer ject to the External Loading phononic crystal. We numerically and experimentally show that by choosing the appropriate set of contact angles, the In the present talk I will discuss the recent results concern- impact excitation can either be localized and transmitted ing the basic problem of energy transfer emerging in the with minimum attenuation, or it can be highly dispersed 2D mechanical meta-materials with the internal, resonat- leading to strong attenuation. Moreover, we use asymp- ing inclusions. This study is mainly devoted to the analysis totic analysis and show that there can be countable in- of special response regimes manifested by the 2D resonant finity of such contact angles. We close the discussion by energy transfer induced by the various types of external highlighting the key characteristics of the mechanisms that forcing. facilitate strong attenuation of incident impulse. These in- clude low-frequency to high-frequency (LF-HF) scattering, Yuli Starosvetsky and turbulence-like cascading in this ordered system. We Technion, Israel Institute of Technology thus envision that these adaptive, cylinder based Phononic [email protected] crystals, in conjunction with conventional impact mitiga- tion mechanisms, could be used to design highly tunable and efficient impact manipulation devices. MS51 Globally Attracting Synchrony in a Network of Os- cillators with Strong Inhibitory Pulse Coupling Rajesh Chaunsali University of Washington, Seattle The synchronization tendencies of networks of oscillators [email protected] have been studied intensely in the context of fireflies, car- diac cells, Josephson junctions, laser arrays, chemical os- Eunho Kim cillators, hybrid dynamical systems, pulse-coupled sensor University of Washington networks and neutrino flavor oscillations. We assume 1) all- [email protected] to-all pulse-coupled oscillators, 2) the effect of a pulse is in- dependent of the number of oscillators that simultaneously Matthew Toles, Jinkyu Yang emit a pulse, 3)the phase resetting is a phase delay that University of Washington, Seattle is a monotonically increasing with slope everywhere less 186 DS17 Abstracts

than one, (which implies the phase resetting has a strongly and recovery of sleepiness occurs sufficiently quickly, two destabilizing discontinuity at the threshold for pulse emis- sleep cycles per day, a nap as well as nighttime sleep, may sion which is strongly destabilizing and 4) the normalized occur. To investigate the transition between one and two delay (the phase resetting) lies everywhere to the left of the sleep cycles per day, we analyzed bifurcations in a model line y =2φ − 1, where φ is the phase. We then prove that for human sleep/wake dynamics as the time constants re- synchrony is globally attracting for small conduction de- lated to the build up and recovery of sleepiness are de- lays. Absorptions are allowed, therefore clusters can form. creased. We found that the system exhibits an incremen- The requirement for the slope to be everywhere less than tal increase in the number of sleep cycles per day. Using one precludes order switching, so the firing order, once es- a one-dimensional map to represent the dynamics of the tablished, must remain constant. Therefore the only pos- system, we relate this map to a normal form for a piece- sible solutions are globally synchrony and cluster solutions wise continuous system which undergoes a border collision with a fixed firing order. The strategy is to prove the for- bifurcation, and we provide numerical evidence for period- mer stable and all other possible attractors to be unstable. adding behavior. This analysis has implications for under- We then extend these results to sparse networks. standing the dynamics of the transition from napping to non-napping behavior in early childhood. Carmen Canavier Louisiana State University Health Sciences Center Cecilia Diniz Behn, Kelsey Kalmbach [email protected] Colorado School of Mines Applied Math & Statistics Ruben Tikidji-Hamburyan [email protected], [email protected] George Washington University [email protected] Victoria Booth University of Michigan [email protected] MS51 Entrainment Maps: A New Tool for Understanding Properties of Circadian Oscillators MS51 Farey sequences in Periodically Forced 2- Entrainment implies that an endogenous oscillator has Dimensional Integrate-and-Fire Models matched its period to that of an external periodic forcing and has established a stable phase relationship with the In this talk we consider general two-dimensional integrate- forcing signal. The process of circadian entrainment has and-fire models with a subthreshold attracting equilibrium been studied extensively using tools from oscillator theory, and driven by a periodic forcing. Instead of considering in particular phase response curves (PRCs) that measure the firing-map, we use the stroboscopic one, which is the the change in the phase of an endogenous limit cycle oscil- usual tool for periodic smooth systems. It becomes a planar lation (typically in constant darkness or DD) induced by a discontinuous map defined in different partitions given by perturbation (typically a light pulse) as a function of the the number of spikes performed at each iteration. Periodic phase at which the perturbation is applied. Such PRCs orbits may bifurcate via tangent grazing or non-smooth can be constructed for light pulses of arbitrary strength grazing. We show that, in the latter case, the map may and duration. However, for a PRC to accurately predict become a quasi contraction, and hence possesses periodic properties of entrainment to periodic light pulses, the per- orbits whose symbolic dynamics are in the Farey tree. We turbations must be weak or brief enough that the oscilla- also show that, when periodic orbits bifurcate close to a tor would relax back to the DD limit cycle attractor before tangency, bistability may occur. the next pulse arrives. We show that PRCs do not accu- rately predict the phase of entrainment in a model of the Albert Granados Drosophila circadian clock subjected to photoperiods with COMPUTE Department substantial amounts of both light and dark, such as 12:12 Technical University of Denmark light:dark (LD) cycles. In this talk, we introduce entrain- [email protected] ment maps, which are one-dimensional maps that are not based on perturbing the DD oscillator and thus are able to Gemma Huguet accurately predict the phase of entrainment for any pho- Universitat Politecnica de Catalunya toperiod. [email protected]

Casey Diekman Department of Mathematical Sciences MS52 New Jersey Institute of Technology Gene Expression Dynamics with Stochastic Bursts: [email protected] Exact Results for a Coarse-Grained Model

Amitabha Bose We describe a theoretical framework to analyze the dy- New Jersey Institute of Technology namics of gene expression with stochastic bursts. Begin- [email protected] ning with an individual-based model which fully accounts for the messenger RNA (mRNA) and protein populations, we propose an expansion of the master equation for the MS51 joint process. The resulting coarse-grained model is a re- A Map-Based Approach to Understanding Circa- duced system describing only the protein population while dian Modulation of Sleep fully accounting for the effects of discrete and fluctuating mRNA population. Closed form expressions for the sta- A homeostatic need for sleep increases with time awake and tionary distribution of the protein population and mean decreases during sleep. In typical adult human parameter first-passage times of the coarse-grained model are derived. regimes, this homeostatic sleep drive produces one night- Large-scale Monte Carlo simulations show that the analysis time sleep episode per day. However, when the build up accurately describes the individual-based process account- DS17 Abstracts 187

ing for mRNA population, in contrast to the failure of com- computing the optimal trajectory for genetic evolution tra- monly proposed diffusion-type models. This is joint work jectories leading to the emergence of a-typical bacterial with Yen Ting Lin. genotypes. In particular, we consider the stochastic dy- namics of histograms of bacterial populations with n geno- Charles R. Doering types in a given fitness landscape. We consider radical University of Michigan shifts in the genetic composition of large cell populations Mathematics, Physics and Complex Systems and use the large-deviations approach to derive explicit [email protected] cost function for paths connecting the initial and the final histograms. We then demonstrate that the minimizer of this cost function is the most likely evolutionary trajectory MS52 leading to the emergence of the final histogram of bacte- Deterministic and Stochastic Effects in the Assem- rial genotypes. We also develop a backward shooting nu- bly of Multi-species Microbial Communities merical algorithm for effectively computing the most likely population trajectory linking any two successive genotype Microbial communities play an essential role in determining fixations (histograms) and to estimate the probabilities of both human health as well as the health of the planet. Over these long term transitions. Numerical examples illustrat- the last decade tremendous progress has been made in char- ing the approach will also be presented. acterizing these complex microbial communities, but the lack of experimentally tractable model systems has made Ilya Timofeev it difficult to discern the rules governing microbial com- University of Houston munity assembly and function. In this talk I will describe [email protected] our recent experimental efforts to develop a bottom-up ap- proach to community assembly. We have measured all pair- Robert Azencott wise competitive outcomes among 20 isolates from a single Department of Mathematics grain of soil, where we find a remarkably hierarchical struc- University of Houston ture with no rock-paper-scissors interactions. We find that [email protected] simple assembly rules incorporating pairwise competitive outcomes are surprisingly successful in predicting the out- come of multi-species competition. Finally, we have begun Brett Geiger to characterize community assembly within the gut of the University of Houston worm C. elegans, where we find that stochastic coloniza- [email protected] tion can drive strong heterogeneity between the microbial communities in different worms. MS53 Jeff Gore Canards in Stiction: On Solutions of a Friction Os- Massachusetts Institute of Technology cillator by Regularization [email protected] We consider the problem of the friction oscillator using the stiction model of friction. This friction law has a discon- MS52 tinuity between the dynamic and the static regime. The Large Deviations for Gaussian Processes with De- discontinuity set has a sticking region in which the for- lay ward solution is non-unique. In particular, there are special points along these segments where the solution is tangent Dynamical systems driven by nonlinear delay SDEs with to the boundary of the discontinuity set. In order to re- small noise can exhibit important rare events on long solve this uncertainty, we introduce a regularization of the timescales. When there is no delay, classical large devi- vector field and we obtain a multiple-time scale problem. ations theory quantifies rare events such as escapes from Here the special points of the piecewise-smooth problem metastable fixed points. Near such fixed points one can become folded saddles and a canard solution appears. We approximate nonlinear delay SDEs by linear delay SDEs. study the interaction of periodic orbits with the canard and Here, we develop a fully explicit large deviations framework we find that the the regularized problem has solutions that for (necessarily Gaussian) processes Xt driven by linear de- do not appear in the original problem. lay SDEs with small diffusion coefficients. Our approach enables fast numerical computation of the action functional Elena Bossolini controlling rare events for Xt and of the most likely paths Technical University of Denmark transiting from X0 = p to XT = q. Via linear noise local Department of Applied Mathematics and Computer approximations, we can then compute most likely routes Science of escape from metastable states for nonlinear delay SDEs. [email protected] We apply our methodology to the detailed dynamics of a genetic regulatory circuit, namely the co-repressive toggle Morten Brons switch, which may be described by a nonlinear chemical Tech University of Denmark Langevin SDE with delay. Department of Mathematics [email protected] William Ott University of Houston [email protected] Kristian U. Kristiansen Department of Mathmatics Technical University of Denmark MS52 [email protected] Rare Events Reconstruction of Most Likely Evolu- tionary Paths for Bacterial Populations MS53 In this talk we present the large-deviation approach for Overview of Models for Impact and Friction - 188 DS17 Abstracts

Oblique and Spatially Extended Systems deformation of the system from a linear regime where the slow-fast splitting is known exactly, and the full nonlin- This talk will give an overview of recent work at Bristol and ear regime. We show that, providing the ramp function elsewhere on the dynamical systems aspects of the mechan- which defines the homotopy is of Gevrey class 2 and satis- ics of frictional contact. The talk shall cover the strange fies vanishing conditions to all orders at the temporal end dynamics of ”rate and state” friction laws and why these points, the solution of the optimal balance boundary value are better experimentally motivated than pure Coulomb problem yields a point on the approximate slow manifold and related friction models. This is shown to reduce the that is exponentially close to the approximation to the slow non-smooth character of dynamics with friction to that manifold via exponential asymptotics. We also give a nu- of singularly perturbed systems. These results are illus- merical demonstration of the efficacy of optimal balance, trated using the classical sliding block model. It is then showing the dependence of accuracy on the ramp time and shown how things are more subtle if contact is not always the ramp function. This is joint work with G.A. Gottwald maintained. An overview of some recent work on the so- and H. Mohamad. called Painlev´e paradox is given in which there is a cou- pling between tangential and normal degrees of freedom. Marcel Oliver It is shown how to unfold nondeterminism due to so-called Jacobs University Bremen dynamic jam, and reverse chatter using specialized desin- [email protected] gularization methods. Finally the rate and state model is applied to a problem with continuous line contact. Phase plane analysis can then be used to capture both slip and MS54 stick transition waves and pulses as traveling fronts and A Quasi-Lagrangian Approach to Scalar Transport waves, via phase plane analysis. in Complex Flows

Alan R. Champneys, Thbaut Putelat Dynamical systems-based Lagrangian methods can be used University of Bristol to identify key material boundaries and track transports in [email protected], [email protected] ocean and atmospheric flows with coherent structures. At the ocean surface, the velocity field used to compute fluid trajectories is often based on satellite altimetry, which typ- MS53 ically resolves the geostrophically-balanced mesoscale flow. On the Regularization of Impact Without Collision: Contemporary models and observations are beginning to The Painlev´e Paradox and Compliance resolve the sub-mesoscale, which is typically unbalanced and which introduces a degree of complexity that can ren- Abstract Not Available At Time Of Publication. der standard methods cumbersome. In addition, oceanog- raphers are often more interested in the fluxes of scalars S. John Hogan such as heat, vorticity, salt and other biological and geo- University of Bristol chemical tracers than they are in the flux of fluid mate- [email protected] rial. We suggest a new approach that reduces complex- ity through time filtering and that directly addresses non- Kristian U. Kristiansen material, scalar fluxes. The approach is quasi-Lagrangian Department of Mathmatics insofar as it contemplates trajectories of a velocity field re- Technical University of Denmark lated to a scalar flux, usually not the fluid velocity. Two [email protected] examples are explored, the first coming from a canonical example of viscous adjustment along a flat plate and the second from a numerical simulation of a turbulent Antarc- MS53 tic Circumpolar Current in an idealized geometry. Each Dynamic Jamming Singularity of Mechanical Sys- example concentrates on the transport of dynamically rel- tems with Muliple Point Contacts evant scalars, and the second illustrates how substantial material exchange across a baroclinically unstable jet co- Abstract Not Available At Time Of Publication. exists with zero residual buoyancy flux. Peter L. Varkonyi Larry Pratt Budapest University of Technology and Economics Woods Hole Oceanographic Inst. [email protected] [email protected]

MS54 Roy Barkan Optimal Balance via Adiabatic Invariance of Ap- Dept. of Atmospheric and Oceanic Sciences, UCLA, Los proximate Slow Manifolds Angeles [email protected] We analyze the method of optimal balance which was in- troduced by Vi´udez and Dritschel to provide balanced ini- Irina Rypina tializations for two-dimensional and three-dimensional geo- Woods Hole Oceanographic Institution physical flows, here in the simpler context of a finite dimen- [email protected] sional Hamiltonian two-scale system with strong gyroscopic forces. It is well known that when the potential is analytic, such systems have an approximate slow manifold that is de- MS54 fined up to terms that are exponentially small with respect Multiscale Asymptotics for the Madden-Julian Os- to the scale separation parameter. The method of opti- cillation and Tropical-extratropical Interactions mal balance relies on the observation that the approximate slow manifold remains an adiabatic invariant under slow The Madden-Julian Oscillation (MJO) is a planetary-scale deformations of the nonlinear interactions. The method is wave envelope of tropical clouds and precipitation, with formulated as a boundary value problem for a homotopic modulations of hurricanes and tropical cyclones, and with DS17 Abstracts 189

contributions to active and break phases of monsoons. Saulo Orizaga With a wavelength of 10,000-40,000 km and an oscillation University of Arizona period of 30-60 days, it stands at the intersection of weather [email protected] and climate, and its prediction is a major challenge at the frontier of forecasting on weekly and monthly time scales. Multiscale asymptotic models will be presented for inter- MS55 actions between the MJO and extratropical waves. Such Quenched Dynamics in the Symmetric Multicom- interactions include both (i) how the MJO can affect mid- ponent Fch latitude weather and climate, and, in turn, (ii) how extra- tropical waves can influence the initiation and termination Lipid bilayers are typically composed of many species of of MJO events. The asymptotic models are derived from lipids with distinct intrinsic curvatures. We propose a the nonlinear PDEs for the fluid dynamics of the atmo- model for multicomponent lipid bilayers of distinct aspect sphere and the skeleton of the MJO, including the impor- ratio and derive a curvature driven flow the exhibits a tant effects of moisture and convection. quenching phenomena in which the phase separation of lipids along the interface competes temporally with the re- Shengqian Chen laxation of lipid density in the far field. The first process to University of Wisconsin, Madison reach equilibrium quenches the flow in the other process at [email protected] leading order, leading to a degenerate flow at higher order. We present an analysis of the model. Andrew Majda Courant Institute NYU Keith Promislow, Qiliang Wu [email protected] Michigan State University [email protected], [email protected] Samuel Stechmann University of Wisconsin - Madison MS55 [email protected] Self-Organized Patterns in Networks of Electro- chemical Reactions MS54 Experiments with networks of discrete reactive bistable Nonuniqueness of Weak Solutions to the SQG electrochemical elements, organized in regular and non- Equation regular tree networks, are presented to confirm an al- ternative to the Turing mechanism for formation of self- We prove that weak solutions of the inviscid SQG equations organized stationary patterns. The results show that the are not unique, thereby answering an open problem posed pattern formation can be described by identification of do- by De Lellis and Szekelyhidi Jr. Moreover, we show that mains that can be activated individually or in combina- weak solutions of the dissipative SQG equation are not tions. The method was also demonstrated to achieve local- unique, even if the fractional dissipation is stronger than ization of chemical reactions to network substructures and the square root of the Laplacian. This talk is based on a identification of critical sites whose activation results in joint work with T. Buckmaster and S. Shkoller. complete activation of the system. The experiments were performed with a nickel electrodissolution system. Each Vlad C. Vicol unit in the network represented a corroding metal (nickel) Princeton University wire that accommodated a complex reaction system with Department of Mathematics bistable behavior. Coupling was established in the form [email protected] of the charge flow between the wires (due to difference in electrode potential) which affected the rate of metal disso- lution of the coupled electrodes. The experiments repro- MS55 duce all the salient dynamical behavior of a general network Self-assembled Structures in Copolymer-solvent model with a single nonlinearity parameter describing the Mixtures bistability. We also show that similar pattern formation can occur in star networks. The results are relevant for Block copolymers are among the best candidate materi- large random networks that possess locally the tree or star als for synthetic nanoscale self-assembly. The interplay structure with relatively short-range effective interactions between microscopic phase segregation of polymer con- among the elements. stituents and macrophase segregation between polymer and solvent produces a huge variety of morphologies. We Michael Sebek employ a density functional model for mixtures of copoly- Saint Louis University mer and solvent and its associated free boundary problem Department of Chemistry reduction. The simplest types of multilayered equilibria, [email protected] those with translational symmetry (i.e. bilayers, multilay- ers) and radial symmetry (i.e. cylindrical and spherical Istvan Z. Kiss micelles and vesicles), are considered in the context of this Department of Chemistry model. Their existence and stability will be discussed, as Saint Louis University well as hybrid numerical/analytical approaches for their [email protected] construction. In addition, possible scenarios for nonlinear evolution will be illustrated. MS55 Karl Glasner Dynamics and Bifurcations in a Model for Organic The University of Arizona Photovoltaic Cells Department of Mathematics [email protected] Organic photovoltaic (OPV) devices focus much atten- 190 DS17 Abstracts

tion due to their potential to provide economically viable Massachusetts Institute of Technology sources of cheap, lightweight, renewable energy alterna- [email protected] tives. One of the critical components of OPVs is the active layer labyrinthine-type donor/acceptor morphology, which allows relatively high efficiencies by increasing the MS56 surface area for charge generation. Nevertheless, theoret- A Geometric Approach to Dynamical Multiscale ical studies of charge separation and transfer dynamics in Model-Order Reduction OPVs thus far, had neglected the possible morphological evolution during operation. Consequently, a novel mean Any model order reduced dynamical system that evolves a field model is being developed to incorporate the feedbacks modal decomposition to approximate the discretized solu- between donor/acceptor morphologies and spatiotempo- tion of a stochastic PDE can be related to a vector field ral charge properties, i.e., a nonlocal far from equilibrium tangent to the manifold of reduced rank matrices. The theory. Specifically, the focus is on the bifurcations that Dynamically Orthogonal (DO) equations are the canoni- govern the interfacial donor/acceptor properties and rich cal reduced order model for which the corresponding vec- multi-phase structures, i.e., generation and sensitivity of tor field is the orthogonal projection of the original sys- the labyrinthine morphology. The approach is expected to tem dynamics onto the tangent spaces of this manifold. provide a systematic understanding of OPV operation, and The embedded geometry of the fixed rank matrix mani- in particular to shed new light on degradation mechanisms fold is analyzed. Differentiability results for projections related to morphological instabilities. onto embedded manifolds are utilized to obtain the dif- ferential of several truncated decompositions including the Alon Z. Shapira Singular Value Decomposition (SVD). The DO equations Ben-Gurion University of the Negev are shown to be the dynamical system that applies instan- [email protected] taneously the SVD truncation to optimally constrain the reduced solution. The geometric analysis is also used to Nir Gavish discuss truncation errors and provide improved numerical Department of Mathematics time-integration schemes. Technion ITT [email protected] Pierre F. Lermusiaux, Florian Feppon MIT [email protected], [email protected] Arik Yochelis Jacob Blaustein Institutes for Desert Research, Ben-Gurion University MS56 [email protected] Model Order Reduction for Stochastic Dynamical Systems with Continuous Symmetries MS56 Stochastic dynamical systems with continuous symmetries Non-Intrusive Data Driven Reduced-Order Model- arise commonly in nature and can display localized co- ing Via Loewner Framework herent structures, from hurricanes in the atmosphere to rogues waves at the ocean surface or vortices in the wake We presents a data-driven nonintrusive model reduction of marine mammals. Yet, because of their random loca- approach for large-scale systems with linear state depen- tions, these structures are not well captured by current dence. Traditionally, model reduction is performed in an order reduction techniques and a large number of modes intrusive projection-based framework, where the opera- is typically necessary for an accurate solution. In this pre- tors of the full model are required either explicitly in an sentation, we introduce a new framework for efficient or- assembled form or implicitly through a routine that re- der reduction of such systems by combining (i) symmetry turns the action of the operators on a vector. Our nonin- reduction tools from deterministic dynamical systems the- trusive approach constructs reduced models directly from ory with (ii) dimensionality reduction techniques from the trajectories of the inputs and outputs of the full model, uncertainty quantification literature. We demonstrate the without requiring the full-model operators. These trajec- performance of our mixed symmetry-dimensionality reduc- tories are generated by running a simulation of the full tion framework on stochastic solutions of the KdV and 2D model; the method then infers frequency-response data Navier-Stokes equations. from these simulated time-domain trajectories and uses the data-driven Loewner framework to derive a reduced model. Saviz Mowlavi Only a single time-domain simulation is required to derive Massachusetts Institute of Technology a reduced model with the new data-driven nonintrusive [email protected] approach. We demonstrate the propose methodology on benchmark examples and a finite element model of a can- Themistoklis Sapsis tilever beam. Massachusetts Institute of Techonology [email protected] Benjamin Peherstorfer ACDL, Department of Aeronautics & Astronautics Massachusetts Institute of Technology MS56 [email protected] Data-driven Probability Density Function Equa- tions for High-dimensional Stochastic Dynamical Serkan Gugercin Systems Virginia Tech Department of Mathematics We present a new data-driven method to compute the prob- [email protected] ability density function (PDF) of a quantity of interest in high-dimensional stochastic dynamical systems. The key Karen E. Willcox idea is to combine information from sample trajectories DS17 Abstracts 191

with the exact evolution equation that governs the PDF of [email protected] the quantity of interest. Such equation can be derived in rather general cases, e.g., by using the Lundgren-Monin- Novikov (LMN) approach (infinite-dimensional dynam- MS57 ical systems) or the Bogoliubov-Born-Green-Kirkwood- Assessing the Observability of Complex Networks: Yvon (BBGKY) hierarchy of classical statistical mechanics A Nonlinear Theory (finite-dimensional dynamical systems). The closure ap- proximation relies on estimating suitable conditional av- Investigating the dynamics resulting from coupled oscilla- erages from sample trajectories. The effectiveness of pro- tors whose frequencies are close is not new and was in- posed method will be demonstrated through various exam- vestigated in electronics for developing the wireless teleg- ples involving high-dimensional stochastic dynamical sys- raphy. With the emergence of chaotic systems and, more tems. recently, of complex networks, the problem of synchroniz- ing (network of) coupled oscillators became widely studied due to the important applications it has not only in life Daniele Venturi and environmental sciences but also in power grids or so- Department of Applied Mathematics and Statistics cial networks. As soon as a network is considered, the University of California Santa Cruz problem of the number of variables to measure required [email protected] for its study is crucial because it is very often not pos- sible to measure all of them. How to choose them is an important challenge which may depend on the task. We will introduce i) a nonlinear theory to obtain a full observ- MS57 ability of the network dynamics as required for instance for getting a reliable and robust model and/or character- Coupling Optimization Between Dynamical Units izing of its dynamics: a very challenging 13-dimensional Using Ansatz Library and Global Modelling of rational dynamical network will be explicitly treated with Complex Network an analytical validation of our results. ii) A procedure to choose a priori (before any attempt of coupling them) the When dynamical systems (units) are coupled to a complex best variable for full synchronizing (nearly) identical oscil- network several challenging problems, such as finding the lators thus constituting a network will be outlined. We will optimized coupling or obtaining a global model from a lim- show with few examples that the variables required for a ited set of measurements, need to be solved. The Ansatz full synchronizability are not necessarily the same as the Library is a list of transformations applied to a nonlinear ones for a full observability. How observability can be used dynamical system - the unit of the network - into a stan- in oncology will be briefly presented. dard form built from one of its variables and its successive Christophe Letellier derivatives. It can can be used to i) identify a class of non- Coria linear dynamical systems that are dynamically invariant, Universit´edeRouen ii) identify the structure of the parameter space, iii) deter- [email protected] mine how to couple single dynamical systems, and iv) to obtain from one variable a global model expressed in terms of the variables spanning the original state space. Here MS57 we investigated the case of the Hindmarsh-Rose system, a simplified neuronal model, considered as the dynamical Using Delay Coordinates for Quantifying Estima- unit. When two Hindmarsh-Rose systems are coupled, the bility of Model Parameters and State Variables Ansatz Library can be used to show a drop in the dynam- from Observed Time Series ical complexity of the resulting network, especially when In data-driven system identification, values of parameters the two single units have (slightly) different time scales. and not observed variables of a given model of a dynam- The complexity drops further with every added dynamical ical system are estimated from measured time series. We unit. This transition between scales is an analogous to a address the question of estimability and redundancy of pa- transition from microscopic to macroscopic behavior. All rameters and variables, that is, whether unique results can these analytical results are then compared to digital as well be expected for the estimates or whether, for example, dif- as analog simulations. ferent combinations of parameter values would provide the same measured output. To answer these questions we ana- Claudia Lainscsek lyze the null space of the linearized delay coordinates map Salk Institute for Biological Studies [J. Schumann-Bischoff et al., Phys. Rev. E 94, 032221 La Jolla, CA (2016)]. This approach to estimability analysis can be gen- [email protected] eralized to multivariate time series and spatially extended systems or dynamical networks.

Mark Spano Ulrich Parlitz, Jan Schumann-Bischoff, Stefan Luther School of Biological and Health Systems Engineering Max Planck Institute for Dynamics and Self-Organization Arizona State University Research Group Biomedical Physics [email protected] [email protected], jan.schumann- bischoff@ds.mpg.de, [email protected] Christophe Letellier Coria Universit´edeRouen MS57 [email protected] Inference of Dynamics from Network Observations

Terrence Sejnowski Networks of dynamical systems are common in applica- Salk Institute for Biological Studies tions, but rarely can all nodes be observed directly. It 192 DS17 Abstracts

is known that strongly connected networks can be recon- [email protected], [email protected] structed, in theory, from any node. We examine practical limitations on the reconstruction, and propose quantitative measures on inference across networks. MS58 Traveling Waves in Diatomic Fermi-Pasta-Ulam- Timothy Sauer Tsingou Lattices George Mason University, USA - Physica D, Elsevier [email protected] We consider the problem of traveling waves in a diatomic Fermi-Pasta-Ulam-Tsingou lattice. We present recent re- sults on the existence of periodic traveling wave solutions MS58 and the construction of nonlocal solitary wave solutions involving these periodic waves. In particular, we consider Wave Propagation in Models of mRNA Localiza- lattices with higher order terms in their nonlinear spring tion forces, which impose a technically complicated, but concep- tually straightforward, reliance on composition operators. Messenger RNA (mRNA) localization is essential during the development of frog egg cells into embryos. This accu- Timothy E. Faver mulation of RNA at the cell periphery is not well under- Drexel University stood, but is thought to depend on diffusion, bidirectional [email protected] movement and anchoring mechanisms. We test these pro- posed mechanisms using linear and nonlinear PDE models Doug Wright and analysis, informed by numerical parameter estimation. Drexel University Our results yield spreading Gaussian solutions for mRNA Mathematics concentrations and confirm the hypothesis of bidirectional [email protected] transport in localization.

Veronica M. Ciocanel MS58 Brown University veronica [email protected] The Maslov Index and the Spectra of Second Order Elliptic Operators Bjorn Sandstede Division of Applied Mathematics In this talk I will discuss a formula relating the spectral Brown University flow of the one-parameter families of second order elliptic bjorn [email protected] operators to the Maslov index, the topological invariant counting the signed number of conjugate points of certain paths of Lagrangian planes. In addition, I will present Kimberly Mowry formulas expressing the Morse index, the number of neg- Brown University ative eigenvalues, in terms of the Maslov index for sev- kimberly [email protected] eral classes of the second order operators: the θ −periodic Schr¨odinger operators on a period cell Q ⊂ Rn, the elliptic operators with Robin-type boundary conditions, and the MS58 abstract self-adjoint extensions of the Schr¨odinger opera- Nondegeneracy of Antiperiodic Standing Waves for tors on star-shaped domains. This is joint work with Y. Fractional Nonlinear Schr¨odinger Equations Latushkin.

In the stability and blowup analyses for traveling and Selim Sukhtaiev standing waves in nonlinear Hamiltonian dispersive equa- Department of mathematics tions, the nondegeneracy of the linearization about such University of Missouri waves is of paramount importance. That is, one must verify [email protected] that the kernel of the second variation of the Hamiltonian is generated by the continuous symmetries of the PDE. The proof of this property can be far from trivial, especially MS59 when the dispersion admits a nonlocal description where Inertial Particles in Turbulence: Long Transients shooting arguments, Sturm-Liouville theories, and other Due to the History Force ODE methods may not be applicable. In this talk, we dis- cuss the nondegeneracy of the linearization associated with The motion of small inertial particles through a fluid is antiperiodic constrained energy minimizers in a class of de- influenced by different hydrodynamic forces, described by focusing NLS equations having fractional dispersion. Key the Maxey-Riley equations. One of these forces is the Bas- to our analysis is the development of ground state and os- set force — an integral over the particle’s history. We cillation theories for linear periodic Schr¨odinger operators study the effects of the history force for nearly neutrally with antiperiodic boundary conditions. The antiperiodic buoyant particles, exemplified by marine snow, advected by nature of the problem greatly complicates the analysis, as three-dimensional turbulent flow in the presence of grav- linear Schr¨odinger operators with periodic potentials need ity. We find that the presence of the history force in this not have simple antiperiodic ground states even in the clas- system leads to individual trajectories that strongly de- sical (local) case. As an application, we obtain the nonlin- viate from what is observed without memory. However, ear (orbital) stability of antiperiodic standing waves with the main effect of the Basset force is an extraordinarily respect to antiperiodic perturbations. slow convergence towards attractors. For our system with light aerosols we observe a power-law-type (proportional Kyle Claassen, Mathew Johnson to t−1/2) convergence to an asymptotic settling velocity of University of Kansas the center of mass of the particle ensemble, which is found DS17 Abstracts 193

numerically to be the settling velocity in a still fluid. spontaneously decays on a characteristic transient lifetime. This lifetime sharply increases with Reynolds number. We Ksenia Guseva relate the transient dynamics to structures in the state University of Oldenburg, Theoretical Physics/Complex space of plane Couette flow. Based on the numerical anal- Systems ysis of decaying trajectories we discuss the emergence of Oldenburg, Germany the turbulence supporting chaotic saddle and the role of [email protected] the edge-of-chaos during decay. Following the evolution of the chaotic saddle as a function of Reynolds number, Anton Daitche we present a mechanism by which characteristic lifetimes University of Munster increase. Germany [email protected] Tobias M. Schneider Emergent Complexity in Physical Systems Laboratory (ECPS) Ulrike Feudel Ecole Polytechnique Federale de Lausanne University of Oldenburg tobias.schneider@epfl.ch ICBM, Theoretical Physics/Complex Systems [email protected] MS59 Tamas Tel Probing the Edge of Chaos in Fluid Dynamics E¨otv¨os Lor´and University, Budapest, Hungary Institute for Theoretical Physics We investigate the geometry of the edge of chaos and how [email protected] the shape of the edge of chaos changes with increasing Reynolds number. The edge of chaos is the set of basin boundary points that are most immediately accessible from MS59 inside a basin boundary. We will also discuss ways in which Melancholia States in the Climate System a basin can suddenly be destroyed by small perturbations.

Multistability is a ubiquitous feature in systems of geophys- James A. Yorke ical relevance and provides key challenges for our ability to University of Maryland predict a system’s response to perturbations. Near critical Departments of Math and Physics and IPST transitions small causes can lead to large effects and - for all [email protected] practical purposes - irreversible changes in the properties of the system. The Earth climate is multistable: present astronomical/astrophysical conditions support two stable MS60 regimes, the warm climate we live in, and a snowball cli- Extreme Growth of Enstrophy in Incompressible mate. Following an idea developed by Eckhardt and co. for Flows the investigation of multistable turbulent fluids, we study the global instability giving rise to the snowball/warm cli- By solving suitable PDE-constrained optimization prob- mate multistability by identifying the edge state, a saddle lems, we assess the sharpness of fundamental analytic es- embedded in the boundary between the two basins of at- timates for the instantaneous rate of growth of certain traction. The edge state attracts initial conditions belong- Sobolev norms arising in the context of incompressible ing to such a boundary and, while being defined using de- flows defined on two or three dimensional domains without terministic dynamics, is the gate facilitating noise-induced boundaries. In the case of two-dimensional (2D) flows, the transitions between competing attractors. We use a cli- analytic estimates are found to be saturated by families of localized vortex fields parametrized by their H−1 and mate model constructed by coupling a primitive equations 1 model of the atmosphere with a simple diffusive ocean. H norms, providing evidence for the sharpness of the es- We refer to the climatic edge states as Melancholia states timates. These instantaneously optimal vorticity fields are found to also saturate estimates for the finite-time growth and provide an extensive analysis of their features, relating 1 thermodynamical properties to their dynamics and classi- of their H norm under 2D incompressible Navier-Stokes fying them according to their symmetry. We discover cases dynamics. On the other hand, the corresponding estimate for three-dimensional (3D) flows is saturated by a family of where the Melancholia state has chaotic dynamics. We also 0 identify a new stable climatic state characterized by non- localized vortex rings parametrized only by their H norm, trivial symmetry properties. while other families of locally optimal fields are studied. Possible connections between these instantaneously opti- Valerio Lucarini mal fields and the question about finite-time singularity University of Hamburg formation are discussed. Meteorological Institute Diego Ayala [email protected] Dept of Mathematics and Statistics McMaster University Tamas Bodai [email protected] Meteorological Institute University of Hamburg [email protected] MS60 Computer Assisted Proof for the Navier-Stokes Equations: Existence of Periodic Orbits in a MS59 Taylor-Green Flow State Space Structures of Decaying Shear Turbu- lence The last decades saw the emergence and development of several techniques aiming at rigorously validating, a poste- In linearly stable shear flows on a finite domain, turbulence riori, the outputs of numerical simulations. Those methods 194 DS17 Abstracts

(sometimes referred to as validated numerics or rigorous Navier-Stokes turbulence. computations) were already successfully applied to a wide variety of problems, for instance to the study steady states, Brendan Murray, Miguel Bustamante periodic orbits or traveling waves of specific PDEs. How- University College Dublin, Ireland ever, while the field of scientific computing has developed [email protected], many very efficient algorithms to numerically study highly [email protected] complex problems, validated numerics requires somewhat less efficient computations. Therefore we cannot hope to Michele Buzzcotti, Luca Biferale rigorously prove the most complicated solutions that can University of Rome Tor Vergata and INFN be numerically computed. Still, the refinement of the vali- [email protected], [email protected] dated numerics techniques over the last few years, and the ever growing computational power available, make it so we can now start to tackle more and more complex prob- MS60 lems, such as the Navier-Stokes equations. In this talk, The Onset of Turbulence in Large Eddy Simulation I will present more precisely one of those rigorous vali- of Box Turbulence dation methods, based on a computable kind of Newton- Kantorovich argument. I will try to emphasize its scope Over the past three decades, the study of turbulence by of application in fluid dynamics, and show how it can be means of computational dynamical systems theory has used to prove the existence of periodic orbits for the Navier- gathered ample momentum. One central goal is to find Stokes equations with a given Taylor-Green forcing term. simple invariant solutions in Navier-Stokes flow in terms of which turbulence can be decomposed. Ultimately, this should allow us to explain the dynamics behind established Maxime Breden scaling laws such as Kolmogorov’s ”-5/3” law for the en- ENS Paris-Saclay & Universit´eLaval ergy spectrum and the log law for wall-bounded flow. So [email protected] far, this goal has been out of reach because of the large number of degrees of freedom in turbulent flows, easily in Jan-Philippe Lessard the millions for flow with an inertial range, i.e. a suffi- Universit´eLaval ciently large separation of the spatial scales of energy in- [email protected] put and dissipation. In large eddy simulation (LES) the motion on the smallest spatial scales, hypothesized to be unimportant to inertial range dynamics, is modelled by an Jan Bouwe Van Den Berg effective dissipation. The resulting deterministic dynam- VU University Amsterdam ical system describes inertial range dynamics, yet can be Department of Mathematics handled with modern computational methods. We study [email protected] the bifurcation sequence that gives rise to LES turbulence in a Taylor-Green vortical flow at several truncation levels Lennaert van Veen and discuss the peculiarities of LES, the successive break- UOIT ing of symmetries and the development of the inertial range [email protected] energy spectrum.

Lennaert van Veen MS60 UOIT Energy Flux Enhancement via Triad Fourier Phase [email protected] Dynamics in PDEs Tatsuya Yasuda In this work we present a study of Fourier-space phase dy- Imperial College namics in fluid dynamical PDEs with quadratic nonlinear- [email protected] ity where triadic wavevector interactions are responsible for energy transport across scales. We examine in detail Genta Kawahara the dynamics of the triad Fourier phases, defined as tri- Department of Mechanical Science adic linear combinations of individual Fourier phases. The Osaka University individual phases do not explicitly appear in the Fourier [email protected] mode evolution equations in amplitude-phase representa- tion. Instead, the ‘dynamical’ triad phase variables play Susumu Goto an explicit role in governing the time evolution of the Osaka University Fourier amplitude and phase variables. Our study of [email protected] the influence of the dynamics of these triad phase vari- ables on energy transfers include one-dimensional forced systems such as Burgers and Korteweg-de Vries/Burgers MS61 equation [Buzzicotti et al., EPJE, 39(3), 2016] and the Data-Driven Models of Large-Scale, High- two-dimensional Charney-Hasegawa-Mima equation [Bus- Dimensional Neural Recordings tamante et al., PRL , 113(8), 2014], relevant in atmospheric and plasma physics. In these cases we see strong correla- Abstract Not Available At Time Of Publication. tions between the triad phase dynamics and the evolution of the amplitudes of Fourier modes in the triads, with in- Bingni W. Brunton termittent (in time) alignment of triad phases leading to University of Washington maximally efficient energy fluxes throughout the inertial [email protected] range modes of the system. The careful analysis of these systems naturally leads us to a discussion as to how the distribution and dynamics of these Fourier triad phases MS61 may offer insight into energy fluxes in the case of isotropic Data-Driven Discovery of Dynamical System Mod- DS17 Abstracts 195

els for Biological Networks Using Sparse Selection Found in Nature and Information Criteria Abstract Not Available At Time Of Publication. Inferring the structure and dynamical interactions of com- plex systems is critical to understanding and controlling George Sugihara their behavior. As higher fidelity data becomes avail- Scripps Institution of Oceanography able, rapid generation and evaluation of mechanistically University of California San Diego meaningful models from data is increasingly possible. We [email protected] present a data-driven framework for sparse identification of nonlinear dynamical systems (SINDy). SINDy subselects a set of models from the combinatorial possibilities repre- MS62 sented in the feature library. By integrating the Akaike Heteroclinic Switching Between Chimeras Information Criteria (AIC) into the framework, we can rank the models in a principled way. The combined frame- The emergence of collective behavior is a fascinating fea- work also allows us to mitigate measurement error, miss- ture of interacting oscillatory units in nature and technol- ing variables, incomplete feature libraries, and insufficient ogy. We give some recent results [Bick, C. (2017). Hetero- data. To enable discovery of a broader class of functions, clinic switching between chimeras. arXiv:1703.03274] how we have also developed implicit-SINDy, which combines a sequential switching of localized frequency synchrony— compact feature library, implicit formulation, and sparsity that is, localized groups of oscillators will frequency syn- promoting non-convex optimization. The method success- chronize and desynchronize in a prescribed way—arises in fully identifies models for metabolic, regulatory and epi- networks of phase oscillators through generalized, nonsinu- demiological networks. Rapid construction of such mod- soidal coupling. Such dynamics may for example encode els could be leveraged for therapeutic gene modulation, information in real world networks of oscillatory units. metabolic engineering, or disease intervention. Christian Bick Niall M. Mangan University of Exeter, UK Department of Applied Mathematics [email protected] University of Washington, Seattle [email protected] MS62 Nathan Kutz Flexible Information Processing in Complex Net- University of Washington works Dept of Applied Mathematics Flexible information processing fundamentally underlies [email protected] the function of many biological and artificial networks. Yet, how such systems may specifically communicate and Steven Brunton dynamically process information is not well understood. University of Washington In this talk, I will first identify a generic mechanism to [email protected] route information on top of collective dynamical reference states in complex networks akin to radio signals where in- Joshua L. Proctor formation is encoded in the modulation of a carrier signal. Institute for Disease Modeling Switching between collective dynamics induces flexible re- [email protected] organization of information sharing and routing patterns, as quantified by delayed mutual information and transfer entropy measures between activities of networks units. I MS61 will then show how this mechanism provides a basis to On Equation-Free Modeling for Large-Scale Infec- achieve flexible computation in neuronal networks. In par- tious Disease Data ticular, coupled oscillatory Hopfield networks are shown to self-organize their collective reference dynamics so that Equation-free techniques are poised to make substantial only local information with high confidence is broadcasted progress in the analysis of complex systems. In this talk, I across the entire network. This allows the network to per- will discuss a number of equation-free techniques, includ- form coherent multi-modal pattern recognition even under ing the recently developed dynamic mode decomposition conflicting input or context information. These results help (DMD) and sparse identification of nonlinear dynamics understanding and designing information routing and pro- (SINDy). Further, I will demonstrate how these method- cessing across systems where collective dynamics co-occurs ologies can be applied to data sets collected from com- with a communication function. plex systems without a standard set of governing equations, Christoph Kirst specifically focusing on the spread of infectious disease. A Center for Studies in Physics and Biology data-driven understanding of how disease is spreading can Rockefeller University, US help in the design of targeted intervention campaigns in [email protected] resource-limited settings.

Joshua L. Proctor MS62 Institute for Disease Modeling [email protected] Analysis of Cryptocurrencies Networks Using Google Trends Data

Various cryptocurrencies have emerged as possible com- MS61 petitors to fiat currencies, with the underlying blockchain Empirical Dynamics: An Equation-Free Approach technology spreading and gaining recognition. This talk is for Understanding the Nonlinear Complex Systems devoted to the application of nonlinear dynamics methods 196 DS17 Abstracts

to analyse and estimate the proliferation of cryptocurren- this prediction experimentally and show that survival is cies. SIR type models are adapted to describe the take up enhanced due to a combination of disturbance of period-3 and abandonment of the blockchain based technology by oscillations and stochastic re-colonization events. the general population. Publicly available Google Trends data is used for model validation and prediction testing, Shreyas Gokhale reflecting the interest generated by the cryptocurrencies Physics of Living Systems Group discussed. Massachusetts Institute of Technology [email protected] Aleksandra Ross University of Sussex, UK Arolyn Conwill [email protected] Massachusetts Institute of Technology [email protected] MS62 Tanvi Ranjan Coherence-Resonance Chimeras in a Neural Net- Harvard University work tanvi [email protected] We show that chimera patterns can be induced by noise in nonlocally coupled neural networks in the excitable regime. Jeff Gore In contrast to classical chimeras, occurring in noise-free os- Massachusetts Institute of Technology cillatory networks, they have features of two phenomena: [email protected] coherence resonance and chimera states. Therefore, we call them coherence-resonance chimeras[N. Semenova, A. Za- MS63 kharova, V. Anishchenko, E. Sch¨oll, Coherence-resonance chimeras in a network of excitable elements, Phys. Rev. The Emergent Levy Behavior of Stochastic Burst Lett. 117, 014102 (2016)]. These patterns demonstrate Phenomenon in Single-cell Gene Expression the constructive role of noise and appear for intermedi- ate values of noise intensity, which is a characteristic fea- Single-cell gene expression is stochastic at the biochemical ture of coherence resonance. In the coherence-resonance level; it can be represented by a Delbr¨uck-Gillespie pro- chimera state a neural network of identical elements splits cess describing the number of mRNAs and proteins in a volume V ; it is known that in the macroscopic limit of into two coexisting domains with different behavior: spa- →∞ tially coherent and spatially incoherent, a typical property V a deterministic chemical kinetics arises. We de- of chimera states. Moreover, these noise-induced chimera rive another, stochastic L´evy-process limit that agrees with states are characterized by alternating behavior: coherent the empirical equation of Friedman et. al. [Phys. Rev. Lett. 97:168302, 2006]. We establish two types of limits and incoherent domains switch periodically their location. ∝ We show that this alternating switching can be explained in which protein synthesis rate V : One is deterministic by analyzing the coupling functions. corresponding to a cell-extract experiment in which DNA numbers ∝ V (Kurtz limit), another corresponds to a gi- Anna Zakharova ant cell with DNA copy numbers independent of V but ∝ Technische Universit¨at Berlin protein production per DNA V (L´evy limit). The latter Berlin, Germany conceptualizes the stochastic bursts of gene expression. A [email protected] three-stage model with feedback regulation provides by far the most comprehensive analytic theory of the steady-state distribution of the protein concentration in single cells. Im- MS63 plication of the emergent L´evy behavior to biological phe- Synchronization and Survival of Connected Bacte- notypic diversity is discussed. rial Populations Chen Jia Migration plays a vital role in controlling population Department of Mathematical Sciences dynamics of species occupying distinct habitat patches. University of Texas at Dallas While local populations face extinction due to demographic [email protected] or environmental stochasticity, migration from neighbor- ing habitat patches can rescue these populations through Michael Zhang colonization of uninhabited regions. However, a large mi- The University of Texas at Dallas gratory flux can synchronize the population dynamics in [email protected] connected patches and enhance the risk of global extinc- tion during periods of depression in population size. Here, Hong Qian we investigate this trade-off between local rescue and global Department of Applied Mathematics extinction using laboratory populations of E. coli bacteria. University of Washington Our model system consists of mutualistic co-cultures of [email protected] ampicillin resistant and chloramphenicol resistant strains that exhibit period-3 oscillations in the relative population density in the presence of both antibiotics. We quantify MS63 the onset of synchronization of oscillations in a pair of co- Mathematical Modelling of Organelle Transport in cultures connected by migration and show that period-3 Living Cells oscillations are disturbed for moderate rates of migration. These results are consistent with simulations of a mecha- Two basic mechanisms for intracellular transport are nistic model of antibiotic deactivation in our system. The motor-driven trafficking which usually occurs along micro- simulations also predict that the probability of survival of tubules (MTs) and diffusion which is inefficient to move connected populations in high concentrations of antibiotics over long distances. Recently, mathematical modelling ap- is maximized at intermediate migration rates. We verify proaches have been actively developed to understand such DS17 Abstracts 197

intracellular transport. We use live-cell imaging data in years). It appears elasticity is key to the resilience of Fucus Ustilago hyphal cells and develop mathematical models to stands but inhibition is key for Ascophyllum stands. study mechanism underlying spatial organization of molec- ular motors and organelles. In particular, we found that Steven Dudgeon stochastic motility of dynein motors along MTs contribute California State University Northridge to half of its accumulation at hyphal tip to support early [email protected] endosome recycling. The bidirectional transport of early endosome not only facilitates the directed motion of perox- Peter Petraitis isomes along MTs but also enhances their diffusive motion University of Pennsylvania which is so-called active diffusion. Our modelling approach [email protected] also suggests that directed transport and, to a lesser degree, active diffusion contribute to overcome the actin-based po- lar drift forces, to ensure even distribution of peroxisomes MS64 and support their mobility over short and long distances, Emerging Models of Resilience: Introduction respectively. How much disturbance can a system withstand while keep- Congping Lin ing its basic character? This basic question of resilience huazhong university of science and technology matters in many applications, particularly sustainability [email protected] science and natural resource management. Despite its ap- parent simplicity, the question requires careful choices of how to represent disturbance and how to define “basic char- MS63 acter” if one wishes to use dynamical systems to find a Analysis and Simulation of Multiscale Stochastic quantitative answer in metric terms. I will survey existing Intracellular Bio-chemical Reacting Networks methods for measuring resilience, comparing their assump- tions, strengths, and weaknesses. After summarizing key Intracellular reacting networks involving gene regulation approaches and challenges in the field of resilience quan- often exhibits multiscale properties. That includes mul- tification, I will highlight recent mathematical progress in tiple reacting rates, multiple population magnitudes and this area. multi-stability. Direct Stochastic Simulation Algorithm (SSA) would turn out to be inefficient dealing with such Katherine Meyer systems. Schemes such as Nested SSA and Tau-leaping University of Minnesota method have proved to be effective for certain asymptotic [email protected] regimes. Based on the framework of transition path theory (TPT), we extended the probability current between two adjacent reacting states to single reacting states as well as MS64 reacting trajectories, thereby give the definition of transi- Information, Order and the Resilience of Living tion state (TS) as states with maximum velocity strength. Natural and Human Systems I will present recent results on the convergence analysis and applications of the algorithms. A problem of modeling the resilience of a complex adap- tive system is that a shock or persistent change to the sys- Di Liu tem may instigate interactions that have not been seen Michigan State University or imagined before. In ocean ecosystems the interactions [email protected] among species are extensive and flexible. Surprise is fre- quent. DNA and memory are forms of information that channel the behavior of living agents towards order. Hol- MS64 lands learning classifier system, a rule based approach to Emergence and Resilience of a New Alternative modeling behavior, can address evolution through genetics State in the Gulf of Maine or learning. We add the idea that agents acquisition and use of information is constrained by the cost of resolving Sheltered shores in the Gulf of Maine previously contained the uncertainty associated with an opportunity. Agents two alternative states - mussel beds and stands of Asco- then tend to interact with others with whom theyve be- phyllum nodosum. Over the last decade, mussels largely come familiar; they form families, groups, and groups of disappeared. Long-term data show that there are now two groups, creating systematic, hierarchical order. This order distinct rockweed states on sheltered shores, one dominated is far from static; as groups interact they acquire a range by A. nodosum, the other by Fucus vesiculosus.Experi- of experiences. They are able to build upon this knowl- mental clearings to mimic ice scour were established during edge, adapting their behavior to new circumstances. This winter 1996-97 and half of the cleared plots were cleared behavioral flexibility imparts resilience to individuals and again in winter 2010-11. Data sets before and after re- to the overall system. It is what makes these systems so clearing were used to examine development and resilience susceptible to surprising outcomes. With this approach, of both rockweed states. K-means clustering using the ’Be- we reconsider the way humans and fish interact in the fore’ dataset identified 4 groups; Controls, Ascophyllum, overfishing problem, arriving at the broad idea that sus- slow-growing and fast-growing Fucus stands. A discrimi- tainability and resilience require the information needed nant function derived from these clusters assessed resilience for self-organization of natural systems, thereby pointing of each state based on the ’After’ dataset. All Ascophyl- to new ways to govern and design regulations restraining lum stands were assigned as Ascophyllum stands after re- human activity. clearing. Median time for establishment (> 10% cover) was 7 years and full recovery (> 90% cover) took 15-20 years. James Wilson In contrast, recovery of Fucus stands was not as certain; University of Maine only 33% of slow-growing stands but 80% of fast-growing [email protected] Fucus stands were re-assigned after re-clearing. However, median recovery time of Fucus after re-clearing is fast (< 3 Carl P. Simon 198 DS17 Abstracts

University of Michigan [email protected], [email protected] [email protected]

Garrett Nieddu MS64 Montclair State University A Flow-Kick Framework for Exploring Resilience [email protected]

Resilience is a slippery concept that has different meanings in different contexts. From a dynamical systems point of MS65 view, the different meanings of resilience are often about in- Modeling the Dynamics of Interacting Particles by teractions between transient dynamics of a system and dis- Means of Stochastic Networks turbance to the system. In this talk, we subject the flow of an autonomous system of ODEs to regular shocks (“kicks”) Material science have been rapidly developing in recent of constant size and direction. The resulting flow-kick sys- years. A variety of particles interacting according to dif- tems occupy a surprisingly under-explored area between ferent kinds of pair potentials has been produced in exper- deterministic and stochastic dynamics. Natural questions imental works. Looking into the future, one can imagine to ask include: Does the resulting flow-kick system equili- controlled self-assembly of particles into clusters of desired brate? If so, where? Is that a “desirable” region of state structures leading to the creation of new types of materi- space? Does it represent resilience? What are the dynam- als. Analytical studies of the self-assembly involve coping ics near the flow-kick equilibrium? And what can it tell us with difficulties associated with the huge numbers config- about a system subject to stochastic disturbances from a urations, high dimensionality, complex geometry, and un- compact domain? acceptably large CPU times. A feasible approach to the study of self-assembly consists of mapping the collections of Alanna Hoyer-Leizel clusters onto stochastic networks (continuous-time Markov Mount Holyoke College chains) and analyzing their dynamics. Vertices of the net- [email protected] works represent local minima of the potential energy of the clusters, while arcs connect only those pairs of ver- Sarah Iams tices that correspond to local minima between which direct Harvard University transitions are physically possible. Transition rates along Cambridge, MA the arcs are the transition rates between the corresponding [email protected] pairs of local minima. Such networks are mathematically tractable and, at the same time, preserve important fea- Ian Klasky, Victoria Lee, Stephen Ligtenberg tures of the underlying dynamics. Nevertheless, their huge Bowdoin College size and complexity render their analysis challenging and [email protected], [email protected], invoke the development of new mathematical techniques. I [email protected] will discuss some approaches to construction and analysis of such networks.

Katherine Meyer Maria K. Cameron University of Minnesota University of Maryland [email protected] [email protected]

Mary Lou Zeeman Bowdoin College MS65 Department of Mathematics A Kinetic Theory of Birth, Death, and Fission of [email protected] Age-Structured Populations

Classical age-structured mass-action models such as the MS65 McKendrick-von Foerster equation have been extensively Stochastic Population Models: The Dynamics of studied but they are structurally unable to describe Invasion and Extinction stochastic fluctuations or population-size-dependent birth and death rates. We present a semi-Markov stochastic Large populations support many types of contagious dis- framework for populations that incorporate age-dependent eases. Without intervention, it is assumed disease eradi- birth, death, and fission rates. By defining multiparticle cation would not happen. Yet, we see disease fadeout and probability density functions, we derive a BBGKY-like hi- reintroduction cycles in real world data for all but a few dis- erarchy of kinetic equations for the stochastic evolution eases. To accurately model such events, one must include of an aging population undergoing birth, death, and fis- stochasticity, or randomness. This talk will present some sion. Moments of the hierarchy allows us to systematically of the mathematical machinery used to analyze stochastic connect deterministic age-dependent models with existing population models and understand the dynamics not cap- master equation approaches. Our results allow for an in- tured in traditional deterministic modeling. Using these tuitive treatment of the stochastic dynamics of age- and methods, we can predict when random events can drive population-dependent populations applicable to the study a disease to extinction or conversely, reintroduce the dis- of demography, stem cell dynamics, and disease evolution. ease as a large outbreak. Understanding extinction and invasion processes can provide insight on how to best use intervention controls to exponentially improve the proba- Tom Chou bility attaining a desired state. UCLA Departments of Biomathematics and Mathematics Lora Billings, Eric Forgoston [email protected] Montclair State University Department of Mathematical Sciences Chris Greenman DS17 Abstracts 199

University of East Anglia [email protected] [email protected]

MS67 MS65 Toward Understanding the Multi-Scale Coupling in Metastability and Intrinsic Extinction Risk in Fi- Global Oceanic Flows nite, Interacting Populations Large-scale currents and eddies pervade the ocean and play a prime role in the general circulation and climate. The Demographic stochasticity corresponds to random fluctu- 4 coupling between scales ranging from O(10 )kmdown ations due to populations consisting of a finite number of to O(1) mm presents a major difficulty in understanding, individuals whose fates aren’t perfectly correlated. Unlike modeling, and predicting oceanic circulation and mixing, their deterministic counterparts, the asymptotic behavior where our constraints on the energy budget suffer from of models of closed populations experiencing demographic large uncertainties. Identifying the energy sources and stochasticity is often trivial: eventually all populations go sinks at various scales and geographic locations can reduce extinct in the ecological models without immigration, or such uncertainty and yield insight into new parameteriza- all but one allele is lost in the population genetics mod- tions of nonlinear physical processes. To this end, we have els without mutation. These extinction events, however, developed a novel multi-scale analysis framework, which may be preceded by long-term transients i.e. “metastable accounts for the spherical geometry of the problem, and behavior.’ For a very general class of stochastic models, allows one to simultaneously probe the dynamics in both I will discuss recent results on how (i) the log duration space and time. We apply these tools to satellite altimetry of this metastable behavior scales like the “landscape size’ data and to strongly eddying high-resolution simulations provided the mean-field model has an attractor supporting using General Circulation Models (POP and MITgcm). all species or genotypes and (ii) the metastable behavior, We investigate the contribution of various nonlinear mech- for large landscape size, is statistically governed by the at- anisms, such as baroclinic and barotropic instabilities, to tractors of the mean-field dynamics. The results rely on the transfer of energy between scales at various locations, extensive use of large deviation theory, and will be illus- such as in western boundary currents, near the equator, trated with models of competing species and Wright-Fisher and in the deep ocean. models with selection. Hussein Aluie, Mahmoud Sadek Sebastian Schreiber University of Rochester University of California, Davis [email protected], [email protected] [email protected] Matthew Hecht Los Alamos National Laboratory MS66 [email protected] Frictional Instabilities and Squeak in Soft Contacts Geoffrey Vallis Abstract Not Available At Time Of Publication. University of Exeter [email protected] Daniele Dini Imperial College London [email protected] MS67 Averaging, Large Deviations, and Stochastic Pa- rameterization in a Slow-fast two-box Ocean Model MS66 Nonlinear Dynamic Response Predictions of Aero- The author has recently emphasized the non-Gaussian engine Components with Frictional Interfaces quality of subgrid-scale terms in under-resolved ocean models [I. Grooms, A Gaussian-product stochastic Gent- Abstract Not Available At Time Of Publication. McWilliams parameterization, Ocean Modelling 2016]. This talk explores the effects of this kind of non-Gaussian Norbert Hoffmann, Christophe Schwingschakl, Loic Salles noise in a novel low-dimensional system of stochastic ODEs Imperial College, London describing the mean temperature and salinity difference n.hoff[email protected], between the equatorial and polar oceans. The system has [email protected], [email protected] three time scales, corresponding to fast eddy dynamics, medium-fast relaxation of temperature towards the atmo- spheric temperature, and slow salinity evolution. The sys- MS66 tem exhibits a climatological distribution with Gaussian core and non-Gaussian rare event probability. We de- The Rats Whiskers - Transduction of Stick-Slip Dy- velop two levels of parameterizations based on slow-fast namics Via a Tapered Rod averaging theory: a deterministic parameterization and a multiplicative Gaussian parameterization. We explore the Abstract Not Available At Time Of Publication. ability of these parameterized models to reproduce the cli- matological distribution and short-term forecasting of rare Maysam Oldazimi, Cornelius Schwartz events. The results shed light on the qualitative differ- University of Tubingen ences to be expected from deterministic, Gaussian, and [email protected], non-Gaussian stochastic parameterizations in more com- [email protected] plete ocean models.

Thbaut Putelat Ian Grooms University of Bristol Department of Applied Mathematics 200 DS17 Abstracts

University of Colorado, Boulder these spike-adding transitions. [email protected] Mathieu Desroches William Barham INRIA Paris-Rocquencourt University of Colorado, Boulder [email protected] Applied Mathematics [email protected] Vivien Kirk University of Auckland [email protected] MS67 Inertial Particles in a Vortical flow: Dynamics and Data MS68

Two questions will be addressed: (1) What is the dynam- Canards in a Minimal Piecewise-linear Square- ical evolution of a particle with small inertia when placed wave Burster near a vortex center, and (2) if the positions of such a par- ticle are observed, how does the error propagate in tracking In this talk we show how to construct a piecewise-linear the vortex center by assimilating these data as if they were (PWL) approximation of the Hindmarsh-Rose (HR) neu- based on observations of a fluid particle. ron model that is minimal, in the sense that the vector field has the least number of linearity zones, in order to re- Chris Jones, Colin Guider produce all the dynamics present in the original HR model University of North Carolina-Chapel Hill with classical parameter values. This includes square-wave [email protected], [email protected] bursting and also special trajectories called canards, which possess long repelling segments and organise the transitions between stable bursting patterns with n and n +1spikes, MS67 also referred to as spike-adding canard explosions. We pro- Seasonal to Annual Ocean Predictions and the Im- pose a first approximation of the smooth HR model, using a portance of Estimating Uncertainties continuous PWL system, and show that its fast subsystem cannot possess a homoclinic bifurcation, which is necessary Seasonal forecasts of the climate system present a major to obtain proper square-wave bursting. We then relax the challenge. Skilful forecasts on timescales of up to a year assumption of continuity of the vector field across all zones, can have large societal implications. Accurately modelling and we show that we can obtain a homoclinic bifurcation the slowly evolving ocean is essential for such predictions, in the fast subsystem. We use the recently developed ca- as it carries anomalies much longer than the more rapidly nard theory for PWL systems in order to reproduce the fluctuating atmosphere. Aside from accurately modelling spike-adding canard explosion feature of the HR model. the ocean state, it is crucial to understand and account for uncertainties in predictions to enable informed deci- Soledad Fern´andez-Garc´ıa sions. Ensemble simulations are commonly used to pro- University of Sevilla, Sevilla, Spain duce probabilistic forecasts that provide estimates of fore- [email protected] cast uncertainty. In addition to irreducible uncertainties associated with the chaotic climate system, there are three major sources of uncertainty: Initial condition, model, and MS68 observational uncertainty. The 10-month forecast skill of the seasonal forecasting system of the European Centre for Synchronisation of Weakly Coupled Canard Oscil- Medium- Range Weather Forecast will be discussed, fo- lators cusing on the ocean component. Novel stochastic pertur- bation techniques are used to estimate model uncertainties Synchronization has been studied extensively in the con- in highly parametrized oceanic processes. Additionally, ob- text of weakly coupled oscillators using the so-called phase servational uncertainties - associated with the choice of the resetting curve (PRC) which measures how a change of reference reanalysis used for forecast skill evaluation - will the phase of an oscillator is affected a small perturbation. be analysed. Both kinds of uncertainty will be investigated This approach was linked to the work of Malkin, and it has in probabilistic ensemble predictions. been extended to relaxation oscillators. Namely, synchro- nization conditions were established under the weak cou- Stephan Juricke pling assumption, leading to a criterion for the existence Jacobs University Bremen of synchronous solutions of weakly coupled relaxation os- [email protected] cillators. Previous analysis relies on the fact that the slow nullcline does not intersect the fast nullcline near one of its fold points, where canard solutions can arise. In the MS68 present talk we show how to use numerical continuation Three-Timescale Dynamics: Canards and Spike techniques in order to solve the adjoint equations and we Adding show that synchronization properties of canard cycles are different than those of classical relaxation cycles. In partic- In this talk, we will study a minimal three-timescale model ular, maximal canards separate two distinct synchroniza- of van der Pol type. This model displays periodic solu- tion regimes: the Hopf regime and the relaxation regime. tions that mimic bursting dynamics without having two Phase plane analysis of slow-fast oscillators undergoing a clear fast variables. Using geometric singular perturbation canard explosion provides an explanation for this change theory as well as numerical bifurcation methods, we in- of synchronization properties across the maximal canard. vestigate how such bursting orbits grow more spikes as a system parameter is varied. In particular, we focus on two Elif Koksal Ersoz separate canard mechanisms providing key components of INRIA de Paris DS17 Abstracts 201

[email protected] Georgetown University [email protected]

MS68 Ewan Colman Faux Canards and Folded Saddles Georgetown University [email protected] We study the two parameter family of faux canards asso- ciated with the folded saddle singularity within the folded singularity normal form. Recently, rotational behaviour MS69 of folded saddle faux canards has been reported and mer- After the Honeymoon, the Divorce: Unexpected its a closer look at faux canards which have been some- Outcomes of Disease Control Measures what neglected in the literature. We address this gap in canard knowledge providing a comprehensive analysis of We lack effective vaccines for many infections, so disease folded saddle faux canards, both numerically and analyt- control measures often instead attempt to directly reduce ically. We show that for certain values of μ – the eigen- transmission. As an example, the main control measure for value ratio of the associated folded singularity within the the the mosquito-borne dengue virus has been mosquito reduced flow – faux canards may possess rotations about control, e.g. by spraying insecticide aimed at adult insects. the primary faux canard, and that the stable and unstable In this talk we discuss counter-intuitive behavior that can fast manifolds (i.e. the nonlinear stable and unstable fast result when such control measures are employed for a tran- fibre bundles) of the primary faux canard form the bound- sient period against an endemic infection. Building on an aries of sets of solutions with similar rotational behaviour. observation in a recent study of Okamoto et al., we demon- This is in contrast to the folded node case where the stable strate the epidemiologically-troubling result that there can and unstable slow manifolds of the primary weak canard be time windows over which the total number of disease organise the family of weak canards. We develop a numer- cases can exceed the number that would have occurred if ical scheme by which we obtain these fast manifolds and no intervention had been employed: accumulation of sus- locate and characterise the family of secondary faux ca- ceptibles during the control can lead to a large outbreak nards and their bifurcations. We confirm the bifurcations following the end of the control. This outbreak can be so of secondary faux canards via an extended Melnikov anal- severe that all the benefit accrued during control can be ysis where we find an alternating pattern of bifurcations – overcome. We show that this phenomenon can occur in a transcritical and pitchfork – similar to the alternating pat- broad class of infection models and discuss public health tern of bifurcations of secondary weak canards in the case implications, including for clinical trials of proposed con- of the folded node. trol measures.

John Mitry Alun Lloyd University of Sydney North Carolina State University [email protected] alun [email protected]

MS69 Brandon Hollingsworth NC State University Spatial and Social Interactions Drive Transmission [email protected] Dynamics in Ant Colonies

Social behavior in human and animal populations gives mo- Fred Gould bility to infectious disease. In some cases, being in close North Carolina State University proximity of an infected individual can be enough for trans- fred [email protected] mission to occur. On the other hand, diseases that require more intimate contact depend on social behaviors rather Kenichi Okamoto than spatial factors. Here we map the social network of an Yale University ant colony and ask whether ants structure their society in [email protected] a way which mitigates the risk of epidemic disease? Most carpenter ants do not leave the nest. A few foragers ingest the food they find, and the food is transmitted through the MS69 colony on their return, resulting in a complex network of Effectiveness of Unaids Targets and Hiv Vaccina- feeding interactions. In a lab experiment, we filmed several tion Across 127 Countries hours of nest activity. Based on the interactions recorded, we construct temporal networks where nodes are ants and The HIV pandemic continues to evolve, with AIDS-related edges are feeding interactions. Remarkably, the individu- deaths on the rise in regions of Asia and an increasing pro- als in the ant colony are marked by a propensity to seek portion of infections in sub-Saharan Africa are among ado- out new connections rather than repeat previous ones, con- lescents. Simultaneously, advances in antiretroviral ther- trary to what is seen in most other social animals and hu- apy, diagnostic approaches and vaccine development are mans. To address whether this behavior encourages disease providing novel tools for treatment-as-prevention and pro- spread we use a model in which we can adjust the hetero- phylaxis. We developed a mathematical model to evaluate geneity of social tie strength. We then couple this with a the added benefit of an HIV vaccine in the context of goals disease model where transmission depends non-linearly on to increase rates of diagnosis, treatment, and viral suppres- the intensity of the interaction. Our analysis suggests that sion in 127 countries. Under status quo interventions, we a rigid social structure determines the dynamics of com- predict a median of 49 million [1st and 3rd quartiles 44M, municable disease in carpenter ant colonies more so than 58M] incident cases globally from 2015 to 2035. Achiev- spatial restrictions. ing the UNAIDS 95–95–95 target was estimated to avert 25 million [20M, 33M] of these new infections, and an ad- Shweta Bansal ditional 6.3 million [4.8M, 8.7M] reduction was projected Dept. of Biology with the 2020 introduction of a 50%-efficacy vaccine grad- 202 DS17 Abstracts

ually scaled up to 70% coverage. This added benefit of Erik Van Vleck prevention through vaccination motivates imminent and Department of Mathematics ongoing clinical trials of viable candidates to realize the University of Kansas goal of HIV control. [email protected]

Jan Medlock Jesse Nippert Oregon State University Kansas State University Department of Biomedical Sciences [email protected] [email protected] Maged Nosshi Abhishek Pandey University of Kansas Department of Biostatistics, Yale University [email protected] [email protected] Zak Ratajczak Alyssa S. Parpia, Amber Tang, Laura A. Skrip, Alison P. University of Virginia Galvani [email protected] Yale University [email protected], [email protected], [email protected], [email protected] MS70 Unifying Ecological Models for Understanding and Prediction MS69 Rationally Time-shifting Risk in Response to Zika Ecological theory surrounding populations and communi- ties of biological species is embedded in a series of basic but The morbidity of Zika infection varies depending on the age distinct mathematical models, including differential equa- of infection. Infection is usually mild, but is associated with tion models for logistic growth, competition, and preda- serious birth defects during pregnancy. Thus, costs from tion. The basic models parallel the historical development infection for women of child-bearing age have the potential of ecology over the twentieth century. However, distinc- to be much larger than for younger and older woman. This tions among the models are only apparent – all convert to raises a conundrum similar to Rubella’s – early infection a common quadratic form differing only in the signs of their greater susceptibility may reduce median infection ages and parameters. Considering this common form reveals ecolog- costs. In this talk, I’ll pose a time-dependent differential ical dynamics that have long been missing or ignored in population game model of Zika transmission and numeri- the existing subset of equations, yet are necessary to un- cally solve for the rational social-distancing strategy. The derstand certain dynamics crucial in the modern world, solutions will be compared to optimal public policies, and including the global population of our own species. Here costs of cooperation will be discussed. we explain the common form, expose the formerly missing parts, bring out some mathematical points remaining un- Timothy Reluga developed, and show how the models explain and predict Pennsylvanian State University ecological observations at the Cedar Creek field station and [email protected] in the world.

Clarence Lehman MS70 University of Minnesota Stability of Tree-Grass Interactions under Woody [email protected] Encroachment

Woody encroachment of grasslands is occurring and accel- MS70 erating across the globe. Dynamical systems approaches Toward a Mathematical Theory of Resilience are often used in assessing the stability of the woody and grass end member states. Here, we investigate the roles Many ecological systems exhibit multiple equilibrium of precipitation frequency and intensity and fire frequency states, and questions arise regarding how resistant or re- on controlling the rate of the conversion of grasslands to silient they are to perturbations or to changing environ- woody dominated systems. We investigate these interac- mental conditions. The classical concept of hysteresis is tions using data from the Konza Prairie Long Term Eco- often used to explore whether a system will return to its logical Research Site in North-Central Kansas. A low- previous state if the environment is returned to its previ- dimensional model with stochastic precipitation and fire ous condition. In this lecture we discuss the notion of the disturbance is introduced to examine the complex inter- ”intensity” of an attractor and examine how it can be used actions between precipitation and fire as mechanisms that to quantify resilience. An example of plant community re- may suppress or facilitate increases in woody cover. We use sponses to nutrient changes will be explored. Lyapunov exponents to ascertain the relative control ex- erted on woody encroachment through these mechanisms. Richard McGehee, Katherine Meyer Our results indicate that precipitation frequency is a more University of Minnesota important control on woody encroachment than the in- [email protected], [email protected] tensity of individual precipitation events. Fire, however, exerts a much more dominant impact on the limitation of encroachment over the range of precipitation variability MS70 considered here. Parameter Estimation for Land-Surface Models

NathanielA.Brunsell In this talk we consider a projected shadowing based data Dept of Geography & Atmospheric Science, U Kansas assimilation technique for simultaneous state space and pa- [email protected] rameter estimation. Application of these techniques is con- DS17 Abstracts 203

sidered for land-surface models ranging from a low dimen- Massey University sional water balance model with stochastic forcing for fire [email protected] and precipitation to a complex model, Noah-MP, that al- lows for thousands of parameter combinations. Mike R. Jeffrey University of Bristol Erik Van Vleck mike.jeff[email protected] Department of Mathematics University of Kansas [email protected] MS71 Pausing: Revealing Time Indeterminacy in Piece- NathanielA.Brunsell wise Smooth Systems Dept of Geography & Atmospheric Science, U Kansas [email protected] Continuity in a dynamical system is known to not be a sufficient property to guarantee the uniqueness of its solu- tions. Here we consider piecewise smooth systems where MS71 ambiguity arises when the temporal determinacy of a so- Interconnection and Multiple-Investment Strate- lution is lost - pausing. In phase-space this corresponds gies in Energy Markets to the same spatial trajectory being traversed in differ- ent times, as the solution pauses arbitrarily long at par- This talk proposes a general economics model for the sup- ticular points. This potentially mimics behaviour seen in ply and demand of a commodity in a domestic market earthquakes and static friction models, where periods of when investments are required for supporting it. Piece- dynamic activity are separated by seemingly static phases wise smooth and hybrid dynamical systems have been in- of unpredictable length. We study piecewise smooth sys- creasingly used in Engineering and Applied Sciences. More tems containing temporal (and sometimes spatial) ambi- recently, these systems appeared also in Economics and So- guities that can be partly resolved through application of cial Science, mainly in Sustainability Development, Bioe- parameter rescalings and asymptotic balancing; except at conomics, and new knowledge areas. Usually, the problem particular parameter values where the uncertainty persists. of one surface dividing the state space into two different These are our pausing points. We present examples of the regions is considered. In this paper, several switching sur- phenomena occurring at a fold-fold singularity in a system faces are taken into account, since our application lies on drawn from a gene regulation model, and in the impacting this multi-surface situation. When several markets are con- system of the classical Painlev´e paradox. nected, complex networks (in the dynamics and structure) naturally appear. Finally, stochasticity is introduced into Simon C. Webber the system to model the risk aversion of investment agents. University of Bristol This is done through Markov Chains. This combination of Department of Engineering Mathematics deterministic paths and stochasticity leads to the so-called [email protected] Piecewise-Deterministic Markov Processes. Depending on the risk behavior, simulations show several decision pat- terns. Mike R. Jeffrey University of Bristol Gerard Olivar mike.jeff[email protected] Department of Electrical and Electronics Engineering Universidad Nacional de Colombia, sede Manizales Paul Glendinning [email protected] the University of Manchester [email protected] Johnny Valencia Calvo Tecnologico de Antioquia [email protected] MS71 Coherent Behaviour in Non-Smooth Neural Net- Oscar Emilio Molina Diaz works Universidad del Quindio [email protected] Certain neural systems show computation through pat- terned activity: persistent localised activity, in the form of bumps, has been linked to working memory, whilst the MS71 propagation of activity in the form of waves has been as- Desynchronising Collections of Oscillators by Using sociated with binocular rivalry tasks. Individual neurons Two-Fold Singularities typically exhibit an all-or-nothing response, dependent on the summation of signals they receive from the rest of the In this talk I will present possibly the first practical ap- network. This fact, coupled with the desire to understand plication of a two-fold — a singular point in a nonsmooth coherent patterns of activity across the network has re- system of differential equations at which the long-term dy- sulted in the widespread use of non-smooth neural mod- namics is, in a sense, infinitely sensitive to random per- els that greatly simplify the complex dynamics of individ- turbations. By applying a well-chosen control signal one ual cells. Whilst these descriptions often provide tractable can create a two-fold in a generic oscillatory system that models of neural tissue, their non-smooth nature presents has the effect of randomising the phase of the motion in a its own mathematical challenges. We will show how lo- highly efficient manner. Such control signals could be used calised bumps of activity and travelling waves are gener- to break the synchrony of collections of oscillators with ated in a synaptically coupled neural network and how they several advantages over more traditional methods. lose stability through bifurcations of both smooth and non- smooth type. This will be done through the construction of David J. Simpson interface equations that take advantage of the non-smooth Institute of Fundamental Sciences nature of the model. We will examine how this can be 204 DS17 Abstracts

used to explain the experimentally observed activity in grid blowflies revisited. Nature, 287:17–21, 1980], cells, which are a significant component of the brain’s rep-  −αx(t−τ) resentation of spatial location within an environment. Fi- x (t)=−γx(t)+px(t − τ)e , p,α,γ,τ >0 nally, we will present an approach to the asses impact of is proposed to model oscillations in lucilia cuprina pop- temporal and spatial noise in neural networks using numer- 1 ical coarse-graining procedures. ulations. In this model, α is the size of the population at which it achieves maximal reproduction success, γ is Kyle Wedgwood the adult death-rate, and parameter p is the maximal egg- University of Exeter production rate. We report on the effect of considering two [email protected] maturation sites with different maturation times. That is, we consider Daniele Avitabile  −δ1τ1 −δ2τ2 x = −γx+p qe f(x(t − τ1)) + (1 − q)e f(x(t − τ2)) . School of Mathematical Sciences University of Nottingham [email protected] Our main interest is to investigate how the stability and oscillatory properties of the blowflies equation are affected by the heterogeneity of the maturation sites. Joshua Davis, Stephen Coombes University of Nottingham Gabor Kiss, Gabor Kiss [email protected], University of Szeged [email protected] [email protected], [email protected]

MS72 MS72 Delay-Induced Dynamics in An El Ni˜no Southern Stability in a Neural Network with Distributed- Oscillation Model: A Bifurcation Analysis Delay Coupling We consider a conceptual model for the El Ni˜no South- In this talk I will present a Hopfield-type neural network ern Oscillation system in the form of a delay differential model, where one sub-system receives a delayed input from equation that includes feedback loops between the ocean another subsystem [ B. Rahman, K.B. Blyuss & Y. N. and atmosphere. A bifurcation analysis demonstrates that Kyrychko Dynamics of neural systems with discrete and qualitatively realistic behaviour results from the interplay distributed time delays, SIAM Journal on Applied Dynam- between delayed feedback and seasonal forcing. Moreover, ical Systems, 14 (4), pp. 2069–2095 (2015).]. The model the model features the phenomenon of folding invariant includes a combination of both discrete and distributed tori, which may be interpreted as a form of climate tip- delays, where distributed time delays represent the neural ping. feedback between the two sub-systems, and discrete delays describe the neural interactions within each of the two sub- Andrew Keane systems. Stability properties are investigated for different The University of Auckland commonly used distribution kernels, and the results are [email protected] compared to the corresponding results on stability anal- ysis for networks with no distributed delays. I will show Bernd Krauskopf how boundaries of the stability region of the trivial equi- University of Auckland librium can be obtained analytically for the cases of delta, Department of Mathematics uniform and gamma distributions. Direct numerical sim- [email protected] ulations that confirm analytical findings will also be pre- sented. Claire M. Postlethwaite University of Auckland Yuliya Kyrychko [email protected] University of Sussex, UK [email protected]

MS72 MS73 Poincar´e-Bendixson-Type Theorems for Equations with State-Dependent Delay Meanfield Coupling of Expanding Circle Maps

We will discuss classes of scalar-valued differential equa- We study the synchronization phenomena that arises when tions with state-dependent delay and monotone feedback increasing the interaction strength in globally coupled cir- for which Poincar´e-Bendixson-type theorems do and do cle maps. Specifically, we focus on the case when the in- not hold. This work builds, most particularly, on work dividual site dynamics are doubling maps which interact of Mallet-Paret and Sell from the 1990s. by a mean field diffusive coupling of strength ε.Inthis setting and for finitely many sites, two distinct bifurca- Benjamin Kennedy tion values of the coupling strength have been identified in Gettysburg College the literature, corresponding to the emergence of contract- Department of Mathematics ing directions (Koiller & Young, Nonlinearity, 2010) and, [email protected] specifically for N = 3 sites, to the loss of ergodicity (Fer- nandez, Journal of Stat. Phys. 2014). In our recent paper (B´alint & S´elley, Journal of Stat. Phys. 2016), on the one MS72 hand, we reconsider these results and provide an interpre- Nicholson Blowflies Dynamics in a Heterogenous tation of the observed dynamical phenomena in terms of Environment for Juveniles the synchronization of the sites. On the other hand, we initiate a new point of view which focuses on the evolution In [WSC Gurney, SP Blythe, and RM Nisbet. Nicholson’s of distributions and allows to incorporate the investigation DS17 Abstracts 205

of a continuum of sites. In particular, we observe phenom- phenomena, most notably by the formulation of a center ena that is analogous to the limit states of the contracting manifold theorem for networks. An interesting caveat is regime of N = 3 sites. In this talk, we would like to present the fact that these symmetries seldom come from groups, these results and describe possibilities for generalization to but rather from monoids and other algebraic structures. other values of N, other types of site dynamics and other This opens the door to very interesting generalizations of forms of the coupling. equivariant dynamics and representation theory.

Peter Balint Eddie Nijholt,BobRink Institute of Mathematics VU University Amsterdam Budapest University of Technology and Economics [email protected], [email protected] [email protected] Jan Sanders Fanni Mincsovicsne Selley Free University of Amsterdam Budapest University of Technology and Economics Tne Netherlands Department of Stochastics [email protected] [email protected]

MS73 MS73 Maps Coupled in Networks. The Interplay Be- Generation of Stable Traveling Waves in Unidirec- tween Structure and Dynamics tional Chains of Idealized Neural Oscillators Recent results reveal that typical real-world networks have Phase oscillator ensembles exhibiting a preferred direction various levels of connectivity. These networks exhibit emer- of coupling are a much studied topic of mathematical neu- gent behaviour at various levels. For instant, massively roscience, where they provide a framework for understand- connected nodes can synchronise, while less connected ing the generation and propagation of electrical impulses nodes do not. Striking examples are found in the brain, across brain tissues. We report here on some recent suc- where synchronisation between highly connected neurons cesses in establishing stability of traveling wave solutions coordinate and shape the network development. These (TW) in feedforward chains of idealized neural oscillators phenomena remain a major challenge. We will discuss featuring a pulse emission / Type-I phase response (PRC) consider expanding maps coupled on networks with var- interaction. In prior work, the smooth version of this model ious connectivity scales and perform a probabilistic dimen- was studied through a combined numerical / analytical sion reduction to describe the network dynamics. We show methodology: an iterative fixed point scheme was used to that, at large levels of connectivity, the high-dimensional verify existence of TW; these findings then supported a rel- network dynamics can be reduced to a few macroscopic evant abstract hypothesis that proved sufficient to establish equations. This reduction provides the opportunity to ex- global stability of the solution. We have since completed plore the coherent properties at various network connec- a full mathematically-rigorous study of a piecewise affine tivity scales. This is a join work with M. Tanzi and S. van version of this model, establishing all of the observed phe- Strien. nomenology (both existence and stability of TW) analyti- cally. In addition, we now have an understanding of how a Tiago Pereira robust TW solution in this setting may be generated with ICMC-University of Sao Paulo a variety of external forcing stimuli, including such that Sao Carlos are structurally distinct from the consequent wave. This is [email protected] an apparent feature of this class of models.

Stanislav M. Mintchev MS74 The Cooper Union A Model for Diffusive Gas Dynamics Away from New York Thermodynamic Equilibrium [email protected] We find that the Boltzmann equation is a simplification of Bastien Fernandez a multimolecular random jump system, which is different Laboratoire de Probabilit´es et Mod`eles Al´eatoires from the realistic gas dynamics. We correct the difference CNRS - Universit´eParisDiderot-UPMC via a multiscale homogenization formalism, which equips [email protected] the Boltzmann equation and its fluid dynamics closures with a spatial diffusion term. As a result, the nonequilib- rium Grad closure becomes well posed for boundary value MS73 problems, and can be used for practical problems in the Hidden Symmetry in Coupled Cell Networks same manner as the Navier-Stokes equations. For the Cou- ette flow, we find that the diffusive Grad closure captures Dynamical systems with a network structure often dis- both the Knudsen boundary layer and the nonzero heat play remarkable similarities with equivariant systems (that flux in the direction of the flow. is, systems possessing a symmetry). For example, a net- work structure often seems to force the existence of flow- Rafail Abramov invariant subspaces, and the linearized system on such a Department of Mathematics, Statistics and Computer space often comes with a degenerate spectrum. As a result, Science many network ODE’s support generic bifurcations that are University of Illinois at Chicago unheard of for general vector fields (a phenomenon that is [email protected] again well-known for symmetric systems). We present a set-up where network ODE’s are indeed presented as (sub- systems of) symmetric systems. This in turn yields rela- MS74 tively straightforward explanations for the aforementioned Dynamics of Optical Pulses Riding on a Back- 206 DS17 Abstracts

ground, An Exact Approach Rensselaer Polytechnic Institute [email protected], [email protected], The study of scalar and vector nonlinear Schr¨odinger [email protected] (NLS) and Maxwell-Bloch (MB) systems with non-zero boundary conditions at infinity has received renewed in- Peter R. Kramer terest recently. This talk will report on recent results on Rensselaer Polytechnic Institute focusing scalar and vector NLS and MB equations with Department of Mathematical Sciences non-zero boundary conditions. It will be shown how the [email protected] inverse scattering transform can be constructed in both cases, and a number of explicit soliton solutions will be discussed. Gregor Kovacic Rensselaer Polytechnic Inst Gregor Kovacic Dept of Mathematical Sciences Rensselaer Polytechnic Inst [email protected] Dept of Mathematical Sciences [email protected] David Ca Shanghai Jiao Tong University, China, Gino Biondini Courant Institute, New York University, USA. State University of New York at Buffalo [email protected] Department of Mathematics biondini@buffalo.edu Maxwell Jenquin, Alexander Mayer Rensselaer Polytechnic Institute Daniel Kraus [email protected], [email protected] SUNY Buffalo dkkraus@buffalo.edu MS75 Sitai Li Bellerophon States: Coexistence of Quantized, State University of New York at Buffalo Time Dependent, Clusters in Globally Coupled Os- sitaili@buffalo.edu cillators

Ildar R. Gabitov From rhythmic physiological processes to the collective be- Department of Mathematics, University of Arizona haviors of technological and natural networks, coherent [email protected] phases of interacting oscillators are the foundation for the emergence of the system’s cooperative functioning. We un- veil the existence of a new of such states, occurring in glob- MS74 ally coupled nonidentical oscillators in the proximity of the An Openfoam Implementation of the Nonequilib- point where the transition from the system’s incoherent rium Diffusive Gas Dynamics Model: Testing and to coherent phase converts from explosive to continuous. Validation In such a state, oscillators form quantized clusters, where they are neither phase- nor frequency-locked. Oscillators’ We implement a novel scheme in the high-level finite- instantaneous speeds are different within the clusters, but volume description defined in OpenFOAM atop C++, thus they form a characteristic cusped pattern and, more im- abstracting over a rich variety of meshes and boundary con- portantly, they behave periodically in time so that their ditions with guaranteed physical units. We discuss the pro- average values are the same. Given its intrinsic specular cedure to define and compile the custom solver, and demon- nature with respect to the Chimera states, the phase is strate a low-abstraction open-source toolchain to define termed the Bellerophon state. We provide analytical and and analyze standard compressible flow problems against numerical description of the microscopic and macroscopic multiple solvers. details of Bellerophon states, thus furnishing practical hints on how to seek for the new phase in a variety of experimen- Jasmine T. Otto tal and natural systems. Department of Mathematics, Statistics and Computer Science Hongjie Bi University of Illinois at Chicago Department of Physics, East China Normal University, [email protected] Shangha [email protected]

MS74 Xin Hu Effective Dispersion and Linearization in Wave- Key Lab of Nanodevices and Applications-CAS Like Systems Chinese Academy of Sciences, Suzhou 215123, China [email protected] Effective dispersion in extended systems may be generated by the increasing nonlinearity, with resonant wave-wave in- Stefano Boccaletti teractions appearing or disappearing, and resonant mani- CNR-Istituto dei Sistemi Complessi folds deforming. This occurs even in systems with no linear [email protected] dispersion. In such a subcase of the MMT model, due to its symmetry, we calculated energy and wavenumber spectra both directly and from the wave-turbulence theory asso- Xingang Wang ciated with the effective dispersion relation. For the NLS Zhejiang University, China equation, we computed this relation to be a quadratic. [email protected]

Madison Wyatt, Katelyn J. Leisman, Michael Schwarz Yong Zou, Zonghua Liu, Shuguang Guan DS17 Abstracts 207

East China Normal University, Shanghai 200241, China Keck Laboratory for Network Physiology [email protected], [email protected], guan- Physics Department, Boston University [email protected] [email protected]

MS75 MS76 Anomalous Critical and Supercritical Phenomena Speed Modulated Social Influence in Evacuating in Explosive Percolation Pedestrian Crowds

The emergence of large-scale connectivity on an underlying Models of pedestrian motion such as the social force model network or lattice, the so-called percolation transition, has and the anticipatory collision avoidance model are often a profound impact on the systems macroscopic behaviors. validated in terms of their ability to faithfully reproduce There is thus great interest in controlling the location of the an evacuation situation, where individuals exit a confined percolation transition to either enhance or delay its onset space through a single exit. Most pedestrian models are and, more generally, in understanding the consequences of able to reproduce salient features of pedestrian evacuation such control interventions. Here we review explosive per- such as the faster-is-slower effect, which represents the sig- colation, the sudden emergence of large-scale connectivity nificantly longer time taken by a rushing crowd to evacuate that results from repeated, small interventions designed than a crowd which is relatively less hurried. The social to delay the percolation transition. These transitions ex- influence in such models is represented in the form of a re- hibit drastic, unanticipated and exciting consequences that pelling action based on proximity or time-to-collision, and make explosive percolation an emerging paradigm for mod- all individual agents are modeled to have the same ten- elling real-world systems ranging from social networks to dency to exit. We show through experiments that in sce- nanotubes. narios where individuals may have different tendencies to Raissa D’Souza exit, difference in speed exerts an additional social influ- UC Davis ence. Specifically, we analyze trajectory data of crowds of [email protected] individuals with different tendencies to exit and show the presence of social influence where slower individuals tend to speed up in the presence of those that are rushing. These MS75 results motivate an update to pedestrian motion models in Recent Advances in Explosive Percolations and order to include social influence due to variability in speeds. their Applications Percolation has long served as a model for diverse phe- Sachit Butail nomena and systems. The percolation transition, that is, Northern Illinois University the formation of a giant cluster on a macroscopic scale, [email protected] is known as one of the most robust continuous transi- tions. Recently, however, many abrupt percolation tran- Abhishek Bhatia sitions have been observed in complex systems. To illus- Indian Institute of Technology Delhi trate such phenomena, considerable effort has been made [email protected] to introduce models and to construct theoretical frame- works for explosive, discontinuous, and hybrid percolation Elham Mohammadi transitions. Experimental results have also been reported. Northern Illinois University In my talk, I will review results of such effots, percolation [email protected] models, their critical behaviors and universal features, and real-world phenomena. MS76 Byungnam Kahng Department of Physics and Astronomy Lagrangian Coherent Structures as Sub-mesoscale Seoul National University Transport Barriers in Amospheric Flows [email protected] Lagrangian coherent structures (LCS) in two-dimensional flows have been studied in the context of transport in fluid MS75 dynamics. However, for geophysical systems a small ver- Mechanism of Explosive Synchronization in Net- tical velocity can lead to nontrivial three-dimensional mo- worked Oscillators tion of airborne particles. Examples of such particles in- clude biological populations affecting agriculture and haz- In the past several years, explosive synchronization in net- ardous outputs from both man-made and natural disasters. worked oscillators has became a hot topic. Many models The pathways and barriers in the lower atmosphere, from and aspects of explosive synchronization have been studied. ground level to a kilometer altitude and over a horizontal In our work, we focus on the mechanism of explosive syn- scale of several kilometers–which bridges the scale of, for chronization and find a mechanism called ”suppress rule” example, local terrain to the larger regional scale–are still which is similar to the mechanism of explosive percolation. unclear. This requires exploring relevant spatiotemporal Taking advantage of this ”suppress rule”, we can gener- scales related to advection in the space of 3D + time. In ate explosive synchronization in different kinds of network this talk, we will present the application of finite-time Lya- (fully connected, random, scale free and multi-layer net- punov exponent-based three-dimensional LCS to address work). Through the further research, we find other mecha- questions of transport using historical data sets from satel- nism (such as competition) can also lead to explosive syn- lite observations, local field measurements and the Weather chronization. In my talk, I will introduce the mechanisms Research and Forecasting (WRF) model. of explosive synchronization we found. Peter Nolan Xiyun Zhang Virginia Polytechnic Institute and State University 208 DS17 Abstracts

[email protected] [email protected], [email protected]

MS76 MS77 Interactional Dynamics of Same-Sex Marriage Leg- On How Turing Triggers Biochemical Spot Dynam- islation in the United States ics in a Plant Root Hair Initiation Model Pattern formation aspects in a 2D reaction-diffusion (RD) Understanding how people form opinions and make deci- sub-cellular model characterising the effect of a spatial gra- sions, at individual and population scales, is a complex dient of a plant hormone distribution on a family of G- phenomenon that depends on both personal practices and proteins associated with root-hair (RH) initiation in the interactions with others. Recent availability of real-world plant cell Arabidopsis thaliana are analysed. The acti- data has enabled opinion formation analysis, which can vation of these G-proteins, known as the Rho of Plants shed light on phenomena that impact physical and social (ROPs), by the plant hormone, auxin, is known to promote sciences. Public policies exemplify complex opinion for- certain protuberances on root hair cells, which are crucial mation spanning individual and population scales, and a for both anchorage and the uptake of nutrients from the timely example is the legalization of same-sex marriage in soil. The mathematical model for the activation of ROPs the United States. Here, we seek to understand how this is- by the auxin gradient is an extension of the model pro- sue captures the relationship between state laws and Senate posed by Payne and Grierson [PLoS ONE, 12(4), (2009)], representatives as it depends on geographic and ideologi- and consists of a two-component generalised Schnakenberg cal factors. Using a distance-based correlation metric, we RD system with spatially heterogeneous coefficients on a study how physical proximity and state-government ide- 2D domain. The components in this system model the ology may be used to extract patterns in state-law adop- tion and senatorial support of same-sex marriage. Results spatio-temporal dynamics of the active and inactive forms demonstrate that proximal states have a strong similar- of ROPs. As will be shown, Turing bifurcation plays a key ity in both the state-law and the senators’ opinions, and role on the triggering phenomenon that gives place to such states with similar state-government ideology are predic- a localised structures. By using a singular perturbation tors of the senators’ opinions. Moreover, we find from analysis to study 2D localised spatial patterns of active a time-lagged correlation that state-laws show maximum ROPs, it is shown that the spatial variations in the model, dependencies on senators’ opinions from one year prior. due to the auxin gradient, lead to a slow spatial alignment Thus, change in opinion is not only a result of negotiations of the localised regions of active ROPs along the midline among the individuals, but also reflects inherent spatial of the plant cell. and political similarities. We build a social impact model Victor F. Brena-Medina of state-law adoption in light of these results and verify its Universidad Nacional Autonoma de Mexico predictive power. University of Bristol [email protected] Subhradeep Roy, Nicole Abaid Virginia Polytechnic Institute and State University [email protected], [email protected] Michael Ward Department of Mathematics University of British Columbia MS76 [email protected] A Persistent Homology Approach Towards a Ther- modynamic Description of Insect Swarms MS77 After 1952: Alan Turing’s Later Work on Pattern Various animals from birds and fish to insects tend to form Formation aggregates, displaying self-organized collective swarming behavior. Due to their frequent occurrence in nature and Alan Turing’s paper ‘The chemical basis of morphogenesis’ their implications for engineered, collective systems, these [Phil.Trans.R.Soc.Lond.B 237, 37–72 (1952)] remains systems have been investigated and modeled thoroughly hugely influential despite being his only published work in for decades. Common approaches range from modeling the field of mathematical biology. In this talk I will discuss them with coupled differential equations on the individ- the later development of his ideas as revealed by lesser- ual level up to continuum approaches. We present an al- known, but readily available, archive material. This mate- ternative, topology-based approach for describing swarm- rial shows that Turing studied differential equation mod- ing behavior at the macroscale rather than the microscale. els (motivated by various biological phenomena including We study laboratory swarms of Chironomus riparius, a fly- phyllotaxis and the unicellular marine organisms Radio- ing, non-biting midge. To obtain the time-resolved three- laria) that present a significantly more complicated math- dimensional trajectories of individual insects, we use a ematical challenge than the well-known analysis contained multi-camera stereoimaging and particle-tracking setup. in the 1952 paper. In taking on this challenge, Turing’s To investigate the swarming behavior in a topological work anticipates (i) the description of patterns in terms of sense, we employ a persistent homology approach to iden- modes in Fourier space and their nonlinear interactions, (ii) tify persisting structures and features in the insect swarm the construction of the well-known model equation usually that elude a direct, ensemble-averaging approach. We are ascribed to Swift & Hohenberg, published 23 years after able to identify features of sub-clusters in the swarm that Turing’s death, and (iii) the use of symmetry to organ- show behavior distinct from that of the remaining swarm ise computations of the stability of symmetrical equilibria members. The coexistence of sub-swarms with different corresponding to spatial patterns. It is also notable that features resembles some non-biological systems such as ac- Turing’s later work appears to have been carried out sepa- tive colloids or even thermodynamic systems. rately from, and in parallel with, related developments in fluid mechanics. Michael Sinhuber, Nicholas T. Ouellette Stanford University Jonathan Dawes DS17 Abstracts 209

University of Bath wide array of applications. In seeking to understand the [email protected] mechanisms of pattern formation, studying the effect of parameters on the formation and evolution of complex spatio-temporal patterns offers valuable insight into under- MS77 lying mechanisms driving the system. However, determin- Patterns in the Starch-Iodine Reaction ing these parameters can be difficult and computationally expensive. We believe that the parameters influence the The reaction between starch and iodine has interested peo- dynamic data in a way that is detectable by persistent ho- ple since 1812, including a young chemically intrigued Alan mology. Using a finite-dimensional vector representation of Turing. The interaction of iodine vapor with a starch solu- persistence diagrams called persistence images, we are able tion can produce a variety of patterns such as stripes and to leverage machine learning techniques to classify data by hexagons. A model to explain this reaction which makes parameters and to consider the temporal evolution of the use of the cooperative binding of iodine molecules to the pattern. The reduction of the dynamic data using persis- starch helix will be presented. tent homology still retains a remarkable amount of infor- mation that is useful for parameter classification and we Derek Handwerk are able to achieve good classification results for simulated Colorado State University data. [email protected] Rachel Neville Colorado State University MS77 [email protected] Phyllotaxis: The Hows Rather Than the Whys

Phyllotaxis, the arrangement of phylla, leaves, flowers, flo- Patrick Shipman rets, bracts, on the shoot apical meristems (SAMS) of Department of Mathematics plants has mystified and fascinated natural scientists for Colorado State University more than two thousand years. Recently there has been [email protected] considerable progress in understanding the observed behav- iors. This talk will review the progress in understanding MS78 what the biochemical processes are which give rise to what is seen and will contrast the results with those obtained Witness Complexes for Time Series Analysis by teleological approaches based on the idea that plants d A scalar time-series can be “unfolded” into R by delay organize their SAMS to optimize access to light and/or coordinate reconstruction. In the best case, this gives an nutrients. What is remarkable is that in many respects attractor that is topologically equivalent to that of the un- both approaches lead to similar patterns which suggests derlying dynamical system. We can then compute the per- that perhaps in many contexts nature uses pattern form- sistent homology of the data using a variety of complexes, ing processes to achieve optimal outcomes. e.g., Cech,ˇ Vietoris-Rips, or alpha. To be more computa- Alan Newell tionally efficient we use a witness complex: it can provide University of Arizona a sparser representation and yet be faithful to the homol- Department of Mathematics ogy. Topologically accurate delay reconstruction requires [email protected] choice of appropriate values for a d and a time delay. In practice, these must be estimated from the data, and the estimation procedures are heuristic and often problematic. MS78 Recent work of Garland et al. demonstrates that accu- Combinatorial Approximation and Discrete-Time rate persistent homology computations are possible from Dynamics witness complexes built from delay reconstructions with d below that demanded by the theory. Following this, we in- Researchers have used Conley index theory as a base for troduce novel witness relations that incorporate time and computer-assisted proof techniques for dynamical systems. explore the robustness of the resulting homology with re- In this talk, we will explore some recent work in automat- spect to choice of delay. We utilize the witness map, a ing approaches for uncovering highly complicated, often multivalued map on the witness complex induced by the chaotic, dynamics in discrete-time systems governed by it- temporal ordering of the data and velocity estimates. The erated maps. I will also briefly discuss challenges in ex- new relations seek to inhibit data points from witnessing tending these techniques to systems described solely by landmarks traveling in disparate directions and that are time series measurements and give sample computations on distinct branches of an attractor, as these can suggest for illustration. a false connection, due to the particular reconstruction.

Sarah Day Nicole Sanderson The College of William and Mary University of Colorado [email protected] Department of Mathematics [email protected]

MS78 Classification of Pattern-Forming Systems Using MS78 Persistence Topological Data Analysis of Stochastic Collective Motion Complex spatial-temporal patterns can be difficult to char- acterize quantitatively, but examples of such patterns We introduce computational persistent homology, the are ubiquitous. For example, the anisotropic Kuromoto- workhorse of topological data analysis, to a dynamical sys- Sivashinky equation which has been derived to model tems audience. Then, we apply this tool to analyze a sem- pattern-forming systems driven far from equilibrium in a inal model of collective motion due to Vicsek et al. (1995). 210 DS17 Abstracts

The model describes an ensemble of self-driven particles. a purely data-driven framework in which, first, an intrin- At each moment, each particle’s heading is determined by sic representation is derived without prior knowledge on interactions with neighbors, subject to noise. Using time the system by applying diffusion maps. Second, we show series of the particles’ positions and headings, we assign a that the dynamics of the constructed representation are ap- topological signature to the evolving ensemble. This signa- proximately linear, even for highly non-linear systems, and ture identifies dynamical events that traditional methods propose two filtering frameworks based on a linear observer do not. Finally, we pose open questions related to topolog- and based on the Kalman filter. These filtering schemes en- ical signatures averaged over many simulations of the noisy able to incorporate the inherent dynamics and time depen- model. No previous knowledge of topology is required. dencies between consecutive system observations into the diffusion maps coordinates. We demonstrate the benefits Chad M. Topaz of our approach on simulated data and on real recordings Dept. of Mathematics, Statistics, and Computer Science from various applications. Macalester College [email protected] Tal Shnitzer, Ronen Talmon Technion [email protected], [email protected] MS79 Extensions of Koopman Operator Theory: Jean-Jacques Slotine Stochastic Dynamical Systems and Partial Differ- MIT ential Equations [email protected]

We extend the theory of Koopman operators to the case of random dynamical systems. We show that the stochas- MS80 tic Koopman operator for random linear systems, defined Noise-induced Transitions between Alternative using expectation over the state space of the underlying Stable Periodic Orbits of a Periodically-forced Sys- random process, admits eigenfunctions related to expec- tem: Is there a Preferred Phase for Tipping? tation of eigenvectors of the system. We use this to, via conjugacy, provide a theory for a large class of random We consider a simple periodically-forced 1-D Langevin dynamical systems. We also describe an extension of the equation which possesses two stable periodic orbits in the Koopman operator theory for partial differential equations, absence of noise. We ask the question: is there a preferred where we define the notion of eigenfunctionals of the Koop- phase of the periodic-forcing and a most likely transition man operator and describe consequences for reduced order path between the stable orbits? The question arose in the modeling of the dynamics. setting of a conceptual Arctic sea ice model in a bistable region, where we were interested in whether it is more vul- Igor Mezic nerable to noise in the winter or summer. We compare University of California, Santa Barbara results for our model problem obtained in three different [email protected] ways: (1) calculating statistics of the tipping time obtained from path-wise analysis, (2) computing optimal transition path using a least-action principle, and (3) examining fea- MS79 tures of the steady state solution of the associated Fokker Reconstruction of the Koopman Operator’s Spec- Planck equation for the probability density. Results for the tral Measure from Data preferred tipping phase are compared with the determinis- tic aspects of the problem, and interpreted for the Arctic In data-driven modeling using the Koopman operator, it is sea ice model. often implicitly assumed that the operator either has a pure point spectrum or the observables under consideration are Yuxin Chen in the span of the operator’s eigenfunctions. In this talk, we Northwestern University discuss methods of reconstructing the operator’s spectral [email protected] measure from data with an emphasis on the continuous part of the spectrum. Mary Silber University of Chicago Ryan Mohr Dept. of Engineering Sciences and Applied Mathematics University of California, Santa Barbara [email protected] [email protected] John Gemmer MS79 Wake Forest University Manifold Learning with Contracting Observers for Department of Mathematics Data-Driven Dynamical Systems Analysis [email protected]

High dimensional signals generated by dynamical systems Alexandria Volkening arise in many fields of science. For example, many biomed- Division of Applied Mathematics ical signals can be modeled by few latent physiologically- Brown University related variables measured by a large set of noisy sensors. alexandria [email protected] In such applications the goal is to identify the latent in- trinsic variables, which describe the true, intrinsic state of the system. We approach the problem from a geometric MS80 analysis standpoint by applying manifold learning. Our Sequential Escapes for Network Dynamics main assumption is that the accessible high dimensional data (the observations of the system) lie on an underly- It is well known that the addition of noise in a multi- ing nonlinear manifold of lower dimensions. We present stable system can induce random transitions between sta- DS17 Abstracts 211

ble states. Analysis of the transient dynamics responsi- [email protected] ble for these transitions is crucial to understanding, for example, the evolution of epileptic seizures. We consider sequential transitions, also called escapes, of nodes in di- MS81 rected networks. We assume that each node has two stable Spatio-temporal Canards in Neural Field Models states, one of which is only marginally stable. All nodes start in this marginally stable state and once a node has Canards are special solutions to ordinary differential equa- escaped we assume that the transition times back are as- tions that follow invariant repelling slow manifolds for long tronomically large by comparison. Using numerical and time intervals. In realistic biophysical single cell models, theoretical techniques we explore how escape times of the canards are responsible for several complex neural rhythms network are effected by changes in network structure, noise observed experimentally, but their existence and role in amplitude and coupling strength. spatially-extended systems is largely unexplored. We de- scribe a novel type of coherent structure in which a spatial Jennifer L. Creaser pattern displays temporal canard behaviour. Using interfa- The University of Exeter cial dynamics and geometric singular perturbation theory, Centre for Predictive Modelling in Healthcare we classify spatio-temporal canards and give conditions for [email protected] the existence of folded-saddle and folded-node canards. We find that spatio-temporal canards are robust to changes Jennifer L. Creaser in the synaptic connectivity and firing rate. The theory University of Exeter, UK correctly predicts the existence of spatio-temporal canards EPSRC Centre for Predictive Modelling in Healthcare with octahedral symmetries in a neural field model posed [email protected] on the unit sphere. This is joint work Mathieu Desroches and Edgar Knobloch. Peter Ashwin Daniele Avitabile University of Exeter, UK School of Mathematical Sciences [email protected] University of Nottingham [email protected] Krasimira Tsaneva-Atanasova University of Exeter [email protected] MS81 A Robust Neural Integrator Based on the Interac- tions of Three Time Scales MS80 Abstract Not Available At Time Of Publication. Rate-Induced Tipping Through Hopf Bifurcations in Fast/Slow Systems Bard Ermentrout University of Pittsburgh We analyze rate-dependent tipping in a fast/slow system. Department of Mathematics In the reduced co-moving system we find a Hopf bifurcation [email protected] as the rate parameter increases. This implies extra struc- ture as the system moves from tracking a quasi-static equi- librium to tipping, and poses new questions about what MS81 defines a tipping point. In this kind of rate-induced tipping Ducks in Space: From Nonlinear Absolute Insta- phenomenon, the state tracks the quasi-static equilibrium bility to Noise-sustained Structures in a spiral corresponding to the emerging periodic orbit in the Hopf bifurcation of the reduced system. A subcritical pattern-forming system with nonlinear advec- tion in a bounded domain is recast as a slow-fast system in Jonathan Hahn space and studied using a combination of geometric singu- University of Minnesota lar perturbation theory and numerical continuation. Two [email protected] types of solutions describing the possible location of sta- tionary fronts are identified, one of which is present for all values of the bifurcation parameter while the other is present for zero or sufficiently small inlet boundary con- MS80 ditions but only when the bifurcation parameter is large Non Smooth Tipping from Paleoclimate enough. For slightly larger inlet boundary condition a con- tinuous transition from one type to the other takes place as the bifurcation parameter increases. The origin of the two A recently-developed conceptual climate model incorporat- solution types is traced to the onset of convective and abso- ing Earth’s energy balance, ice-albedo feedback, and the lute instability on the real line. The role of canard trajecto- mass-balance principle has been shown to exhibit limit cy- ries in the transitions between these states is clarified and cles reminiscent of glacial cycles. The model is nonsmooth the stability properties of the resulting spatial structures with a ”switch” that is activated when the ice mass balance are determined. Front location in the convective regime is reaches a certain threshold. In this talk, I will present the highly sensitive to the upstream boundary condition and model, the tipping behavior that leads to large oscillations, its dependence on this boundary condition is studied using and I will discuss its physical interpretation. a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical Esther Widiasih properties of the system subjected to random or stochastic Mathematics and Science Subdivision boundary conditions at the inlet are interpreted using the University of Hawai‘i West O‘ahu deterministic slow-fast spatial-dynamical system. This is 212 DS17 Abstracts

joint work with D. Avitabile, M. Desroches and M. Krupa. tently infected by HIV, is a major barrier to cure HIV infection. Extensive efforts have focused on shock and Edgar Knobloch kill strategies to purge the latent reservoir. The idea is University of California, Berkeley - to shock the latently infected cells using latency reversing Nonlinearity, Institute of Physics agents (LRAs) to activate HIV gene expression, and then [email protected] these cells can be killed rapidly via viral cytopathic effects or immune-mediated cell death. Here, we focus on the combination therapies involving both LRAs and immuno- MS81 therapeutics that active HIV expression in latently infected Role of Gap Junctions in a Neuron Astrocyte Net- cells and boost immune-mediated killing to purge the reser- work Model voir. We construct mathematical models to describes the dynamics of latently infected cells under the combination A detailed biophysical model for a neuron/astrocyte net- therapies. We show that in addition to shock and kill rates, work is developed to explore mechanisms responsible for a previously understudied parameter, the length of time the initiation and propagation of cortical spreading depo- that HIV antigen is expressed and can be recognized by im- larizations and the role of astrocytes in maintaining ion mune effector cells, plays an important role in determining homeostasis, thereby preventing these pathological waves. the efficacy of HIV latency reversal strategies. Therefore, In particular, we consider the neuroprotective role of astro- in addition to increasing shock and kill rates, drug develop- cyte gap junction coupling. The model demonstrates that ment should focus on drugs that maximize the duration of a syncytium of electrically coupled astrocytes can main- HIV expression and antigen presentation both during and tain a physiological membrane potential in the presence after treatment. We further estimate the potential efficacy of an elevated extracellular potassium concentration and of a recently developed immune-therapeutics based on the efficiently distribute the excess potassium across the syn- Dual-Affinity Re-Targeting proteins based on experimen- cytium. This provides an effective mechanism for delaying tal data, and develop treatment strategies based on the or preventing the initiation of spreading depolarizations. characteristics of the therapeutics. David H. Terman The Ohio State University Ruian Ke Department of Mathematics North Carolina State University [email protected] ruian [email protected]

MS82 MS82 Skipping a Pill to Prevent Drug Resistance? Impact of Innate Immune Response on Spatial Dy- namics of Acute In-host Viral Infection The antiviral drug oseltamivir has proven very effective in reducing influenza symptom intensity and accelerating recovery. However oseltamivir’s utility can be compro- The innate immune response, particularly interferon pro- mised by the emergence of drug resistant strains. We duction and signaling, represents the first line of defense employ a hybrid deterministic and stochastic simulation against viral invasions. Previously, in-host viral dynam- model which combines models of drug pharmacokinetics, ics and the innate immune response have been studied pharmacodynamics, viral kinetics and immune responses mostly as a spatially homogeneous system. Though valid to investigate oseltamivir use accross a distribution of pa- for some diseases and tissue types, this assumption of ho- tient types. Via simulation, we investigate the interplay be- mogeneous mixing ignores the biological reality of the spa- tween drug efficacy and innate immune responses that can tial distribution of cells and diffusion of viruses and inter- prevent or accelerate emergence of drug resistant strains. feron molecules in an organ or epithelium for many other Treatment regimen initiation during the incubation period viral systems. is typically found to be the main factor leading to resis- tance emergence in a patient: resistant strains are selected Michael Lavigne, Hayley Russell, Ruian Ke when treatment is too late to prevent de novo mutation to North Carolina State University a drug resistant variant, but still early enough to suppress [email protected], n/a, ruian [email protected] development of immune responses that would control both drug sensitive and drug resistant strains. Thus, although conventional wisdom says improper use of antivirals, in- MS82 cluding dose skipping, leads to drug resistance, skipping a Targeting the Achilles Heel of HIV Reservoir Per- dose may actually reduce probability of emergent drug re- sistence sistance by allowing for immune response development. We will explore non-traditional treatment regimens aimed at HIV cure remains elusive due to a reservoir of latently maximizing treatment effectiveness while minimizing the infected memory CD4+ T-cells which persist despite risk of drug resistance emergence. decades of antiretroviral therapy (ART). Using mathemat- Jessica M. Conway ical model fit to existing data, we demonstrate that after Pennsylvania State University one year of ART, most reservoir cells are generated via pro- [email protected] liferation rather than new viral infection. We discuss meth- ods to differentiate the percentage of proliferation events which are antigen driven rather than programmed home- MS82 ostasis among different reservoir T cell subsets. We con- Modeling Combination Therapies Using Latency clude by modeling antiproliferative therapy as a means to Reversing Agents and Immunotherapeutics to reduce reservoir volume. Cure Hiv Joshua Schiffer The existence of latent reservoir, a population of cells la- Fred Hutchinson Cancer Research Center DS17 Abstracts 213

jschiff[email protected] cur in the tissue depth, but measurements are restricted primarily to the tissue surfaces. We will discuss recent advances in using data assimilation to use observations MS83 sparse in time and space to reconstruct cardiac electrical Cardiac Re-Entry Evolution in MRI-based Models dynamics. We show ways of handling discrepancies be- of the Heart tween the dynamics of the observed system and the nu- merical model used, including parameter value differences, Despite over a century of study, mechanisms of cardiac ar- dissimilar wave shapes, and spatial heterogeneity in the rhythmias are poorly understood. The recently discovered observed system. In addition, we will discuss how observa- new phenomenon of dissipative vortices interaction with tions can be used to estimate parameter values and show sharp variations of thickness in excitable layer suggests a results in the cardiac setting. cardiac re-entry interaction with fine anatomical features such as pectinate muscles and terminal crest in atria, which might cause considerable displacement of the re-entry’s es- Elizabeth M. Cherry tablished localisation compared to where it was first ini- Rochester Institute of Technology tiated. With the recent advance in DT-MRI technology, School of Mathematical Sciences more detailed models of heart anatomy are becoming avail- [email protected] able for inclusion into anatomically realistic computer sim- ulations of cardiac arrhythmias, providing a fascinating in- Darby Cairns silico test bed with unimpeded access to the whole heart Rochester Institute of Technology at greater spatial and temporal resolutions than in exper- [email protected] iment, to test the effects of a patient anatomy on cardiac re-entry dynamics. Nicholas LaVigne Cornell University Irina Biktasheva [email protected] University of Liverpool [email protected] Nathan Holt Rochester Institute of Technology Vadim N. Biktashev [email protected] College of Engineering, Mathematics and Physical Sciences Flavio H. Fenton University of Exeter Georgia Institute of Technology [email protected] fl[email protected]

Arun Holden Matthew J. Hoffman University of Leeds, UK Rochester Institute of Technology [email protected] [email protected]

Sanjay R. Kharche University of Exeter MS83 [email protected] Reconstruction of Cardiac Dynamics from Cardiac Eleftheria Pervolaraki Deformation Mechanics University of Leeds, UK [email protected] The self-organizing, dynamic pattern forming processes that underlie the arrhythmic electrical activity during car- Girish Ramlugun diac fibrillation remain insufficiently understood due to the Auckland BioEngineering Institute, Auckland, New absence of requisite in-depth imaging techniques, capable Zealand of visualizing the fibrillatory electrical patterns through- [email protected] out the volume of the cardiac muscle. Here, we show that intracardiac wave phenomena can be reconstructed Gunnar Seemann by imaging and analyzing the fibrillatory contractile de- Karlsruhe Institute of Technology formation mechanics of the cardiac muscle. Combining Germany optical mapping, ultrasound and computational modeling, [email protected] we show that the arrhythmic electromechanical activity of the heart exhibits closely correlated electrical and mechan- Bruce smaill ical dynamics. During fibrillation in isolated hearts, we Auckland BioEngineering Institute, Auckland, New observed that electrical spiral wave rotors can be accom- Zealand panied by rotational elasto-mechancial patterns, which like [email protected] fingerprints of vortex activity depict characteristic dynamic properties of fibrillation and reveal the underlying struc- tural organization of the wave dynamics evolving inside MS83 the cardiac muscle. In summary, three-dimensional cardiac Incorporating Data into Models of Cardiac Electri- deformation mechanics can reveal elasto-mechanical topo- cal Dynamics logical defect lines or lines of mechanical phase singularity, which reflect vortex wave dynamics such as scroll wave ac- Cardiac tissue displays a broad range of dynamical behav- tivity. In particular, the combination of the imaging data iors, including bifurcations and spiral and scroll waves of with computational modeling of cardiac electromechanics electrical activity. However, understanding experimental can provide solutions to the inverse problem in which elec- results is challenging because complex behavior can oc- trical wave phenomena are computed from mechanical de- 214 DS17 Abstracts

formation. Russell Jeter Georgia State University Jan Christoph, Stefan Luther [email protected] Max Planck Institute for Dynamics and Self-Organization Research Group Biomedical Physics Vladimir Belykh [email protected], [email protected] Lobachevsky State University of Nizhny Novgorod, Russia [email protected] MS83 Cardiac Response to Low-Energy Field Pacing MS84 Challenges the Standard Theory of Defibrillation A Consensus Dynamics with Delay-Induced Insta- bility can Self-Regulate for Stability via Agent Re- The electrical response of myocardial tissue to periodic grouping field stimuli has been considered as the basis for low-energy anti-fibrillation pacing (LEAP), potentially more effective Dynamics of many multi-agent systems is influenced by than traditional single high-energy shocks. In conventional communication/activation delays D. In the presence of de- models, an electric field produces a highly non-uniform re- lays, there exists a certain margin called the delay margin sponse of the myocardial wall, with discrete excitations, DM determining the largest delay with which the system or hot spots (HS), occurring at cathodal tissue surfaces or stability holds (D ¡ DM). This margin depends strongly on large coronary vessels. We test this prediction using novel agents dynamics and the agent network. In this talk, three 3D tomographic optical imaging. Experiments were per- key elements, namely, the delay margin, network graph, formed in pig ventricular wall stained with near-infrared and a distance threshold conditioning two agents connec- voltage-sensitive fluorescent dye DI-4-ANBDQBS. The 3D tivity are considered in a multi-agent consensus dynamics coordinates of HS were determined using alternating tran- under delay D. We report that when the dynamics is un- sillumination. The peak HS distribution is located deep stable under this delay (D ¿ DM), its states can still remain inside the heart wall and the depth is not significantly af- bounded, even for arbitrarily large threshold values, pre- fected by field polarity. We did not observe the strong colo- venting agents to disperse indefinitely. This mechanism can calization of HS with major coronary vessels anticipated also make the system recover stability in a self-regulating from theory. Yet, we observed considerable lateral dis- manner, mainly induced by complete network separation placement of HS with field polarity reversal. Models that and enhanced delay margin. Under certain conditions, un- deemphasized lateral intracellular coupling and accounted stable consensus dynamics can even keep separating into for resistive heterogeneity in the extracellular space showed smaller stable subnetwork dynamics until all agents stabi- similar HS distributions to the experimental observations. lize in their respective subnetworks. The HS distributions within the myocardial wall and the significant lateral displacements with field polarity rever- Rifat Sipahi sal are inconsistent with standard theories of defibrillation. Department of Mechanical and Industrial Engineering Extended theories based around ephaptic mechanisms may Northeastern University be necessary. [email protected]

Arkady Pertsov Min Hyong Koh SUNY Upstate Medical University Northeastern University [email protected] [email protected]

MS84 MS84 Bistable Gaits and Wobbling Induced by Spotting Leaders in Groups of Self-propelled Par- Pedestrian-Bridge Interactions ticles

Several modern footbridges around the world have experi- In animal groups, collective behavior affords several advan- enced large lateral vibrations during crowd loading events. tages in avoiding predators, foraging, and mating. Collec- The onset of large-amplitude bridge wobbling has gen- tive motion manifests into highly coordinated maneuvers, erally been attributed to crowd synchrony; although, its determined by local interactions among individuals. Un- role in the initiation of wobbling has been challenged. To derstanding the mechanisms behind these interactions is study the contribution of a single pedestrian into over- central to elucidate the hows and whys collective behavior. all, possibly unsynchronized, crowd dynamics, we use A particularly critical question is to detect and classify a bio-mechanically inspired inverted pendulum model of leader-follower relationships in the group. In the technical human balance and analyze its bi-directional interaction literature, several methods have been proposed to recon- with a lively bridge. Through theory and numerics, we struct static interaction networks in coupled dynamical sys- demonstrate that pedestrian-bridge interactions can induce tems, including linear correlation analysis, Granger causal- bistable lateral gaits such that switching between the gaits ity, transfer entropy, and event synchronization. While can initiate large-amplitude wobbling. We also analyze the these analyses have been helpful in reconstructing network role of stride frequency and the pedestrian’s mass in hys- models from neuroscience to public health, rules on the teretic transitions between the two types of wobbling. Our most appropriate method to use for a specific dataset are results support a claim that the overall foot force of pedes- lacking. Their application to the study of collective mo- trians walking out of phase can cause significant bridge tions is particularly challenging due to the time-varying vibrations. nature of the underlying interaction networks. In this talk, we demonstrate the possibility of spotting leaders in Igor Belykh a group from raw positional data, in an authentic data- Department of Mathematics and Statistics driven, model-free approach. Our approach cogently com- Georgia State University bines estimations from two or more network reconstruction [email protected] methods to maximize accuracy. We test our framework on DS17 Abstracts 215

synthetic data of groups of self-propelled particles where for epidemic models with waning immunity. We introduce a single agent acts as a leader and we offer insight on the a mathematical model formulated by a system of delay dif- application of the approach to study fish schooling. ferential equations in order to give a possible explanation of periodic outbreak of mycoplasma pneumoniae observed Violet Mwaffo, Maurizio Porfiri in Japan. Stability analysis and numerical studies for pe- Dept. of Mechanical and Aerospace Engineering riodic solutions will be presented together with biological New York University Polytechnic School of Engineering interpretations. We also discuss global asymptotic stabil- vmwaff[email protected], mporfi[email protected] ity of SIS type epidemic model. The talk is based on a collaboration work with G. Kiss, R. Omori, G. Rost and G. Vas. MS84 Synchronization and Pinning Control of Smooth Yukihiko Nakata and Piecewise-smooth Networks with Noise Shimane University Japan This talk will be focused on the study of synchronization in [email protected] networks of nonlinear dynamical systems affected by noise. Specifically, we will provide sufficient conditions for syn- chronization, which explicitly link together network topol- MS85 ogy, nodes’ dynamics and noise diffusion. We will first Constant, Distributed and State Dependent Delays consider smooth nodes dynamics and then present some in Models of Waning and Boosting of Immunity extensions to networks of nodes with piecewise-smooth dy- namics. After introducing the key theoretical results, we Abstract Not Available At Time Of Publication. discuss how the theoretical results can be used to devise synchronization schemes based on pinning (controlling) the Gergely Rost dynamics of a fraction of the network nodes. The effec- Bolyai Institute, University of Szeged tiveness of the methodology will be validates on a set of Szeged, Aradi v´ertank tere 1, H-6720 Hungary relevant applications in Smart Cities and Systems Biology. [email protected]

Giovanni Russo MS85 IBMIreland Research Lab, Dublin, Ireland Feedback Stabilizability with Time-Delayed Feed- [email protected] back

Mario Di Bernardo Time-delayed feedback control is one of the most success- University of Bristol ful methods to discover dynamically unstable features of a Dept of Engineering Mathematic dynamical system in an experiment. This approach feeds [email protected] back only terms that depend on the difference between the current output and the output from a fixed time T ago. Thus, any periodic orbit of period T in the feedback- MS85 controlled system is also a periodic orbit of the uncontrolled Gevrey Properties for Center Manifolds of Delay system, independent of any modelling assumptions. It has Differential Equations been an open problem whether this approach can be suc- cessful in general, that is, under genericity conditions sim- We consider a system of delay equations of the form ilar to those in linear control theory (controllability), or if there are fundamental restrictions to time-delayed feedback x˙(t)= (t) control. We show that, in principle, there are no restric- y˙(t)=ay(t)+by(t − h)+ (t)F (x(t),y(t),y(t − h), (t)) tions. This paper proves the following: for every periodic ˙(t)=0, orbit satisfying a genericity condition slightly stronger than classical linear controllability, one can find control gains where F is a holomorphic function of its variables and con- that stabilize this orbit with extended time-delayed feed- ditions are imposed on a and b which guarantee that the back control. While the papers techniques are based on center manifold of this system is two dimensional. Using linear stability analysis, they exploit the specific proper- techniques similar to the ones used in [M. Canalis-Durand, ties of linearizations near autonomous periodic orbits in J.-P. Ramis, R. Sch¨afke, and Y. Sibuya. Gevrey solutions nonlinear systems, and are, thus, mostly relevant for the of singularly perturbed differential equations. J. Reine analysis of nonlinear experiments. Angew. Math., 2000.] we show that the center manifold is Gevrey asymptotic to its Taylor series. The Gevrey prop- Jan Sieber erty can be seen as something in between infinite smooth- University of Exeter ness and being holomorphic. As such this result improves [email protected] what is known from general center manifold theory.

Karel Kenens MS86 Hasselt University Piecewise-Smooth Networks Belgium [email protected] The aim of this talk will be to discuss the problem of study- ing convergence in networks of piecewise smooth dynam- ical systems. This class of networks arises often in those MS85 applications where systems affected by discontinuities on Delay Equations for Epidemic Models with Waning a macroscopic timescale are coupled together. Examples Immunity include coupled mechanical systems with impacts and fric- tion (such as those in automotive drivelines), networks of In this talk we discuss dynamical aspects of delay equations power electronic converters, mobile robots etc. I will start 216 DS17 Abstracts

by proposing a classification of such networks in terms of ”Synchronization in on-off stochastic networks: windows of the source of their discontinuities and then present our lat- opportunity”, IEEE Transactions on Circuits and Systems est results on studying convergence and the emergence of I: Regular Papers (2015)] and discrete-time [O. Golovneva, synchronous collective behaviour in these networks. The et al., ”Windows of opportunity for synchronization in cou- approach is based on the use of global results from differ- pled stochastic maps,” Physica D (2017)] oscillators. For ential stability analysis of dynamical systems and contrac- discrete-time oscillators, we analytically demonstrate the tion theory. Other approaches will be also discussed and emergence of windows of opportunity and elucidate their contrasted with Lyapunov based techniques. All theoreti- nontrivial relationship with the stability of synchronization cal derivations will be illustrated by a set of representative under static coupling. applications. Russell Jeter Mario Di Bernardo Georgia State University University of Bristol [email protected] Dept of Engineering Mathematic [email protected] Olga Golovneva New York University Polytechnic School of Engineering [email protected] MS86 Functional Asynchronous Networks Maurizio Porfiri Dept. of Mechanical and Aerospace Engineering We have developed a theory of asynchronous networks that New York University Polytechnic School of Engineering gives a theoretical and conceptual framework for the study mporfi[email protected] of network dynamics where nodes can evolve independently of one another, be constrained, stop, and later restart, and where the interactions between different components Igor Belykh of the network may depend on time, state, and stochas- Department of Mathematics and Statistics tic effects – features characteristic of networks in tech- Georgia State University nology, neuroscience and biology. Dynamics is typically [email protected] piecewise smooth and there are relationships with Filippov systems. In this talk we describe the theory of functional asynchronous networks and give a decomposition theorem MS86 that allows one to understand the function of the network Community Change-point Detection in Time- in terms of the function of constituent subnetworks. The dependent Networks theorem addresses a question raised by Alon in his book on systems biology: “Ideally, we would like to understand the Time-dependent networks are frequently used to represent dynamics of the entire network based on the dynamics of complex systems in which dynamic interactions occur over the individual building blocks.’ Nonsmoothness is a crucial time. Understanding if, when and how such networks ingredient needed for the result. In networks modelled by change over time can provide valuable insights into how smooth dynamical systems, all nodes are effectively cou- complex systems evolve. However, the variation in indi- pled to each other at all times and information propagates vidual pairwise interactions can be hard to interpret and instantly across the entire network. A spatiotemporal de- is often not particularly meaningful on its own. Instead, composition is only possible if the network dynamics is community detection can be used to identify a more readily nonsmooth and (subsets of) nodes are allowed to evolve interpretable, coarse-grained representation of the system. independently of each other for periods of time. This al- Here we use a hierarchical model to capture stochastically lows the identification of dynamical units, each with its equivalent nodes across multiple resolutions and introduce own function. an efficient spectral method for inferring the community as- signment. We then apply methods from statistical change- Michael Field point detection to detect significant changes in the large- Rice University and Imperial College London scale structure of the network. mikefi[email protected] Michael Schaub Christian Bick Massachusetts Institute of Technology University of Exeter, UK [email protected] [email protected] Leto Peel Universite catholique de Louvain MS86 [email protected] Synchronization in On-off Stochastic Networks: Windows of Opportunity MS87 We study dynamical networks whose topology stochasti- Modelling Movement and Coordination in the Mir- cally changes, on a time scale that ranges from fast to slow. ror Game: From Dyads to Groups Most of the current understanding of synchronization in evolving networks relies on the fast switching hypothesis, This talk will start with a brief overview of the problem where the network dynamics evolves at a much faster time of modelling the emergence of synchronization and coor- scale than the individual units. In this talk, we go beyond dination between humans performing a joint task. Atten- fast switching and reveal unexpected windows of interme- tion will be focussed on the mirror game, a paradigmatic diate switching frequencies in which synchronization in the example where two players need to coordinate each oth- evolving network becomes stable even though it is unsta- ers motion. The use of dynamical systems models will be ble in the averaged/fast-switching network. We consider presented together with the design of virtual players able networks of both continuous-time [R. Jeter and I. Belykh, to interact with a human player. Then, the problem of DS17 Abstracts 217

modelling the emergence of leadership and coordination in tions of the population of dancers committed to different a human ensemble will be described. We will show that strategies (dance motion primitives) as a function of repli- the coordination level of the group depends on the specific cation (commitment to strategies with high fitness) and way in which each individual moves when isolated from mutation (spontaneous switching among strategies). Ex- the others, and on the pattern of their visual couplings. A perimental data suggest that dancers make decisions gov- mathematical model based on networks of nonlinear oscil- erned by an explore-exploit tradeoff in which the group os- lators will be proposed to capture and explain the observed cillates between exploiting multiple coexisting primitives group dynamics. and exploring a single new primitive. We adapt the replicator-mutator dynamics to model the dancers move- Francesco Alderisio ment decisions by introducing a nonlinear fitness function University of Bristol and a feedback controlled bifurcation wherein the mutation Department of Engineering Mathematics rate is driven in a slow time-scale by the population frac- [email protected] tions in the fast time-scale. We show how the adapted replicator-mutator dynamics emulate the oscillatory be- Gianfranco Fiore havior of the dance population in the case of two inter- University of Bristol acting strategies and can be used to study the phasing of gianfranco.fi[email protected] exploration and exploitation in the improvisational dance.

Robin Salesse, Benoit Bardy University of Montpellier Kayhan Ozcimder,BiswadipDey [email protected], [email protected] Princeton University Department of Mechanical and Aerospace Engineering [email protected], [email protected] Mario Di Bernardo University of Bristol Dept of Engineering Mathematic Alessio Franci [email protected] Universidad Nacional Aut´onoma de M´exico M´exico [email protected] MS87 Embedded Dynamics of Multiagent Activity and Rebecca J. Lazier Social Coordination Princeton University Program in Dance Multiagent coordination is a common part of everyday so- [email protected] cial activity. Identifying the dynamic processes that shape and constrain the complex, time-evolving patterns of such Daniel Trueman coordination often requires the development of dynamical Princeton University models to test hypotheses and motivate future research Department of Music questions. Here I review a task dynamic framework for [email protected] modeling multiagent behavior and social coordination, and illustrate the application of this framework using a range NaomiE.Leonard of shepherding and object movement tasks. Emphasizing Princeton University self-organization, I will highlight how the behavioral coor- Department of Mechanical and Aerospace Engineering dination that characterizes many multiagent and social ac- [email protected] tivities emerges naturally from the physical, informational, and biomechanical constraints of an agent-environment task context. Using examples from human-robot interac- MS87 tion and artificial agent research, I will also illustrate how task dynamic models that effectively capture the emergent Behavioral Dynamics of Collective Crowd Behavior and self-organized nature of human social coordination can be used to develop robust human-machine systems. Such The collective behavior of flocks, schools, and human systems have applications in industrial, therapeutic, and crowds is thought to emerge from local interactions be- assistive technologies contexts where close natural interac- tween individuals. There are many models of such col- tion with humans is required. lective motion, but very little experimental evidence. The behavioral dynamics approach [Warren W.H. 2006 The dy- Maurice Lamb, Patrick Nalepka, Rachel W. Kallen, namics of perception and action. Psychological Review MichaelJ.Richardson 113, 358-389] seeks to explain these global patterns by char- Center for Cognition, Action and Perception acterizing the dynamics of the local interactions, and using Department of Psychology, University of Cincinnati them to predict the emergent collective motion. Taking a [email protected], [email protected], local-to-global approach, we have developed a pedestrian [email protected], [email protected] model that captures elementary behaviors such as steering, obstacle avoidance, and pedestrian interactions, based on experiments in virtual reality [Warren W.H., Fajen B.R. MS87 2008 Behavioral dynamics of visually-guided locomotion. Using Evolutionary Dynamics to Model Structured In Coordination: Neural, behavioral, and social dynamics Improvisational Dance (A. Fuchs, V. Jirsa, Eds.), Heidelberg, Springer]. I will fo- cus on recent work that characterizes the visual coupling We investigate the mechanisms of social decision-making between a pedestrian and multiple neighbors in a virtual for a structured improvisational dance in which a group crowd, yielding a novel neighborhood structure. Recipro- of dancers makes a sequence of compositional choices cally, taking a global-to-local approach, we collect motion- among pre-defined dance motion primitives. We use the capture data on human crowds in key scenarios. Patterns replicator-mutator dynamics to model the evolution of frac- of collective motion are analyzed and can be reproduced 218 DS17 Abstracts

using agent-based simulations with just a few elementary tions often present themselves naturally from the analysis. behaviors. The results support the view that crowd be- In principle the analysis can be applied to a system of dif- havior emerges from pedestrian interactions, without in- ferential equations of any size, and the main focus of the ternal models or plans, consistent with principles of self- talk will be on a relatively large system (10 ODEs) de- organization. scribing the toxicological action of a drug called nitisinone, whereby despite its complexity a full description of the evo- William Warren lution of all components can be determined, including a Brown University number of aspects of pharmacological interest. Further ex- Cognitive, Linguistic and Psychological Sciences amples from microbiology and epidemiology will briefly be bill [email protected] presented to demonstrate the broad utility of the analytical approach.

MS88 John P. Ward Predicting the Shapes of Eco-Evolutionary Loughborough University, UK Predator-Prey Cycles Using Fast-Slow Dynamical [email protected] System Theory: Fast Evolution and Slow Ecology, Vice Versa, and Everything in Between MS89 Evolution in predator species, prey species, or both species Quantum Control and the Geometry of Flag Man- can alter the shapes of predator-prey cycles. In particular, ifolds in the absence of evolution, predator oscillations are pre- dicted to lag behind prey oscillations by a quarter period. In this talk we discuss aspects of the physics and mathe- In contrast, evolution in one species can increase the lag matics of a quantum control system interacting with its en- to a half-period and coevolution can reverse the cycles and vironment. In particular we discuss the control of an finite- cause predator peaks to occur before prey peaks. In this dimensional dissipative Lindblad system by considering the talk, I explore how the occurrence of cycles and the ob- geometry of its orbit and interorbit dynamics. This entails served phase lags depend on the relative time scales of the considering the geometry of the system, the structure of the ecological and evolutionary processes. In particular, using Lindblad operator, and the convexity associated with the fast-slow dynamical systems theory, I identify what kinds density equation. Applications are given to constructing of cycles occur in the limit where the ecological dynam- pure states. We will also discuss the general structure of ics are much faster than the evolutionary dynamics and in controlled quantum and nonlinear systems. This includes the opposite limit where the ecological dynamics are much recent work with Rooney and Rangan. slower than the evolutionary dynamics. I then use bifurca- tion theory to identify when cycles are lost/born and how Anthony M. Bloch cycle shape is altered as the relative times scales of the eco- University of Michigan logical and evolutionary variables are changed. This work Department of Mathematics shows how the stabilities of the ecological and evolutionary [email protected] subsystems can be used to predict when cycles will be born or lost and cycle shape. MS89 Michael Cortez Utah State University The Siegel Upper Half Space, Symplectic Reduc- [email protected] tion, and Gaussian Wave Packet Dynamics I will talk about the symplectic geometry behind the dy- MS88 namics of the Gaussian wave packet, particularly the role of the Siegel upper half space in parametrizing the Gaus- Rapid Evolution in Plankton Populations: A Slow- sian wave packet. The Siegel upper half space is the set of Fast Model symmetric complex matrices with positive-definite imagi- Abstract Not Available At Time Of Publication. nary parts. Understanding the geometry of this space gives insights into the description of the dynamics of the Gaus- Frits Veerman sian wave packet. Particularly, I will show that the Siegel University of Edinburgh upper half space is an example of the Marsden–Weinstein [email protected] quotient of symplectic reduction. The result provides a connection between two different formulations of Gaussian wave packet dynamics as Hamiltonian dynamical systems MS88 via symplectic reduction. Applications of Timescale Analysis to Mathemati- cal Models in Toxicology Tomoki Ohsawa The University of Texas at Dallas The modelling of many biological systems typically involves Department of Mathematical Sciences some form of abstraction via proposed sets of reaction path- [email protected] ways of interacting elements, leading naturally to systems of differential equations. It is very common using biologi- cally relevant parameters that the predicted solutions form MS89 clear distinct phases of events as a consequence of there be- Lagrange-Dirac Systems for Nonequilibrium Sys- ing a large/small parameter. Singular perturbation anal- tems ysis is a powerful tool to systematically identify and draw out relevant simplified models for each of these distinct Abstract Not Available At Time Of Publication. phases, enabling the identification of the important and unimportant biological processes involved. Moreover, pa- Hiroaki Yoshimura rameter combinations at which qualitative shifts in solu- Waseday University DS17 Abstracts 219

[email protected] provide a fertile ground for a new generation of mathemat- ical biology.

MS89 Jonathan Swinton Hamel’s Formalism for Constrained Field Theories Deodands Ltd [email protected] Separation of the position and velocity measurements in mechanics originated in Euler’s work on the dynamics of rigid body and fluid. Hamel extended this formalism MS90 from the rigid body setting to arbitrary finite-dimensional Beyond Wave-Pinning; Dynamical Mechanisms for Lagrangian mechanical systems. This talk will intro- Cell Polarity and Interdigitation duce Hamel’s formalism for classical field theories, with an emphasis on the dynamics of constrained continuum- The asymmetrical profile presented by the concentration mechanical systems. of proteins in Eukaryotic cells is the signature of cell po- larisation, a key process in developmental biology. Reac- Dmitry Zenkov tion diffusion models have been proposed in order to de- North Carolina State University scribe this breaking of symmetry. They rely on two qual- Department of Mathematics itatively different mechanisms: mass-conservation (wave- [email protected] pinning models) and diffusion driven instablility (Turing models). In this talk we investigate the connection be- tween both mechaisms by studying a generalised version MS90 of the wave-pinning model, when generic source and loss Self-Organization and Pattern Formation in Auxin terms are added. The new terms give rise to several non- Flux homogeneous spatial equilibrium solutions, namely, peaks, patterns and localised patterns. We fininish the analysis The plant hormone auxin plays a central role in growth and by studying the limit when both non-conservative terms morphogenesis. In the shoot apical meristem the auxin flux vanish, we find a continuous connection between the peak is polarized through its interplay with PIN proteins. Con- spatial solution and the fronts observed in the wave pin- centration based mathematical models of the auxin flux ning model. The last part of this talk is devoted to the permit to explain some aspects of phyllotaxis, where auxin study of the intriguing interdigitation pattern observed in accumulation points act as auxin sinks and correspond to the interface between two adjacent pavement cells on the primordia. Simulations show that these models can re- surface of leaves in plants. produce geometrically regular patterns like spirals in sun- flowers or Fibonacci numbers. We propose a mathematical Nicolas Verschueren Van Rees, Alan R. Champneys study of a related non-linear o.d.e. using Markov chain University of Bristol theory. We will next consider a concurrent model which [email protected], [email protected] is based on the so-called flux hypothesis, and show that it can explain the self-organization of plant vascular systems. MS91 Topological Analysis of Mapper and Multiscale Christian Mazza Mapper University of Fribourg Switzerland Summarizing topological information from datasets and [email protected] maps defined on them is a central theme in topological data analysis. Mapper, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map MS90 defined on the data, and produces a summary of the data Fibonacci Phyllotaxis: Models, Data and Alan Tur- by using a cover of the codomain of the map. This cover, ing via a pullback operation to the domain, produces a simpli- cial complex connecting the data points. Inspired by the Turing worked on his morphogenetic theory over the last concept, we explored a notion of a tower of covers which years of his life (Dawes, 2016). He claimed to be able to induces a tower of simplicial complexes connected by sim- explain the phenomenon of Fibonacci phyllotaxis: he stud- plicial maps, which we call multiscale mapper. We study ied the properties of lattices arising from repeated place- the resulting structure, its stability, and design practical ments of mutually repelling organs, and aimed to show that algorithms to compute its persistence diagrams efficiently. as the patterns deform the stable branch of the resulting In a recent work we discovered further connections between bifurcation pattern maintains Fibonacci structure (Swin- the mapper, its multiscale version and the domain of the ton, 2013). Only after his death was the generic nature of map that defines them. Specifically, we can now answer the Fibonacci phyllotaxis demonstrated in exactly this math- question ”what does a mulitscale mapper reveal?’ more af- ematical sense. The molecular biology of phyllotaxis has firmatively. been increasingly well characterised, with little input from this mathematics. But sunflowers really do commonly have Tamal Dey 89 spirals, and this fact will not be explicable on a molec- The Ohio State University ular basis without such input. One key approach will be [email protected] to examine how plant developmental structure responds to perturbation and defect. Sunflower heads which are not perfect specimens with clear Fibonacci structure are MS91 the most informative for this. Thanks to a citizen science The Conley Index for Sampled Dynamical Systems experiment run by the Museum of Science and Industry, Manchester we can see some of the challenges these models In late 90’ R. Forman [R.Forman, Morse Theory for Cell face (Swinton, 2016). Fibonacci structure in plants should Complexes, Advances in Mathematics (1998)] introduced 220 DS17 Abstracts

the concept of a combinatorial vector field on a CW com- the position of the features in a filtration can sometimes plex and presented a version of Morse theory for such fields. play a more vital role than persistence in the interpreta- Recently the theory has been extended towards topological tion of topological features, even though conventionally the dynamics on CW complexes, in particular the Conley in- latter is used to distinguish between signal and noise. dex theory [M. Mrozek, Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes, Mason A. Porter Foundations of Computational Mathematics, (2016)]. The University of Oxford extension is motivated by the study of dynamics known Mathematical Institute only from a sample as in the case of time series dynam- [email protected] ics. It covers such concepts as attractors, repellers, Morse decompositions, Conley-Morse graphs, stability and con- Heather Harrington tinuation. We will discuss the foundations of the theory, Mathematical Institute the bridges between the classical and combinatorial the- University of Oxford ory [ T. Kaczynski, M. Mrozek, and Th. Wanner, To- [email protected] wards a Formal Tie Between Combinatorial and Classical Vector Field Dynamics, Journal of Computational Dynam- Bernadette Stolz ics,(2016)] and some algorithmic aspects of the theory [ T. University of Oxford Dey, M. Juda, T. Kapela, M. Mrozek, and M. Przybylski, [email protected] Research in progress].

Marian Mrozek MS92 Jagiellonian University, Krakow, Poland [email protected] Dmd-Galerkin Approximation for Nonlinear Dy- namical Systems

MS91 Dynamic Mode Decomposition (DMD) is a recently de- veloped technique for obtaining reduced order models for Applications of Persistence to Time Series Analysis nonlinear dynamical systems. In this talk, we will compare A time series is simply a stream of data. However, the the use of DMD as equation free modeling and Galerkin type of output of this stream can be many different things. projection method through numerical examples. The most commonly studied and most understood form of Alessandro Alla time series is real-valued, however this is certainly not the Department of Scientific Computing only way that time series are presented in practice. For Florida State University, USA instance, a movie can be interpreted as a matrix-valued [email protected] time series. In this talk, we will look at two applications in different domains. Both of these applications utilize persis- tent homology to provide analysis even though the types MS92 of time series under study are quite different. First, we will discuss the phenomenon of chatter in machining dy- Fast and Reliable Extraction of Coherent Features namics. Chatter is the undesirable behavior exhibited by from Models and Data using the Dynamic Laplace a cutting tool which is characterized by large amplitude Operator vibrations that result in non-smooth metal parts, as well as an intense noise. Combining Taken’s embedding theo- Transport and mixing properties of aperiodic flows are cru- rem, persistence, and machine learning methods gives 97% cial to a dynamical analysis of the flow, and often have to accuracy when attempting to predict and prevent this be- be carried out with limited information. Finite-time coher- havior. Second, we will use persistence to quantify a diur- ent sets are regions of the flow that minimally mix with the nal cycle recently observed in IR hurricane data. In this remainder of the flow domain over the finite period of time case, we turn a matrix valued time series into a persistence considered; they thus provide a skeleton of distinct regions diagram valued time series and investigate the resulting around which more turbulent flow occurs. Finite-time co- periodic behavior in persistence diagram space. herent sets manifest in geophysical systems in the forms of e.g. ocean eddies, ocean gyres, and atmospheric vortices. Elizabeth Munch In real-world settings, often observational data is scattered University at Albany and sparse, which makes the already difficult problem of [email protected] coherent set identification and tracking even more chal- lenging. We develop FEM-based numerical methods that rapidly and reliably extract finite-time coherent sets from MS91 models or scattered, possibly sparse, trajectory data. The The Topological Shape of Brexit and Time- numerical methods are based on a dynamic Laplace oper- Dependent Functional Networks ator, whose eigenfunctions identify regions in phase space with persistently small boundary size relative to enclosed Persistent homology is a method from computational al- volume. Oliver Junge will discuss related work in the MS gebraic topology that can be used to study the ”shape” “Set oriented and transfer operator methods for turbulent of data. We consider two examples. In a slightly snarky flows”. (but illustrative) introductory example, we use the weight rank clique filtration and the VietorisRips filtration on a Gary Froyland pair of data sets that are related to the 2016 European UNSW Australia Union ”Brexit” referendum in the United Kingdom. We [email protected] then switch gears and use PH to study ”functional net- works” that we construct from time-series data from both experimental sources (neuroimaging) and synthetic sources MS92 (coupled Kuramoto oscillators). One of our insights is that Extraction and Prediction of Coherent Patterns in DS17 Abstracts 221

Fluid Flows Via Space-Time Koopman Analysis resolution aerial survey photographs taken over the Horn of Africa in the 1940s and 50s represent one of the ear- We discuss a method for detecting and predicting the evo- liest and richest records of vegetation banding. We com- lution of coherent spatiotemporal patterns in incompress- bine such photographs from the early 1950s with present- ible fluid flows. The approach is based on a representa- day satellite imagery, identifying the locations of the old tion of the Koopman operator governing the evolution of images, and comparing the status of the vegetation. For observables in a smooth basis learned from velocity field many areas, changes in the vegetation patterns are surpris- data through the diffusion maps algorithm. This repre- ingly modest, with individual bands remaining identifiable sentation enables the detection of coherent flow patterns with their precursors in the early images. However, others through Koopman eigenfunctions and simulation of the areas have developed substantially since the 1950s, with evolution of observables and probability densities under the bands nearly disappearing from the sites. Many bands in flow map. We present applications in Gaussian vortex flows human-impacted areas exhibit increased upslope coloniza- and chaotic flows generated by Lorenz 96 systems. tion alongside diminished productivity, in apparent con- tradiction with intuition and current theory. In light of Dimitrios Giannakis this, we suggest a reevaluation of modeling frameworks to Courant Institute of Mathematical Sciences take fuller account of human pressures on self-organized New York University vegetation. [email protected] Karna V. Gowda Northwestern University MS92 [email protected] Optimally Time Dependent Modes for the Descrip- tion of Finite-Time Instabilities in Infinite Dimen- Sarah Iams sional Dynamical Systems Harvard University Cambridge, MA High-dimensional chaotic dynamical systems can exhibit [email protected] strongly transient features. These are often associated with instabilities that have finite-time duration. Because of this Mary Silber finite-time character of these transient events it is hard to University of Chicago detect the associated subspaces where these ‘live’ based Dept. of Engineering Sciences and Applied Mathematics on long term averages or information about the statistical [email protected] steady-state. Here we utilize a recently developed frame- work, the optimally time-dependent (OTD) modes, to ex- tract a time-dependent subspace that spans the modes as- MS93 sociated with transient features associated with finite-time instabilities. This time-dependent subspace encodes the Predictability of Critical Transitions temporal correlation of these transient events. Most im- portantly, the effectiveness of the method does not depend Critical transitions in multi-stable systems have been dis- on the physical dimensionality of the system or the intrin- cussed as models for a variety of phenomena ranging from sic dimensionality of the underlying attractor. Specifically, the extinctions of species to socio-economic changes and we show that the OTD modes, under appropriate condi- climate transitions between ice-ages and warm-ages. From tions, converge exponentially fast to the eigendirections of bifurcation theory we can expect certain critical transitions the Cauchy-Green tensor associated with the most intense to be preceded by a decreased recovery from external per- finite-time instabilities. The case where eigenvalue crossing turbations. The consequences of this critical slowing down occurs may lead to instantaneous interruption of this con- have been observed as an increase in variance and auto- vergence and such cases are thoroughly presented and ana- correlation prior to the transition. However especially in lyzed through specific examples. We apply our analysis for the presence of noise it is not clear, whether these changes the parsimonious computation of Finite-Time Lyapunov in observation variables are statistically relevant such that Exponents for high dimensional systems and demonstrate they could be used as indicators for critical transitions. In numerically the validity of the theoretical findings. this contribution we investigate the predictability of criti- cal transitions in conceptual models. We study several fast Themistoklis Sapsis slow-systems under the influence of external noise. We fo- Massachusetts Institute of Techonology cus especially on the statistical analysis of the success of [email protected] predictions and the overall predictability of the system. The performance of different indicator variables turns out to be dependent on the specific model under study and the MS93 conditions of accessing it. Furthermore, we study the in- Dynamics and Resilience of Vegetation Bands in fluence of the magnitude of transitions on the predictive the Horn of Africa: Comparing Observation and performance. Modeling Sarah Hallerberg Bands of vegetation alternating with bare soil in rhyth- Network Dynamics, Max Planck Inst. Dynamics and mic succession have been well known in the Horn of Africa Self-Org. and elsewhere since the 1950s. Modeling efforts over the [email protected] past two decades have sought to account for the emer- gence of these patterns via a self-organizing interaction Christian Kuehn between vegetation and water resources. A number of Technical University of Munich predictions have been made using such models, but many [email protected] important predictions pertaining to vegetation resilience under environmental and human pressure have been un- Nahal Sharafi, Xiaozu Xhang derconstrained by a lack of long-term observation. High- Max Planck Institute 222 DS17 Abstracts

[email protected], [email protected] [email protected]

Jean-Luc Thiffeault MS93 Dept. of Mathematics Partial Eclipse of the Heart: Early Warning Signs University of Wisconsin - Madison from Sparse Observations [email protected] The Pediatric Early Warning Score (PEWS) was designed to ensure that pediatric hospital patients receive assistance MS94 before their conditions constitute an emergency. PEWS Weakly Three-Dimensional Transport by Vortices has significantly decreased the rate of pediatric cardiac ar- in SQG Flows rest outside of the ICU and subsequently the pediatric mor- tality rate. However, approximately 15% of deteriorating The surface quasi-geostrophic (SQG) equations are a model patents are missed entirely by PEWS, and in some studies for low-Rossby number geophysical flows in which the dy- the median lead time is only 30 minutesinsufficient time namics are governed by potential temperature dynamics on to prep an ICU and implement a preventive response. Ef- the boundary. The model can be used to explore the transi- forts have shifted to finding robust methods of determin- tion from two-dimensional to three-dimensional mesoscale ing at-risk patients earlier in the process. Cerner Math geophysical flows. In the approximation of linear strat- has developed an improved early warning system called ification and f-plane, we examine the dynamics of SQG PEWS*. Where PEWS only uses the most current obser- vortices and the resulting three-dimensional flow including vation, PEWS* utilizes past observations. By utilizing a at first order in Rossby number (O(Ro)). This requires multivariate time series rather than static observations, we solving an extension to the usual QG equation to compute are able to provide 2-6 hours additional warning. the velocity corrections. It is straightforward to obtain the vertical velocity, but difficult to find the O(Ro) horizon- Andrew Roberts tal corrections. The newly-developed Finite Time Braid- Department of Mathematics ing Entropy (FTBE) from Thiffeault & Budisic is used to Cornell University quantify the chaotic mixing induced by three point vortices. [email protected] We also consider the exact SQG vortex solution developed by Dritschel (2011) from the limit of a QG ellipsoid of con- MS93 stant potential vorticity and its transport properties. Bifurcations in the Dynamic Budyko Model with Stefan Llewellyn Smith Diffusive Heat Transport Department of MAE University of California, San Diego We consider Budyko’s energy balance climate model, with [email protected] diffusive meridional heat transport, coupled with a dy- namic ice line equation. We present bifurcations in the coupled model with parameters aligned with the modern Cecily Taylor climate. Adjusting the governing equations to model the UCSD extensive glacial episodes of the Neoproterozoic Era, we [email protected] show there exists a stable equilibrium with the ice line in tropical latitudes. We also present several bifurcation sce- MS94 narios appearing in this latter model. Trajectory Encounter Number As a Diagnostic of James Walsh Mixing Potential in Fluid Flows Oberlin College [email protected] Fluid parcels can exchange water properties when coming in contact with each other, leading to mixing. The tra- jectory encounter number, which quantifies the number of MS94 fluid parcel trajectories that pass close to a reference trajec- Coherency in Braids of Trajectories As a Machine- tory over a finite time interval, is introduced as a measure Learning Problem of the mixing potential of a flow. Regions characterized by low encounter numbers, such as cores of coherent eddies, A sparse set of Lagrangian trajectories can be used to con- have low mixing potential, whereas turbulent or chaotic re- struct a topological braid to represent the planar incom- gions characterized by large encounter numbers have high pressible flow that generated the trajectories. In the braid mixing potential. The encounter number diagnostic was setting, only topological information about trajectories is used to characterize mixing potential in 3 flows of increas- retained, and their precise spatial coordinates ignored. We ing complexity: the Duffing Oscillator, the Bickley Jet, and approach the problem of defining and identifying coher- the altimetry-based velocity in the Gulf Stream Extension ent structures with the braid representation as the start- region. An additional example was presented where the en- ing point. After a review of previous results, we formulate counter number was combined with the u-star-approach of the machine learning problem of identifying which trajec- Pratt et al., 2016 to characterize the mixing potential for a tories can be grouped into coherent bundles, based on the specific tracer distribution in the Bickley Jet flow. Analyt- effect their motion has on stretching of topological mate- ical relationships were derived connecting encounter num- rial curves (loops). We propose one approach for solving ber to diffusivity for purely-diffusive flows, and to shear this machine learning problem and demonstrate its results and strain rates for linear shear and linear strain flows, on a fluid flow that contains both chaotic and regular tra- respectively. It is shown that in a diffusive regime the en- jectories. counter number grows as a square-root of time, whereas in a linear shear and strain flows the encounter number is Marko Budisic proportional to time. Department of Mathematics Clarkson University Irina Rypina DS17 Abstracts 223

Woods Hole Oceanographic Institution ability distribution. In general, those probability distribu- [email protected] tions have fat tails accounting for the enhanced probabil- ity of extreme events. An even more extreme noise can Larry Pratt be realized as an on-off intermittency, which consists of Woods Hole Oceanographic Inst. many small perturbations interspersed with large events. [email protected] We study tipping behavior in bistable systems subject to such perturbations. We show that intermittent noise leads qualitatively to similar stability properties as known for MS94 Gaussian noise (N-tipping) expressed as the Arrhenius law Winding Angle Distribution of 2D Brownian Mo- for the mean escape time. However, quantitatively the tion with Vortical Flow same attractor appears to be more stable with respect to intermittent noise than for Gaussian noise. We compare Let the winding angle distribution with respect to the ori- our results to the case of a single perturbation, which can gin of the standard 2D Brownian motion be θ(t). As proved be considered as E-tipping (event-tipping). To this end we by Spitzer, 2θ(t)/ ln(t) converges in distribution to stan- compute the minimal perturbation necessary to leave the dard Cauchy, for infinitely large t. We show that, in the attractor, i.e. the minimal distance between attractor and presence of an additional tangential flow induced by a vor- the boundary of its basin of attraction. The algorithm used tex at the origin, N1(t)/θ(t) converges to Gamma(1/2,1/2), to compute this minimal perturbation is based on an op- where N1(t) is some normalizer. In particular, θ(t)/N1(t) timization procedure introduced to estimate the minimal is always in the direction of the flow and has heavy tail seed leading from a laminar to a turbulent state in fluid just as in Spitzer’s original case, due to the large winding mechanics. near the origin. To regularize the heavy tail, we also study winding outside an obstacle and inside an annulus. In the Ulrike Feudel former case, θ(t)/N2(t)is given in terms of an elliptic√ theta University of Oldenburg function, while in the latter case, (θ(t) − At)/B t is stan- ICBM, Theoretical Physics/Complex Systems dard Gaussian, where N2(t) is some normalizer and A, B [email protected] are some constants. Lukas Halekotte Huanyu Wen University of Oldenburg ICBM, University of Wisconsin - Madison Theoretical Physics/Complex Systems Department of Mathematics [email protected] [email protected] James A. Yorke MS95 University of Maryland Departments of Math and Physics and IPST Extreme Value Analysis in Dynamical Systems: [email protected] Two Case Studies

Extreme Value Theory for deterministic dynamical systems is a rapidly developing area of research. This contribution MS95 presents numerical analyses designed to show particular Predicting Extreme Optical Pulses in Laser Sys- features of extreme value statistics (EVS) in dynamical sys- tems tems, and also to explore the validity of the theory. We find that formulae that link the EVS with geometrical proper- ties of the attractor hold typically for high-dimensional sys- In this presentation I will discuss the predictability of ultra- tems whether a so-called geometric distance observable or high pulses emitted by optically perturbed semiconductor a physical observable is concerned. In very low-dimensional lasers. Time-delayed optical feedback and continuous-wave settings, however, the fractality of the attractor prevents optical injection configurations will be considered, both of the system from having an extreme value law (EVL), which which produce a chaotic laser output with occasionally rare might well render the evaluation of EVS meaningless and so ultra-high pulses [J. A. Reinoso et al, ”Extreme intensity ill-suited for application. Only for distance observables can pulses in a semiconductor laser with a short external cav- we recover an EVL exactly, even in a low-dimensional set- ity, Phys. Rev. E 87, 062913 (2013) and C. Bonatto et ting, when lumped block maxima data are considered mea- al, ”Deterministic optical rogue waves”, PRL 107, 053901 suring the distance from various points all over the attrac- (2011)]. I will consider experimental and simulated data in tor. The latter corresponds to an averaging and thereby order to show that nonlinear tools of time-series analysis smoothing of unsmooth extreme value distributions. can allow for a certain degree of predictability of the like- lihood of occurrence of extreme intensity fluctuations. In Tamas Bodai the time series of the laser intensity, symbolic patterns can Meteorological Institute be identified, which are more likely or less likely to occur University of Hamburg before the high intensity pulses. The more likely patterns [email protected] canbeusedas”earlywarning”ofanupcomingextreme fluctuation, while the less likely patterns can be used as an indicator of an upcoming ”safe time window” where MS95 extreme fluctuations are unlikely to occur. This symbolic Tipping Due to Extreme Events approach could provide useful information about the pre- dictability of extreme fluctuations in the output signals of Extreme events can be considered as very rare events hav- many real-world complex systems. ing a large impact on a physical or natural system. Such events can be either recurrent or can happen as a single Cristina Masoller large perturbation. In case of recurrent events they might Universitat Polit`ecnica de Catalunya (UPC be characterized as noise with a certain non-Gaussian prob- Departament de F´ısica i Enginyeria Nuclear (DFEN) 224 DS17 Abstracts

[email protected] [email protected]

MS95 MS96 Spatiotemporal Dynamics in the Space of Persis- Extreme Events in Coupled Systems with Different tence Diagrams Delays The celebrated theorem of Takens, which lays the math- Extreme events are rare, recurrent and aperiodic phenom- ematical foundation for the process of dynamic attractor ena which have a large impact on the system. While pre- reconstruction from a time series by the method of delays, vious research has identified stochasticity, progressive spa- has since been generalized and extended by various au- tial synchronization, and inhomogeneity of the system to thors. In particular, Robinson proved a partial extension of be factors which may cause such events; the impact of time delay reconstruction to the setting of infinite-dimensional delays — an important factor which often shapes the dy- dynamical systems possessing finite-dimensional attractors namics of such systems — has not been analyzed for its by showing that a prevalent set of real-valued ’observation role in extreme event generation. We investigate a deter- functions’ can be used to create an injective map from the ministic system of identical FitzHugh-Nagumo oscillators attractor in Hilbert space to delay-coordinate space. More coupled by multiple delay coupling and study the role that recently, tools from computational topology have been used such a coupling plays on extreme event generation. The in- in a host of data-driven applications to summarize topo- vestigations reveal that delay coupling might indeed cause logical characteristics of scalar fields into compact repre- extreme events in such relaxation oscillators even in the sentations called persistence diagrams. If these fields are absence of all previously reported causal factors. We also evolving in time—for instance, under a dissipative PDE—a note that the generation of extreme events in our system persistence transformation induces continuous dynamics in is a manifestation of the intermittency, which is induced in the space of persistence diagrams, and so may be regarded a system when a period doubling cascade leading to chaos as a diagram-valued observation function. In this talk, we collides with a period-adding cascade in an interior crisis. discuss a first numerical investigation of these induced dy- The study also underlines the role of an invariant synchro- namics with a focus on dynamic and structural properties nization manifold and its transverse stability in shaping the of the underlying attractors. mechanism generating the extreme event. The intricate dy- namics occurring on this manifold leads to long transients Francis Motta before the system converges to a fixed point or the chaotic Duke University attractor. This interplay also results in the formation of Department of Mathematics riddled basins of attraction with tongue-like structures em- [email protected] bedded in them.

Arindam Saha MS96 IISER Kolkata, India How to Wake the Homoclinic Snake on the Surface [email protected] of a Ferrofluid

Ulrike Feudel In 2017 it is 50 years ago when Cowley & Rosensweig re- University of Oldenburg ported on the instability of a horizontal layer of ferrofluid ICBM, Theoretical Physics/Complex Systems subjected to a vertical magnetic field. The flat layer be- [email protected] comes unstable due to a subcritical bifurcation and a reg- ular hexagonal pattern of spikes emerges. The field re- gained considerable interest, when localized states of soli- MS96 tary spikes were observed in the hysteretic regime of the magnetic field, where hexagons coexist with the flat layer. The Phase Structure of Grain Boundaries The localized states can be generated by pulses of a local magnetic field (Richter et al. PRL 2015). Likewise, they I will present numerical and analytical results on grain arise after a specific pulse sequence of the global field (Goll- boundaries in the Swift-Hohenberg and Cross-Newell equa- witzer et al. NJP 2010), which allows to access the vicinity tions. It is well known that as the angle made by the roll of the unstable branch of the bifurcation. In this way, a patterns on each side of this line defect is decreased, dislo- sequence of localized patches of hexagons was generated in cations appear at the core of the grain boundary. Under- experiment, indicating for homoclinic snaking, which was standing this transition is an interesting problem since it corroborated in numerics (Lloyd et al. JFM 2015). So far, provides an example of defect formation in a system that the experiment gives evidence for slanted snaking, charac- is variational and therefore more amenable to analysis. I teristic for a conserved quantity (Firth et al. PRL 2007), will show numerical results of the Swift-Hohenberg equa- like the volume of ferrofluid in the vessel. It remains to be tion that aim to analyze the phase structure of far-from- investigated how larger vessels or a horizontal field compo- threshold grain boundaries and connect these observations nent may modify the snaking in experiments. to properties of the associated phase diffusion equation, the regularized Cross-Newell equation. This work is part Reinhard Richter of a long-term project whose goal is to understand the role University of Bayreuth played by phase derivatives in the creation of defects in [email protected] pattern forming systems, and is joint with Nick Ercolani and Nikola Kamburov. MS96 Joceline Lega Time-Dependent Spatiotemporal Chaos in University of Arizona, USA Pattern-Forming Systems with Two Length DS17 Abstracts 225

Scales [email protected], [email protected]

Nonlinear resonant three-wave interactions between two waves with the same wavelength and a third wave with MS97 a different wavelength can lead to steady patterns (stripes The Extreme Weather-Causing Patterns of the or rectangles), to simple time-dependent patterns (travel- Midlatitudes ling waves) or to chaotic dynamics, depending on the de- tails of how the three waves transfer energy between the Identifying and understanding the dynamics of atmo- two wavelengths. In a large domain, waves with the two spheric circulation patterns that cause extreme events are wavelengths and any orientation allowed by the domain crucial for better prediction and projection of weather ex- can interact in much more complex ways, potentially lead- tremes in the midlatitudes. Accurate linear response func- ing to steady superlattice patterns or quasipatterns, or to tions (LRFs) of global climate models (GCMs) can be used time-dependent spatiotemporal chaos. We investigate the to overcome some problems related to studying the tur- conditions that encourage spatiotemporal chaos, and illus- bulent midlatitude circulation by providing a framework trate what can happen in the particular case of a two-layer to control the turbulent flow for hypothesis testing, iden- reaction–diffusion system. tify dynamical modes of the system, and evaluate statis- tical methods that can be applied to observational data Alastair M. Rucklidge [Hassanzadeh and Kuang 2015 GRL; Ma et al 2016 J. At- Department of Applied Mathematics mos Sci.]. We compare and contrast LRFs calculated using University of Leeds Greens functions [Hassanzadeh and Kuang 2016 J. Atmos. [email protected] Sci. Part I], Fluctuation-Dissipation Theorem, and Lin- ear Inverse Modeling (LIM)/Dynamic Mode Decomposi- Priya Subramanian, Jennifer Castelino tion (DMD) for an atmospheric GCM. We show how non- University of Leeds normality of the LRF can complicate using the data-driven [email protected], [email protected] methods [Hassanzadeh and Kuang 2016 J. Atmos. Sci. Part II] and then employ the accurate LRF calculated us- ing Greens functions to interpret the Principal Oscillatory MS97 Patterns obtained using LIM/DMD. The LIM/DMD is ap- plied to the observational data from the Northern Hemi- Predicting the Intraseasonal Precipitation Mon- sphere and the connection of the modes to extreme events soon Through a Low-Order Nonlinear Stochastic such as heat waves and cold spells are discussed. Model with Intermittency Pedram Hassanzadeh We assess the limits of predictability of the large-scale Mechanical Engineering patterns in the boreal summer intraseasonal variability Rice University (BSISO) as a measure of precipitation. A recent devel- [email protected] oped nonlinear data analysis technique, nonlinear Lapla- cian spectrum analysis, is applied to the precipitation data, de?ning two spatial modes with high intermittency associ- MS97 ated with the BSISO time series. A recent developed data- No Equations, No Parameters No Variables - Data driven physics-constrained low-order modeling strategy is and the Reconstruction of Normal Forms for Para- applied to these time series. The result is a 4-D system metrically Dependent Dynamical Systems with two observed BSISO variables and two hidden vari- ables involving correlated multiplicative noise through the Abstract Not Available At Time Of Publication. nonlinear energy-conserving interaction. With the opti- mal parameters calibrated by information theory, the non- Ioannis Kevrekidis Gaussian fat tailed probability distribution functions, the Dept. of Chemical Engineering autocorrelations and the power spectrum of the model sig- Princeton University nals almost perfectly match those of the observed data. An [email protected] ensemble prediction scheme including an e?ective online data assimilation algorithm for determining the initial en- semble of the hidden variables shows the useful prediction MS97 skill is at least 30 days and even reaches 60 days in some years. The ensemble spread succeeds in indicating the fore- Rogue Waves and Large Deviations in the Non- cast uncertainty. Twin experiment indicates the signi?cant linear Schroedinger Equation with Random Initial skill of the model in determining the predictability limits Data of the BSISO. Finally, a practical spatial-temporal recon- The problem of appearance of rogue waves is investigated struction strategy is developed and the predicted patterns within the context of the modified nonlinear Schr¨odinger have consistent skill as the time series. (MNLS) equation with random initial data. Specifically, we identify the initial condition within a random set of Nan Chen given statistics that is most likely to create a large dis- New York University turbance of the solution of MNLS within a certain time. [email protected] This allows us to estimate the probability of appearance of such disturbances, and to estimate the tail of the dis- AMajda tribution of the solution amplitude. To make contact with NYU real observations, we assume that the initial condition is [email protected] Gaussian distributed, with a spectrum taken from experi- mental data. The method proposed builds on results from Sabeerali Thelliyil, Ajaya Ravindran large deviation theory and is transportable to other deter- NYU Abu Dhabi ministic dynamical systems with random initial conditions. 226 DS17 Abstracts

the slow and fast timescales that govern synthesis and se- cretion process in the hypothalamus. Analysis of our model Eric Vanden-Eijnden, Tobias Grafke, Giovanni Dematteis demonstrates how bistability arises in certain parameter Courant Institute regimes, in which one of the two oscillating stable states New York University is characterized as the ”diseased” state. Identifying the [email protected], [email protected], demat- most relevant parameters in shaping the bistable structure [email protected] will provide insights on what determines the susceptibility of an individual to stress-related disorders. Moreover, un- derstanding the underlying mechanisms of transitions be- MS98 tween the two stable states can guide us in improving exist- Bipolar Disorder Dynamics: Multiscale Mathemat- ing treatment protocols. In particular, our model explains ical Approaches for Translational Benefits how lower baseline cortisol level observed in PTSD sub- jects can be induced by the stress response elicited during Bipolar disorder is a chronic, recurrent mental illness char- a traumatic event in novel, intensity, and timing-dependent acterised by episodes of depressed and manic mood. Be- ways. Our model also suggests a mechanism whereby ex- tween these episodes, irregular mood fluctuations cause posure therapy may act to normalize downstream dysregu- significant sickness and ill-health. In this talk I will in- lation of the HPA axis associated with stress disorders such troduce the paradigm of a mechanistic hierarchy to under- as PTSD. stand bipolar dynamics. This necessities the use of both multi scale mathematical (relaxation oscillations; Markov Lae Un Kim chains) and statistical (maximum likelihoods) approaches. Dept. of Biomathematics I will emphasize the need for a cogent stochastic frame- UCLA works to advance the translational aspects of the work to [email protected] achieve greater clinical benefits to these mathematical ap- proaches to psychiatry. Maria D’Orsogna CSUN Michael Bonsall [email protected] Dept. of Zoology Oxford University Tom Chou [email protected] UCLA Departments of Biomathematics and Mathematics [email protected] MS98 Using Mathematical Models of Biological Processes in Genome-wide Association Studies of Psychiatric MS98 Disorders Modeling Neural Circuit Dysfunction in Schizophrenia: Effects of Disrupted Excitation- Genome-wide association studies have implicated a large Inhibition Balance number of genes in psychiatric disorders such as bipolar disorder and schizophrenia. These genes, however, are usu- Mechanistic understanding of mental disorders is limited ally weakly-associated and involved in diverse biological by the stark explanatory gap between levels of analysis: processes, thereby obscuring mechanisms for how such dis- disease-related mechanisms occur at the level of neurons orders arise. To better elucidate mechanisms, disorders can and synapses, whereas symptoms occur at the level of cog- be tested for association with a set of genes that represent nition and behavior. I will present how biophysically-based a biological pathway. Even so, mechanisms can remain elu- computational modeling of neural circuits can be harnessed sive if ultimately a specific biological function is important. to probe how synaptic disruptions implicated in psychi- We propose and analyze a statistical test to implicate bi- atric diseases can induce alterations in cognition and large- ological functions in psychiatric disorders. Our approach scale neural network dynamics. In particular, I will dis- relies on mathematical models of biological processes to cuss recent studies exploring the effects of disruption in the predict how strongly genes contribute to a certain biologi- synaptic balance between excitation and inhibition, which cal function. These predictions are used to assign weights is a leading hypothesis for cortical circuit dysfunction in to genes, which are then incorporated into a statistical test schizophrenia. on genetic data. To demonstrate, we use this statistical test to explore the role of calcium signaling in bipolar dis- John Murray order. Yale University School of Medicine [email protected] Amy Cochran Dept. of Mathematics University of Michigan MS99 [email protected] Complex Dynamics in a Conceptual Model of El Ni˜no

MS98 ENSO (El Ni˜no Southern Oscillation) is a climate phe- A Multiple Timescale Model of H-P-A Axis Dy- nomenon in the tropical Pacific ocean. Most of the time, namics: Onset, Timing, and Exposure Therapy of upwelling in the eastern Pacific near South America makes Stress Disorders ocean temperatures there cold and prevailing winds blow from east to west. At the end of some years, this pat- The hypothalamus-pituitary-adrenal (HPA) axis is a neu- tern changes and produces global effects on the weather. roendocrine system that regulates the secretion of the pri- This lecture will discuss the dynamics of a three dimen- mary stress hormone called cortisol. We develop a delay sional vector field that has been proposed as a conceptual equation model for HPA axis dynamics which distinguishes model of ENSO and its variability. Roberts et al. studied DS17 Abstracts 227

the model as a multiple time scale slow-fast dynamical sys- theory, averaging theory and geometric singular perturba- tem. This lecture will discuss its dynamics and some of the tion theory to develop an averaging method for folded man- mathematical problems that have arisen in its analysis. ifolds of limit cycles. In so doing, we devise an analytic scheme for the identification and topological classification John Guckenheimer of torus canards. We demonstrate the predictive power of Cornell University our results in a model for hormone induced calcium oscil- [email protected] lations in liver cells, where we use our torus canard theory to explain the mechanisms that underlie a novel class of bursting rhythms. MS99 Complex Action Potential Firing in Developing In- Theodore Vo nerHairCells Boston University [email protected] Inner Hair Cells (IHCs) are the proper receptor cells of hearing and are connected to the afferent nerves. In mam- mals, sound transduction by IHCs generates a receptor po- MS99 tential whose amplitude and phase drive auditory nerve Timescales and Mechanisms of Sigh-like Bursting firing. Immature IHCs differ in several aspects, and in and Spiking in Models of Rhythmic Respiratory particular in their electrophysiological characteristics and Neurons behaviour, from mature IHCs. Hence, from a functional perspective and to obtain insight into the development of Neural networks generate a variety of rhythmic activity hearing, it is important to understand the factors control- patterns, often involving different timescales. One exam- ling the behavior of immature IHCs. In order to address ple arises in the respiratory network in the preB¨otzinger this question we developed a mathematical model of elec- complex of the mammalian brainstem, which can gener- trical activity and calcium dynamics. Our model repro- ate the eupneic rhythm associated with normal respiration duces experimentally observed patterns of immature IHCs as well as recurrent low-frequency, large-amplitude bursts activity, such as spiking and bursting. Using the model associated with sighing. Two competing hypotheses have we studied the involvement of intracellular calcium stores been proposed to explain sigh generation: the recruitment in regulating the intracellular calcium signal, important of a neuronal population distinct from the eupneic rhythm- for local dynamic fine-tuning of the IHCs membrane filter. generating subpopulation or the reconfiguration of activity We also performed a numerical bifurcation analysis, which within a single population. Here, we consider two recent revealed the existence of a complex but structured set of computational models, one of which represents each of the periodic attractors with multiple-spike solutions. This is hypotheses. We use methods of dynamical systems theory, not surprising as often observed in models of excitable cells such as fast-slow decomposition, averaging, and bifurcation the variables in the immature IHCs model vary on differ- analysis, to understand the multiple timescale mechanisms ent timescales. Given the multi-timescale character of our underlying sigh generation in each model. In the course of model we also employed fast-slow analysis with a view to our analysis, we discover that a third timescale is required investigate further the complex periodic behaviour of the to generate sighs in both models. Furthermore, we iden- model. tify the similarities of the underlying mechanisms in the two models and the aspects in which they differ. Krasimira Tsaneva-Atanasova University of Exeter Yangyang Wang [email protected] Mathematical Biosciences Institute The Ohio State University Daniele Avitabile [email protected] School of Mathematical Sciences University of Nottingham Jonathan E. Rubin [email protected] University of Pittsburgh Department of Mathematics Harun Baldemir [email protected] Department of Mathematics University of Exeter [email protected] MS100 Unsteadily Manipulating Internal Flow Barriers

MS99 Typical flows contain internal flow barriers: specialized Generic Torus Canards and a Novel Class of Burst- time-moving entities which demarcate distinct motions. ing Rhythms Examples include the boundary between an oceanic eddy and a nearby jet, the edge of the Antarctic circumpolar Torus canards are solutions of slow/fast systems that al- vortex, or the interface between two fluids which are to ternate between attracting and repelling manifolds of limit be mixed together in an microfluidic assay. These barriers cycles of the fast subsystem. Since their discovery in 2008, are Lagrangian in the sense that they move with the flow, torus canards have been shown to mediate the transi- and their location at each time is usually determined by tion from rapid spiking to bursting in several paradigm performing diagnostic calculations using one of the many computational neural models. So far, torus canards have methods which are commonly referred to as Lagrangian only been studied numerically, and their behaviour inferred Coherent Structure (LCS) analysis. The ability to con- based on classical averaging and geometric singular pertur- trol the locations of these barriers in a user-specified time- bation methods. This approach, however, is not rigorously varying (unsteady) way can profoundly impact fluid trans- justified since the averaging method breaks down near a port between the coherent structures which are separated fold of periodics – exactly where the torus canards origi- by the barriers. In this talk, the unsteady Eulerian velocity nate. In this work, we combine techniques from Floquet required to achieve a specified Lagrangian time-variation is 228 DS17 Abstracts

explicitly derived. The excellent accuracy of the method indeed exist and govern the key transport characteristics is demonstrated using the Kelvin-Stuart cats-eyes flow and of the region. finite-time Lyapunov exponents. This is a promising first approach in the quest for being able to control internal Margaux Filippi flow barriers—and the transport across them—in general Massachusetts Institute of Technology unsteady flows. Department of Mechanical Engineering [email protected] Sanjeeva Balasuriya University of Adelaide Alireza Hadjighasem [email protected] ETHZ [email protected] MS100 Matt Rayson, Gregory Ivey Go With the Flow, on Jupiter and Snow, Coher- University of Western Australia ence From Video Data Without Trajectories [email protected], [email protected] Viewing a data set such as the clouds of Jupiter, coher- ence is readily apparent to human observers, especially the Thomas Peacock Great Red Spot, but also other great storms and persis- Massachusetts Institute of Technology tent structures. There are now many different definitions [email protected] and perspectives mathematically describing coherent struc- tures, but we will take an image processing perspective MS100 here. We describe an image processing perspective infer- ence of coherent sets from a fluidic system directly from Phase Space Structures in Velocity Space for Glid- image data, without attempting to first model underlying ing and Falling Bodies flow fields, related to a concept in image processing called motion tracking. In contrast to standard spectral methods Even the most simplified models of falling and gliding bod- for image processing which are generally related to a sym- ies provide rich nonlinear dynamical systems for study. A metric affinity matrix, leading to standard spectral graph better understanding of such systems will give new insight theory, we allow a nonsymmetric affinity which arises nat- into the biomechanics of animal gliders and the design of urally from the underlying arrow of time. We develop an engineered systems such as descending spacecraft, supply anisotropic, directed diffusion operator corresponding to airdrops, and robotic gliders. Taking a global view of the flow on a directed graph, from a directed affinity matrix dynamics, we find an attracting invariant manifold that developed with coherence in mind, and corresponding spec- acts as a barrier to trajectories in velocity space. This tral graph theory from the graph Laplacian. Our examples invariant manifold presents a higher-dimensional analogue will include partitioning the weather and cloud structures of terminal velocity onto which the dynamics of a glider of Jupiter, and a local to Potsdam, N.Y. lake-effect snow settle. In this work, we present theoretical and numer- event on Earth, as well as the benchmark test double-gyre ical techniques for approximating the invariant manifold system. and discuss approaches to think about falling and gliding motion in terms of the underlying phase space structures. Erik Bollt These phase space structures can be leveraged to design Clarkson University efficient control strategies for minimally actuated systems. [email protected] Gary K. Nave Virginia Tech Abd Alrahman R. Almomani Department of Biomedical Engineering and Mechanics Clarkson University [email protected] Department of Mathematics [email protected] Isaac Yeaton Virginia Tech MS100 [email protected] The Detection of Lagrangian Transport Structures in a Coral Reef Atoll: A Field Experiment Shane D. Ross Virginia Tech The flow circulation around coral ecosystems influence bi- Engineering Mechanics program ological processes such as nutrient exchange and larval [email protected] spawning, which determine population growth rates. Un- derstanding reef connectivity is thus crucial to study pop- ulation dynamics, as well as the resilience of coral reefs MS101 to the environmental stressors associated with climate Continuous Data Assimilation for Geophysical change and other anthropogenic activities. Approaches Flows Employing Only Surface Measurements from Lagrangian data analysis can provide useful infor- mation about the key transport structures that govern reef We prove that data assimilation by feedback control can be connectivity. In October 2016, we conducted a field ex- achieved for the three-dimensional quasi-geostrophic equa- periment to test predictions of key Lagrangian Transport tion using only large spatial scale observables on the dy- Structures in the vicinity of Scott Reef, an atoll system in namical boundary. On this boundary a scalar unknown the Timor Sea. From the surface velocity data obtained (buoyancy or temperature of a fluid) satisfies the surface through a numerical ocean model, we ran predictive anal- quasi-geostrophic equation. The feedback nudging is done ysis and identified a strong,transient Lagrangian feature on this 2D model, yet ultimately synchronizes the stream aligned with the spring tide. We deployed two arrays of function of the three-dimensional flow. The main ana- surface drifters and demonstrated that this feature does lytical difficulties involved in ensuring the synchronization DS17 Abstracts 229

property arise from the presence of a nonlocal dissipative University of Exeter operator in the surface quasi-geostrophic equation. This Mathematics research Institute necessitates the derivation of various boundedness and ap- [email protected] proximation properties for the observation operators used in the feedback nudging. MS101 Michael S. Jolly A Nonautonomous Spectral Stability Theory for Indiana University Ordinary Differential Initial Value Problem Solvers Department of Mathematics [email protected] In this talk we analyze the time-dependent Lyapunov sta- bility of the numerical approximation of a time-dependent solution of an ordinary differential equation (ODE) initial MS101 value problem (IVP) by one-step and general linear meth- Connections Between Rate-Induced Tipping and ods. We give an example of an exponentially decaying, Nonautonomous Stability Theory time-dependent, linear ODE for which an implementation of the implicit Euler method with adaptive step-size se- We discuss the phenomenon of rate-induced tipping, where lection fails to produce a decaying numerical solution. To changing a parameter past a critical rate causes the system explain this phenomenon we employ Lyapunov and Sacker- to ‘tip’ or diverge from an attractor, within the frame- Sell spectral theory to derive conditions under which the work of nonautonomous stability theory. We explore rate numerical solution of a bounded and uniformly exponen- induced tipping in several one dimensional model nonau- tially stable trajectory of a nonlinear ODE IVP is bounded tonomous systems using finite time Lyapunov exponents and uniformly exponentially stable. These results are used and Steklov averages. Our results show that these numer- to develop a step-size selection strategy based on nonau- ical stability spectra are effective diagnostics for determin- tonomous stability and accuracy. ing rate-induced tipping points in systems with parameters that are asymptotically constant or are linear. We then ap- Andrew J. Steyer ply these stability spectra to a conceptual climate model. University of Kansas [email protected]

Alanna Hoyer-Leitzel Erik Van Vleck Mathematics Department Department of Mathematics Bowdoin College University of Kansas [email protected] [email protected]

Alice Nadeau University of Minnesota MS102 [email protected] A Differential Equation with State-Dependent De- lay Describing Stem Cell Maturation - Stability Andrew Roberts and Oscillations Department of Mathematics Cornell University The talk will be about a two component differential equa- [email protected] tion with implicitly defined state dependent delay and neg- ative feedback. The equation describes the regulated matu- Andrew J. Steyer ration process of a stem cell population. In [Ph. Getto and University of Kansas M. Waurick, A differential equation with state-dependent [email protected] delay from cell population biology, J. Diff. Eqs. 260 (7) 6176-6200, 2016] we have recently established smoothness conditions for the functional defining the right hand side such that theoretical results [H.O. Walther, The solution MS101 1 Early-Warning Indicators for Rate-Induced Tip- manifold and C -smoothness for differential equations with ping state-dependent delay, J. Diff. Eqs, 195 2003 46-65] can be applied to achieve well-posedness and a linearized stabil- A dynamical system is said to undergo rate-induced tip- ity theorem. Currently research topics include global ex- ping when it fails to track its quasi-equilibrium state due istence, characteristic equations and oscillations via Hopf to an above-critical-rate change of system parameters. We bifurcation and fixed point theorems. We are also apply- study a prototypical model for rate-induced tipping, the ing pseudo spectral methods developed in [D. Breda, O. saddle-node normal form subject to time-varying equilib- Diekmann,M. Gyllenberg, F. Scarabel and R. Vermiglio, rium drift and noise. We find that both most commonly Pseudospectral Discretization of Nonlinear Delay Equa- used early-warning indicators, increase in variance and in- tions: New Prospects for Numerical Bifurcation Analysis, crease in autocorrelation, occur not when the equilibrium SIAM J. Appl. Dyn. Syst. 15 (1) 1-23 2016, D. Breda, Ph. drift is fastest but with a delay. We explain this delay Getto, J. Sanchez Sanz and R. Vermiglio, Computing the by demonstrating that the most likely trajectory for tip- Eigenvalues of Realistic Daphnia Models by Pseudospectral ping also crosses the tipping threshold with a delay and Methods SIAM J. Sci. Comp. 37 (6) (2015) A2607-A2629]. therefore the tipping itself is delayed. We find solutions of the variational problem determining the most likely tipping path using numerical continuation techniques. The result Philipp Getto is a systematic study of the most likely tipping time in the Bolyai Institute plane of two parameters, distance from tipping threshold University of Szeged and noise intensity. [email protected]

Paul Ritchie Dimitri Breda 230 DS17 Abstracts

Department of Mathematics and Computer Science [email protected] University of Udine [email protected] MS102 Gergely Rost Lyapunov-Razumikhin Techniques for State- Bolyai Institute, University of Szeged Dependent Delay Differential Equations Szeged, Aradi v´ertank tere 1, H-6720 Hungary [email protected] We present theorems for the Lyapunov and asymp- totic stability of the steady state solutions to general Francesca Scarabel state-dependent delay differential equations (DDEs) us- University of Helsinki ing Lyapunov-Razumikhin methods. These theorems Department of Mathematics and Statistics build upon the previous work of Hale and Verduyn-Lunel francesca.scarabel@helsinki.fi [Springer-Verlag, New York, 1993], and Barnea [SIAM J. Appl. Math, 17(4):681–697, 1969] which were mainly Tibor Krisztin aimed at equations with simpler delay terms (e.g. con- Bolyai Institute stant and time-dependent delays), and are less applicable University of Szeged to state-dependent DDEs such as the following model equa- [email protected] tion, − − Istvan Balazs u˙(t)=μu(t)+σu t a cu(t) . University of Szeged [email protected] For fixed a and c, the stability region Σ∗ of the zero so- lution to this model problem is known, and it is the same for both the constant delay (c = 0) and state-dependent Yukihiko Nakata delay (c = 0) cases. Using our results we can prove the Shimane University asymptotic stability of the zero solution to this model prob- Japan lem in parts of Σ∗, considerably expanding upon the work [email protected] of Barnea who proved Lyapunov stability for the simpler μ = c = 0 constant delay case. Similar techniques are MS102 used to derive a condition for global asymptotic stability of the zero solution to the model problem, and bounds on Computing the Unstable Manifolds of Delay Dif- periodic solutions when the zero solution is unstable. ferential Equations

In this talk I will discuss how one can compute high or- Felicia Magpantay der Taylor approximations of the unstable manifold (of Department of Mathematics equilibrium and periodic solution) of a Delay Differential University of Manitoba Equation. This approach is based on the Parametrization [email protected] Method for ODEs and not only describes the unstable man- ifold, but also encapsulates the dynamics thereon. Finally, I will briefly sketch how the resulting parametrization can MS103 be made rigorous using a computer-aided proof. Computation and Stability of Waves in Second Or- der Evolution Equations Chris M. Groothedde VU University Amsterdam [email protected] The main topic of this talk are traveling waves for semi- lineardampedwaveequations.Weshowhowthefreez- ing method generalizes from first to second order evolution MS102 equations by transforming the original PDE into a partial differential algebraic equation (PDAE). Solving a Cauchy Modelling Myelopoiesis with State-Dependent De- problem via the PDAE generates a comoving frame, in lay Differential Equations which the profile varies as little as possible, and an al- gebraic variable which approximates its speed. Under suit- The production of white blood cells is modelled from able stability conditions the profile and the speed converge hematopoietic stem cells (HSCs) through proliferating and to their limiting values. A rigorous theory of this effect maturing precursors to circulating neutrophils. The main is presented in one space dimension, building on recent cytokine that regulates this process is Granulocyte Colony results by J. Rottmann-Matthes on nonlinear stability of Stimulating Factor (G-CSF). G-CSF regulates the differen- waves in first order hyperbolic systems. Numerical exam- tiation rate of HSCs, the proliferation rate during mitosis, ples demonstrate the applicability of the method. We also the maturation time, and the rate at which mature neu- show generalizations of the method to rotating waves in trophils are released into circulation from the bone mar- several space dimensions. For this case the nonlinear sta- row. We model the variable maturation time via an age- bility theory is still largely unexplored. structured PDE model with variable ageing rate and show how this results in a model for myelopoiesis as a system of state-dependent delay differential equations. We discuss Wolf-Juergen Beyn the application of this model to study dynamical diseases Bielefeld University (cyclical neutropenia), infection immune response and pe- [email protected] ripheral blood stem cell harvesting and transplantation. Simon Dieckmann, Christian Doeding Tony R. Humphries Department of Mathematics McGill University Bielefeld University Mathematics and Statistics [email protected], [email protected] DS17 Abstracts 231

bielefeld.de in the Schr¨odiner case, the recent results of Alexeeva, Barashenkov, Sukhorukov, Kivshar and Bludov, Konotop, Malomed. In our analysis, we rely exclusively on instability MS103 index counting methods. Defect Induced Target Waves in Reaction Diffusion Systems Milena Stanislavova University of Kansas, Lawrence It is known that in reaction diffusion systems the phase Department of Mathematics modulation of coherent structures is modeled by a viscous [email protected] eikonal equation. Using this phase equation we show the existence of target waves in reaction diffusion systems in R2, which are generated by the presence of a localized per- MS104 turbation (the defect). Viewed from the point of view of Dynamical Proxies of North Atlantic Predictability perturbation theory this problem presents two difficulties: and Extremes it is a perturbation problem beyond all orders, and more- over the linear operator has a zero eigenvalue embedded in Atmospheric flows are characterized by chaotic dynamics the essential spectrum. The first technical difficulty is over- and recurring large-scale patterns. These two character- come using techniques from matched asymptotics, while istics point to the existence of an atmospheric attractor the second hurdle is surpassed using the Fredholm proper- defined by Lorenz as: the collection of all states that the ties of the Laplacian in the setting of Kondratiev spaces and system can assume or approach again and again, as op- a decomposition of these spaces into polar modes. Similar posed to those that it will ultimately avoid. The average results have been shown using spatial dynamics in the radi- dimension D of the attractor corresponds to the number of ally symmetric case, and the theory of Schr¨odinger opera- degrees of freedom sufficient to describe the atmospheric tors in the non-radial case. The novelty of this approach is circulation. However, obtaining reliable estimates of D has that this technique is also applicable to integro-differential proved challenging. Moreover, D does not provide infor- equations, where the linearization is now a radially sym- mation on transient atmospheric motions, such as those metric convolution operator. leading to weather extremes. Using recent developments in dynamical systems theory, we show that such motions Gabriela Jaramillo can be classified through instantaneous rather than average The University of Arizona properties of the attractor. The instantaneous properties [email protected] are uniquely determined by instantaneous dimension and stability. Their extreme values correspond to specific at- mospheric patterns and match climate extreme events. MS103 Spectral Stability of Solutions to the Vortex Fila- Davide Faranda ment Hierarchy LSCE, CEA Saclay l’Orme des Merisiers, CNRS [email protected] The Vortex Filament Equation (VFE) is part of an inte- grable hierarchy of filament equations. Several equations in this hierarchy have been derived to describe vortex fil- MS104 aments in various situations. Inspired by these results, we The Extremal Index for the AR(1) Process: Finite develop a general framework for studying the existence and Size Considerations the linear stability of closed solutions of the VFE hierarchy. The framework is based on the correspondence between I begin by discussing a few heuristic interpretations of the the VFE and the nonlinear Schr¨odinger (NLS) hierarchies. extremal index, namely (i) a measure of loss of iid degrees Our results establish a connection between the AKNS Flo- of freedom, (ii) the multiplicity of a compound Poisson quet spectrum and the stability properties of the solutions point process, (iii) the sojourn time of an excursion. I of the filament equations. We apply our machinery to so- then consider finite-size effects of the extremal index in the lutions of the filament equation associated to the Hirota AR(1) process with Gaussian noise which, as the sample equation. We also discuss how our framework applies to size goes to infinity, shows no clustering. soliton solutions. Nicholas Moloney Stephane Lafortune London Mathematical Laboratory College of Charleston [email protected] Department of Mathematics [email protected] MS104 Thomas Ivey Stochastic Bifurcation in Random Logistic Maps College of Charleston [email protected] A topological bifurcation point of stochastic dynamics is introduced illustrating noise-induced chaos in a random logistic map. We identify two different transition points MS103 characterized by the zero-crossing points of the supremum Stability of PT Symmetric Ground States for of the dichotomy spectrum and the Lyapunov exponent. Schrodinger and Klein-Gordon Equations in The associated three phases are characterized as a random Higher Space Dimensions periodic attractor, a random point attractor, and a random strange attractor, respectively. The dichotomy spectrum We consider PT Schr¨odinger and Klein-Gordon equations characterizes uniform hyperbolicity of random attractors in higher dimensional spaces. After the construction of based on all possible asymptotic behaviour, while the Lya- the ground states, we proceed to systematically and com- punov exponent characterizes stability of random attrac- pletely characterize their spectral stability. This extends, tors based on the most likely asymptotic behaviour. A 232 DS17 Abstracts

solvable example of noise-induced bifurcation and an ex- [email protected] ample of numerically computed dichotomy spectrum for the random logistic map are presented as well. MS105 Yuzuru Sato Signal Detection by Active, Noisy Oscillators on RIES/Department of Mathematics the Brink of Self-Oscillation Hokkaido University [email protected] Many physical systems employ limit-cycle oscillators to de- tect periodic signals, but their performance is often lim- ited by noise. For example, vertebrate hearing is based on the mechanosensory hair bundle that detects sound- MS104 induced vibrations within the ear. We study the response Extreme Value Theory in Dynamical Systems of noisy oscillators operating near supercritical or subcrit- ical Hopf bifurcations to periodic forcing. A noisy oscil- We present in this note a new application of Extreme Value lators frequency-detuned sensitivity and degree of entrain- Theory (EVT) to Coupled Lattice Maps (CLM) on a finite ment, as well as its bandwidth are maximized when it self- lattice. Actually a first result in this direction was given oscillates near a Hopf bifurcation. Owing to three distinct in the paper, where the authors considered two coupled mechanisms, the sensitivity and entrainment of a Hopf os- interval maps and applied their theory of rare events for cillator peak as a function of the noise level. We confirmed open systems with holes to investigate the first entrance of several of these predictions experimentally by periodically the two components into a small strip along the diagonal, forcing hair bundles held near the two types of Hopf bi- which is equivalent to the synchronisation of the two-side furcation. We describe how to determine the identity and network up to a certain tolerance. We will show that that location of a bifurcation from observations of a bundles dis- synchronisation process could be interpreted and quantified placement. A hair bundle is most sensitive and most easily by computing the asymptotic distribution of the maximum entrained by a periodic stimulus when it spontaneously os- of a suitable random process. cillates near a Hopf bifurcation. Moreover, the addition of noise can improve a bundles ability to detect the stim- ulus. This work suggests that auditory systems rely on Sandro Vaienti self-oscillatory elements operating near a Hopf bifurcation. CPT Marseille [email protected] Daibhid O Maoileidigh, Joshua Salvi, AJ Hudspeth The Rockefeller University MS105 [email protected], [email protected], [email protected] Nonlinear Dynamics of Inner Ear Hair Cells

The inner ear constitutes a remarkably sensitive biological MS105 detector. The first step in auditory processing is performed Nonlinear Micromechanics of the Organ of Corti in by hair cells, which act as transducers that convert minute the Low-Frequency Region of the Cochlea mechanical vibrations into electrical signals that can be sent to the brain. The hair cells operate in a viscous envi- The organ of Corti in the mammalian inner ear houses ronment, but can nevertheless sustain oscillations and am- the mechanosensitive inner and outer hair cells. Outer plify incoming signals. The thermodynamic requirements hair cells can provide mechanical forces that can amplify indicate the presence of an underlying active process that a weak sound stimulus, which is then detected by the in- pumps energy into the system. We explore experimentally ner hair cells. However, the micromechanics of how me- the response of oscillatory hair bundles to signals spanning chanical amplification is achieved by the organ of Corti a broad range of frequencies and amplitudes, to study their remains unclear. In particular, the apical organ of Corti is phase-locking properties. We demonstrate the presence of shaped differently from the basal organ, indicating a differ- Arnold Tongues, with significant overlaps between the syn- ent micromechanical functioning that remains poorly un- chronization regimes. Secondly, we demonstrate the pres- derstood. Here we combine mathematical modeling and ence of chaos in the innate motility exhibited by hair cell laser-interferometric recordings to show that the apical or- bundles. Poincare maps constructed from driven bundle gan of Corti’s micromechanics contains a strong nonlinear- oscillations indicate a quasiperiodic transition from chaos ity with respect to length changes of the outer hair cells. to order with increasing amplitude of mechanical forcing. This nonlinearity allows the organ of Corti to operate as The onset of this transition is accompanied by an increase an electromechanical transistor: the elongation of the outer in mutual information between the stimulus and the hair hair cells can sensitively regulate how much sound stimu- bundle, indicative of signal detection. Finally, we explore lation is transmitted to the inner hair cells. These results the interaction between active bundle mechanics and the could, for instance, explain how stimulation of the inner electrical circuit comprised of somatic ion channels, and hair cells can be regulated by efferent nerve fibers. measure the impact of this coupled system on the overall detection performed by the hair cell. Tobias Reichenbach Imperial College Dolores Bozovic, Justin Faber, Yuki Quinones, S. W. F. London Meenderink [email protected] UCLA [email protected], [email protected], Nikola Ciganovic [email protected], [email protected] Imperial College London [email protected] Michael Levy Weizmann Institute Rebecca Warren, Batu Keceli DS17 Abstracts 233

University of Link¨oping University of Bern, Switzerland [email protected], [email protected] [email protected]

Stefan Jacob Karolinska Institute MS106 [email protected] Faster and More Accurate Computations for Cer- tain Highly Oscillatory Wave Problems Anders Fridberger University of Link¨oping As we can see in many optical applications, faster and [email protected] more accurate solution approximations of highly oscillating problems, including solutions of partial differential equa- tions and beam integrals, are tasks with an overarching MS105 importance. However, they are widely perceived as an ex- tremely difficult or challenging issue in physics computa- Symmetries and Asymmetries in Cochlear Mechan- tions. In fact, a successful numerical method must base on ics an appropriate discretization which overcomes the oscilla- A symmetry is something that stays the same when some- tion. Once the dynamics of a computational strategy is thing else changes. Identifying symmetries in data can pro- designed and correctly understood, the problem of highly vide powerful constraints on models. In this talk, I review oscillatory quadratures may become relatively simple and a handful of important symmetries (and non-symmetries) that the precision of calculations may even increase as the evident in responses from the mammalian cochlea and dis- frequency of oscillation grows. This talk will focus at a cuss their consequences for models of cochlear mechanics. monochromatic laser traveling in a linear medium with re- fractive index that varies slowly on the scale of an opti- Christopher Shera cal wavelength. Several interesting numerical methods will University of Southern California be introduced and discussed for evaluating corresponding [email protected] highly oscillatory wave problems. Some simulation results will be given.

MS106 Qin Sheng Bifurcation of Nonlinear Bound States from Department of Mathematics Eigenvalues and from Spectral Intervals in PT- Baylor University symmetric Systems Qin [email protected]

In PT-symmetric nonlinear systems, like the Gross- Pitaevskii equation with a complex PT-symmetric poten- MS106 tial, bound states with real propagation constants (i.e. Nonlinear Dynamics of Parity-tme (PT) Symmet- standing solitary waves) have been previously found nu- ric Lasers merically. We present a rigorous proof of their bifurcation in two important cases: bifurcation from a simple eigen- The introduction of parity-time symmetry formalism lead value and from a spectral interval. In the case of the bifur- to the discovery of new and unimaginable phenomena in cation from a simple real linear eigenvalue we work in the optics. The first PT laser was introduced a few years general abstract setting ago. The new experiments have gained substantial atten- tion from the theoretical point of view. Current models are Aψ − f(ψ)=λψ, limited to the lasing in the steady-state regime or under as- sumption of stationary population inversion. In this work, with ∈ R (small), a Lipschitz continuous f and with we propose a study of both spatial and temporal dynamics a closed linear operator A densely defined on a Hilbert of the two coupled micro-ring PT symmetric resonators. space and having a non-empty resolvent set. In the more The proposed models were obtained from the basic equa- difficult case of the bifurcation from a spectral interval we tions of the semi-classical laser theory. consider the one dimensional stationary Gross-Pitaevskii (GP) equation Alexey Sukhinin Southern Methodist University  2 −ψ + V (x)ψ + σ(x)|ψ| ψ = λψ, x ∈ R [email protected] with periodic PT-symmetric coefficients V and σ.Wepro- Jianke Yang vide explicit asymptotic expansions of the bound states Department of Mathematics and Statistics with error estimates. The proofs rely in both cases on the University of Vermont Banach fixed point theorem restricted to a symmetric sub- [email protected] space. In the case of the periodic GP equation we work in Bloch variables. MS106 Tomas Dohnal Long-time Dynamics and Interaction of Ultrashort TU Dortmund Light Pulses [email protected] The propagation dynamics and interaction of ultrashort Dmitry Pelinovsky light pulses is presented. The pulse propagation behav- McMaster University ior is studied by numerical simulations of mathematical Department of Mathematics models at realistic physical conditions. The spatiotempo- [email protected] ral dynamics of ultrashort light pulses within dispersive equations like generalized (3+1)D nonlinear Schrodinger Petr Siegl equation and (3+1)D nonlinear envelope equations as gov- 234 DS17 Abstracts

erning equations as well as at ionization free and ioniza- Dynamical Systems tion regimes is revealed and summarized. The complex structure and the presence of terms with different physical In this talk, I will introduce a toolbox to compute spec- sense requires the coordinate splitting to be preceded by tral properties of the Koopman operator for measure- splitting by physical factors (processes). In contrast to the preserving dynamical systems. The algorithm is based on coordinate splitting this kind of splitting can be exact in a periodic approximation of the underlying dynamical sys- some nodes and in the intervals it can be controlled. We tem. Convergence results are briefly discussed and results show that this method is relevant for study of propagating are presented for certain well-known systems. ultrashort localized pulses in nonlinear waveguides. The nonintegrability of the considered dynamical systems and Nithin Govindarajan the missing exact solutions requires to provide the investi- University of California, Santa Barbara gation only numerically. The obtained results are reliable [email protected] and give good predictions for the material quantities and dynamics of the light pulses. MS107 Michail Todorov Faculty of Applied Mathematics and Informatics Network Identification Based on Koopman Opera- Technical University of Sofia, Bulgaria tor Theory mtod@tu-sofia.bg Spectral characterization and global linearization of nonlin- ear systems are two key features of the Koopman operator MS107 framework. These two aspects can be used to derive effi- Hankel Alternative View Of Koopman (Havok) cient numerical methods for studying nonlinear systems. Analysis of Chaotic Systems This is for instance the case in the context of network identification, i.e. the problem of inferring the structure Understanding the interplay of order and disorder in of a network from measurements of its collective dynamics. chaotic systems is a central challenge in modern quanti- We propose two novel techniques for network identification tative science. Toward this goal, approximate linear repre- based on the Koopman operator framework: (1) spectral sentations for nonlinear dynamics have been long sought, network identification method and (2) lifting method. (1) driving considerable interest in modern Koopman opera- Spectral network identification. We show that there is a re- tor theory. We present a universal, data-driven decompo- lationship between the spectrum of the Koopman operator sition of chaos as an intermittently forced linear system. (which is measured from the network dynamics) and the This work combines delay embedding and Koopman the- spectral properties of the network (Laplacian spectrum). ory to decompose chaotic dynamics into a linear model in Then, by inferring these spectral properties, the method the leading delay coordinates with forcing by low energy makes possible the use of a few local measurements to un- delay coordinates; we call this the Hankel alternative view cover global topological properties of the network (e.g. av- of Koopman (HAVOK) analysis. This analysis is applied erage node degree). (2) Lifting method. This method is to the Lorenz system, as well as to real-world examples based on the key idea that, instead of identifying a non- such as the Earth’s magnetic field reversal, and data from linear system in the state space, we can lift the data to ECG, EEG, and measles outbreaks. In each case, the forc- a higher-dimensional space and identify the linear Koop- ing statistics are non-Gaussian, with long tails correspond- man operator. This lifting technique is a general nonlinear ing to rare events that trigger intermittent switching and systems identification method and, in particular, provides bursting phenomena; this forcing is highly predictive, pro- an efficient numerical scheme to infer the topology and the viding a clear signature that precedes these events. The dynamics of (possibly large) networks. activity of the forcing signal demarcates large coherent re- gions of phase space where the dynamics are approximately Alexandre Mauroy linear, from those that are strongly nonlinear. Video ab- University of Liege stract: https://youtu.be/831Ell3QNck [email protected]

Steven Brunton, Bingni W. Brunton University of Washington MS107 [email protected], [email protected] Koopman Operator Framework for Nonlinear Time Series Analysis Joshua L. Proctor Institute for Disease Modeling [email protected] We introduce a Koopman operator based framework for nonlinear time-series analysis, a field which relies heav- ily on nonlinear dynamical systems theory for character- Eurika Kaiser ization of (typically univariate) irregular time series data. University of Washington Spectral properties of infinite dimensional but linear Koop- [email protected] man operator are exploited as a means for nonlinearly de- composing, enhancing and analyzing the measured signals. Nathan Kutz We demonstrate our framework in variety of time series University of Washington applications including forecasting, anomaly detection, in- Dept of Applied Mathematics dexing/retrieval and classification, and for distinguishing [email protected] causality from correlation.

Amit Surana MS107 System Dynamics and Optimization A Toolbox for Computing Spectral Properties of United Tecnologies Research Center (UTRC) DS17 Abstracts 235

[email protected] range of geometric parameters; however, the underlying asymptotic theory fails for deep systems. We also report the behavior of dense vortex populations in systems with MS108 periodic height variations. All of these findings are ob- Filament Tension and Phase Locking of Meander- tained from a bubble-free BZ reaction. ing Scroll Waves Dayton Syme Rotating spiral waves are often not rigidly rotating, but ex- Department of Chemistry & Biochemistry hibit a quasi-periodic motion known as ’meander’. Exam- Florida State University ples include cardiac tissue and the Belousov-Zhabotinsky [email protected] chemical reaction. Using response function theory, we de- rive the drift equations for meandering spiral waves and Oliver Steinbock apply it to the important cases of electroforetic drift (in Department of Chemistry and Biochemistry chemical media) and curvature-induced motion of 3D scroll Florida State University waves. We show that only under certain conditions (when [email protected] averaging is possible), the emerging filament tension prop- erty is found. Otherwise, phase-locking effects or more complex behaviour is found. For a given external field MS108 strength, we discuss the behaviour of the parallel and Periodic Sequence of Stabilized Wave Segments in perpendicular drift velocities in the parameter space of Excitable Media Barkley’s model including the meandering region. Wave segments represent an interesting and important ex- Hans Dierckx ample of spatio-temporal pattern formation in a broad Department of Physics and Astronomy class of nonlinear dynamic systems, so-called excitable me- Ghent University dia. They have been observed, for instance, in cardiac [email protected] tissue, concentration waves in thin layers of the Belousov- Zhabotinsky reaction or during cell aggregation of Dic- tyostelium discoideum. For a given excitability a medium MS108 supports propagation of a wave segment with a selected The Role of Conduction Block in Spiral Breakup size and shape, which is intrinsically unstable. In order and Merger to make this solution observable it has to be stabilized by an adequate noninvasive feedback control. For the case of Spiral wave chaos in excitable media, such as cardiac tis- a solitary propagating wave segments a universal selection sue, is characterized by continuous breakups and mergers rules have been found by use a free-boundary approach. of spiral wave segments. While the linear instabilities of The main aim of our study is to generalize these results traveling waves leading up to wave breakup in different on a case of a periodic sequence of wave segments. To this models are not universal, the breakup itself proceeds in a aim the motion of a stabilized wave segment is numerically universal fasion. It involves conduction block, where a por- studied by use of a generic reaction-diffusion model with tion of the wave fails to propagate, creating two new spiral nonlinear activator-inhibitor kinetic. In addition, the free- cores. We show that this process can be described quanti- boundary approach is applied to determine the wave seg- tatively by following the dynamics of two level sets which ment shape and the speed as functions of the medium pa- define the excitation front and the trailing edge of a refrac- rameters. We hope that the results obtained in this study tory region of a preceding wave. Quite surprisingly, we find are also applicable to the spiral wave dynamics. that spiral wave mergers, where two spiral cores of oppo- site chirality merge and annihilate, also involve conduction Vladimir Zykov block and can be understood with the help of the same two Max Planck Institute for Dynamics and Self-Organization level sets. Hence, the two key phenomena responsible for [email protected] maintaining spiral chaos both involve a universal physical mechanism and allow a universal topological description. Eberhard Bodenschatz Max Planck Institute for Dynamics & Self-Organization Roman Grigoriev [email protected] Georgia Institute of Technology Center for Nonlinear Science & School of Physics [email protected] MS109 Fisher-Kpp Invasion Fronts on Homogeneous Trees Christopher Marcotte and Random Networks Georgia Tech [email protected] We study the dynamics of the Fisher-KPP equation on the infinite homogeneous tree and Erd˝os-R´eyni random graphs. We assume initial data that is zero everywhere except at MS108 a single node. For the case of the homogeneous tree, the Scroll Wave Drift and Interaction in Excitable Sys- solution will either form a traveling front or converge point- tems with Height Variations wise to zero. This dichotomy is determined by the linear spreading speed and we compute critical values of the dif- The Belousov-Zhabotinsky (BZ) solution spontaneously fusion parameter for which the spreading speed is zero and self-organizes three-dimensional vortices, analogous to nat- maximal and prove that the system is linearly determined. ural phenomena like ventricular tachycardia and Dic- We also study the growth of the total population in the net- tyostelium discoideum aggregation. Sharp edges and cor- work and identify the exponential growth rate as a function ners affect the dynamics of these excitation vortices. We of the diffusion constant. Finally, we make predictions for test earlier theoretical predictions regarding the vortex the Fisher-KPP equation on Erd˝os-R´enyi random graphs drift velocity and find excellent agreement over a large based upon the results on the homogeneous tree. We ob- 236 DS17 Abstracts

serve that exponential growth rates on the random network cover an experimentally observed biphasic relationship be- can be bounded by growth rates on the homogeneous tree tween the invasion speed of the tumour and the background and provide an explanation for the sub-linear exponential extracellular matrix density. This is joint work with Lotte growth rates that occur for small diffusion. Sewalt, Kristen Harley and Sanjeeva Balasuriya

Matt Holzer Peter van Heijster Department of Mathematics Mathematical Sciences School George Mason University Queensland University of Technology [email protected] [email protected]

Aaron Hoffman Lotte Sewalt Franklin W. Olin College of Engineering Leiden University aaron.hoff[email protected] [email protected]

Kristen Harley MS109 Queensland University of Technology Contagion Shocks in a Model of Panicking Crowds [email protected]

We consider an agent-based model of emotional conta- Sanjeeva Balasuriya gion coupled with motion that has previously been stud- University of Adelaide ied in the computer science community. The model in- [email protected] volves movement with speed proportional to a fear’ vari- able that undergoes a temporal consensus averaging with other nearby agents. We study the effect of Riemann ini- MS110 tial data for this problem in 1D, leading to shock dynamics that are studied both within the agent-based model as well Optimal Experimental Design for Extreme Event as in a continuum limit. We examine the model under dis- Statistics in Nonlinear Dynamical Systems tinguished limits as the characteristic contagion interaction We propose a computational strategy for the fast evalua- distance and the interaction timescale both approach zero. tion of extreme event statistics associated with nonlinear Here, we observe a threshold for the interaction distance dynamical systems. In particular, our goal is the devel- vs. interaction timescale that produces qualitatively dif- opment of a computational algorithm that adaptively pro- ferent behavior for the system - in one case particle paths vide a series of experimental parameters that rapidly leads do not cross and there is a natural Eulerian limit involving to the accurate determination of the probability density nonlocal interactions and in the other case particle paths function (pdf) for an arbitrary quantity of interest. Since, can cross and one may consider only a kinetic model in the our goal is to capture extreme event statistics, emphasis is continuum limit. We will also discuss recent extensions of given on the accuracy of the method with respect to the the model to two dimensions, and new methods of efficient tails of the pdf. The algorithm is based on the blending of computation for the model. i) an inexpensive, low-fidelity model, which is assumed to Martin Short accurately capture average behavior of the dynamical sys- Georgia Tech tem, but not necessarily extreme events, ii) a Gaussian pro- [email protected] cess regression framework which blends information from the experiment and the low-fidelity model, and iii) a prob- abilistic decomposition synthesis method that provides an MS109 efficient representation of the pdf and its tail structure. The combination of these three components allows for es- On the Evolution of Cancerous Cells timation of the pdf and its tail structure, as well as the Recent advances on a nonlinear model for tumor growth computation of upper and lower bounds. From an initial will be discussed. set of experiments, we adaptively select additional sam- ple points (experimental parameters) so that the posterior Konstantina Trivisa distance between the estimated upper and lower bounds of University of Maryland the pdf is minimized. We illustrate the method to several Department of Mathematics examples of various complexity. [email protected] Mustafa Mohamad MIT MS109 [email protected] Travelling Wave Solutions for a Model of Tumour Invasion Themistoklis Sapsis Massachusetts Institute of Techonology A recent study by Korolev et al. [Nat. Rev. Can- [email protected] cer, 14:371–379, 2014] evidences that the Allee effectin its strong form, the requirement of a minimum density for cell growthis important in the spreading of cancerous tu- MS110 mours. We present one of the first mathematical models of Predicting Extreme Events for Passive Scalar Tur- tumour invasion that incorporates the Allee effect. Based bulence in Two-Layer Baroclinic Flows through on analysis of the existence of travelling wave solutions to Reduced-Order Stochastic Models this model, we argue that it is an improvement on previous models of its kind. We show that, with the strong Allee ef- The capacity of imperfect reduced-order models to capture fect, the model admits biologically relevant travelling wave crucial tracer statistical features such as tracer energy spec- solutions, with well-defined edges. Furthermore, we un- trum and tracer intermittency is investigated. The passive DS17 Abstracts 237

scalar field is advected by a two-layer baroclinic turbu- of the trajectory fragments within the strata with mini- lent flow in atmosphere and ocean regimes with jets and mal communication between them, and combining those vortices. Much simpler and more tractable linear Gaus- averages with appropriate weights to yield averages with sian stochastic models are proposed to approximate the respect to the original underlying process. The result is an complex and high-dimensional advection flow field. The efficient and robust scheme for very general rare event sim- imperfect models prediction skill are improved through a ulation problems. Our framework reveals the full generality judicious calibration about the model errors in autocor- and flexibility of trajectory stratification, and it illuminates relation function using the leading order statistics of the a common mathematical structure shared with the highly background advection flow field, while no additional prior successful, equilibrium umbrella sampling method for free information about the passive tracer statistics is required. energy calculations. A systematic framework of measuring the autocorrelation function with empirical information theory is introduced, Jonathan Weare and optimal model parameters under this unbiased infor- University of Chicago mation measure can be achieved in a training phase before Department of Statistics and the James Franck Institute the prediction. It is demonstrated that the prediction skill [email protected] of the imperfect model can be effectively improved through the calibration strategy with optimal parameters. Cru- cially leading order statistical quantities like the tracer en- MS111 ergy spectrum and fat-tails in the tracer density functions Data-Driven Parameterization of the Generalized in the most important large scales can be captured effi- Langevin Dynamics for Bio-Molecules ciently with accuracy using the reduced-order tracer model in various dynamical regimes of the flow field with distinct This talk is concerned with microscopic models for bio- statistical structures. molecules. Due to the overwhelming number of degrees of freedom in an all-atom model, coarse-grained descrip- Di Qi tion has become a more promising candidate for simulat- New York University ing important biological processes. We present a data- [email protected] driven approach to determine the parameters in a gener- alized Langevin model, which in principle can be derived AMajda from the full dynamics using the Mori-Zwanzig projection NYU formalism. We make explicit connections between the pa- [email protected] rameters and the statistics of the coarse-grain variables. Several examples will be shown to demonstrate the effec- tiveness of the algorithm. MS110 Xiantao Li Identification and Protection Against Critical Con- Department of Mathematics tingencies in Power Systems Pennsylvania State University Power system is the largest, and arguably the most com- [email protected] plex machine ever built by humans. Due to inherent nature of power flows, power system lacks global stability and is MS111 naturally fragile. Large enough disturbances may cause the loss of stability and trigger the cascading failures resulting Data-Driven Modeling and the Mori-Zwanzig For- in major blackouts. Aggressive introduction of renewable malism generation increases the overall stress of the system, so the stability constraints will likely become the main barrier for In many scientific and engineering applications, a fully- transition to clean energy sources. Despite many decades resolved computational model is either unavailable (due of research, stability assessment is still the computational to missing measurements or poorly-understood physics) bottleneck in power grid operation process. Modern tools or prohibitively expensive to use. Data-driven modeling from optimization and dynamical systems provide a unique methods are thus valuable for constructing effective mod- opportunity for addressing these problems with new gener- els, by combining relevant physics with data. In this talk, ation of fast and reliable algorithms. This talk will review I will provide a brief introduction to data-driven modeling the computational challenges the power system operators and model reduction, touching on relevant theoretical and face today and propose specific strategies for characteri- practical issues. I will then discuss a key issue, namely zation of transient stability and impact of uncertainty for how to represent the memory effects that arise when a dy- better screening of critical contingencies and design of cor- namical system is projected onto a subset of its degrees of rective remedial actions. freedom. Connections to the Mori-Zwanzig projection op- erator formalism of nonequilibrium statistical mechanics Konstantin Turitsyn will be discussed. Massachusetts Institute of Technology [email protected] Fei Lu Department of Mathematics, UC Berkeley Mathematics group, Lawrence Berkeley National MS110 Laboratory fl[email protected] Trajectory Stratification for Rare Event Simulation

I will outline a general mathematical framework for trajec- Kevin K. Lin tory stratification of stochastic processes based on the non- Department of Mathematics equilibrium umbrella sampling method. Trajectory strati- University of Arizona fication involves decomposing trajectories of the underlying [email protected] process into fragments limited to restricted regions of state space (strata), computing averages over the distributions Alexandre Chorin 238 DS17 Abstracts

Department of Mathematics likely to affect economies on a global scale. University of California at Berkeley [email protected] Jim Brannan Department of Mathematical Sciences Clemson University MS111 [email protected] Renormalization and Stability of Coarse-Grained Models MS112 The construction of reduced order (coarse-grained) models for complex systems has been a very active area of research Tipping Point Analysis of Dynamical Systems, with in recent years. We will present results about the effect of Applications in Geophysics and Environmental Sci- the dimensionality of the original (full order) system on the ences stability and accuracy of reduced order models. In partic- ular, we want to emphasize that the inability to simulate We apply tipping point analysis for anticipation, detection, the full order system can cause instabilities for the reduced and forecast of tipping points in a dynamical system. De- order models and how renormalization can be used to tame generate fingerprinting indicator is a dynamically derived such instabilities. lag-1 autocorrelation (or short-range scaling exponent of Detrended Fluctuation Analysis [Livina and Lenton, GRL Panos Stinis 2007]), which monitors memory in a series for early warn- Pacific Northwest National Laboratory ing signals. Potential analysis detects a tipping point at [email protected] the time when it happens, which is illustrated in a contour plot mapping the potential dynamics of the system [Liv- ina et al, Climate of the Past 2010; Livina et al, Climate MS111 Dynamics 2011; Livina et al, Physica A 2012; Livina et al, Dimension Reduction for Systems with Slow Re- Physica A 2013]. Potential analysis is also used to forecast laxation time series by extrapolation of Chebyshev approximation coefficients of the kernel distribution, with reconstruction Dynamical systems that relax very slowly to equilibrium of correlations in the data [Livina et al, Physica A 2013]. and can encode long term memory, i.e. the influence of The methodology has been tested on artificial data and the initial conditions can persist. We develop reduced, on geophysical, ecological and industrial sensor datasets stochastic models for high dimensional dynamical systems [Livina et al, Journal of Civil Structural Health Monitor- with slow relaxation. We present a variety of empirical and ing, 2014; Kefi et al, PLoS ONE 2014; Livina et al, Chaos first principles approaches for model reduction, and build 2015; Perry et al, Smart Materials and Structures, 2016], a mathematical framework for analyzing the reduced mod- and proved to be applicable to trajectories of dynamical els. We introduce the notions of universal and asymptotic systems of arbitrary origin [Vaz Martins et al, PRE 2010]. filters to characterize ‘optimal’ model reductions. We dis- cuss how our methods apply to the practically important problem of modeling oil spills. Valerie N. Livina National Physical Laboratory Shankar C. Venkataramani [email protected] University of Arizona Department of Mathematics [email protected] MS112 Raman Venkataramani Stochastic Dynamics of Near-Surface Winds over Seagate Technologies Land [email protected] Understanding the variability of near-surface winds is im- Juan M. Restrepo portant for computing surface fluxes, for characterizing ex- Oregon State University treme wind events, and for assessing the wind power re- Department of Mathematics source. Over land surfaces, the probability distribution of [email protected] near-surface wind speed displays a marked diurnal evolu- tion. In the bottom few tens of meters of the atmosphere, mean wind speeds at night tend to be weaker than dur- MS112 ing the day - but the tails of the nocturnal distribution Unstable Behavior in Capitalistic Economic Sys- are much longer than those of the day. In this talk, I will tems provide evidence that the highly-skewed nature of noctur- nal surface wind speeds is related to the existence of two We present a simple extension of the well-known Solow distinct regimes of the stably stratified nocturnal plane- model of economic growth that describes the statistical be- tary boundary layer. An empirical stochastic model will havior of business cycle fluctuations observed in the U.S. be developed using a Hidden Markov Model analysis of Gross Domestic Product (GDP). The extension shows how long observations of wind and temperature at a tall tower the network structure of economic supply chains induces a at Cabauw, in the Netherlands. The relationship between positive feedback loop in macrodynamics that causes the regime structure and the surface wind speed probability GDP to be very sensitive to small random disturbances distribution will also be investigated using an idealized in consumption and investment. The model suggests that stochastic representation of the near-surface momentum recessions are inevitable in any well developed capitalistic budget. economic system and that increasing globalization of the world economy will result in economic expansions and con- Adam Monahan tractions that are less likely to be locally confined and more University of Victoria DS17 Abstracts 239

[email protected] Forced Beams: An Infinite-Dimensional Analysis

We use invariant manifold results on Banach spaces to con- MS112 clude the existence of spectral submanifolds (SSMs) in a Metastable Phenomena in a Dynamical System class of nonlinear, externally forced beam oscillations . Re- with a Discontinuous Vector Field: Case of Ama- duction of the governing PDE to the SSM provides an ex- zonian Vegetation Model act low-dimensional model which we compute explicitly. This model captures the correct asymptotics of the full, For the tipping elements in the Earth’s climate system, infinite-dimensional dynamics. Our approach is general the most important issue to know is how stable the desir- enough to admit extensions to other types of continuum able state is against random, possibly non-small perturba- vibrations. The model-reduction procedure we employ also tions. We report the results of the stability analysis of the gives guidelines for a mathematically self-consistent mod- metastable fertile forest state in stochastically perturbed eling of damping in PDEs describing structural vibrations. Amazonian Vegetation (AV) model with discontinuous vec- tor field. To solve the problem of non-uniqueness of solu- Florian Kogelbauer tions, that often rise in this type of systems with repulsive ETH Zurich, Switzerland sliding mode, the discontinuous vector field, employing the [email protected] mollification method, was approximated by a smoothened drift coefficient. Relying on the empirical evidence that the environmental studies provide, we have included in the AV MS113 system the symmetric α-stable L´evy perturbations. In our stability analysis we employ mean firs exit time (MFET), Random Perturbations of Periodically Driven Non- escape probability (EP) and stochastic basin of attraction linear Oscillators: Homogenization and Large De- (SBA) which are the quantities that describe the dynamical viations behavior of the system and usually used in the investiga- The recent surge of research articles in energy harvesting tion of the metastable states. Our main conclusions are focuses on the “cantilever beam” devices which are used to that the slightest threat to the forest state stability rep- convert small amplitude mechanical vibration into electri- resents L´evy noise with big jumps of the small intensity cal energy that could be used for electronic devices with (α =0.5, α =1andε =0.1). On the other hand, a L´evy low power requirements. Prototypical beam type nonlin- noise with small jumps (α =1.5) as well as noise with high ear energy harvesting models contain double well poten- intensity (ε =1,α =0.5orα = 1) significantly accelerate tials, external or parametric periodic forcing terms, damp- the transition between forest and savanna states causing ing and ambient broadband additive noise terms. This talk the high instability of the forest. will develop a unified approach to study the dynamics of Larissa I. Serdukova single degree of freedom systems excited by both periodic School of Mathematics and Statistics, and random perturbations. The phase space for a periodi- Huazhong University of Science and Technology cally driven nonlinear oscillator consists of many resonance [email protected] zones. The near resonant dynamics of such systems, in the presence of weak noise, is not well understood. We will study this problem in depth with the aim of discovering MS113 a common geometric structure in the phase space and de- termining the effect of weak noise on the escape from a Exact Model Reduction by a Slow-Fast Decompo- resonance zone. We obtain the large-deviation rate func- sition of Multi-Degree-of-Freedom Vibrations tion for the escape by developing an asymptotic method based on averaging and large deviations. We discuss conditions under which a general nonlinear mechanical system can be exactly reduced to a lower- Navaratnam Sri Namachchivaya, Nishanth Lingala dimensional model that involves only the most flexible de- UIUC grees of freedom. This Slow-Fast Decomposition (SFD) en- [email protected], [email protected] slaves the stiff degrees of freedom exponentially fast to the flexible ones while trajectories converge to a slow manifold. Ilya Pavlyukevich We also find that near equilibria, the SFD gives a mathe- Institut f¨ur Stochastik matical justification for two modal-reduction methods used Friedrich–Schiller–Universit¨at Jena, Germany in structural dynamics: static condensation and modal [email protected] derivatives. These formal reduction procedures, however, are also found to return incorrect results when the SFD conditions do not hold. We illustrate all these results on MS113 mechanical examples. Nonlinear Model Identification and Spectral Sub- George Haller manifolds for Multi-Degree-of-Freedom Mechanical ETH, Zurich, Switzerland Vibrations [email protected] In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to Sten Ponsioen linear modal subspaces of an equilibrium. Amplitude- ETH Zurich frequency plots of the dynamics on SSMs provide the Switzerland classic backbone curves sought in experimental nonlinear [email protected] model identification. We develop here a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating MS113 model fitted to experimental vibration signals. Using Model Reduction to Spectral Submanifolds for examples of both synthetic and real experimental data, 240 DS17 Abstracts

we demonstrate that this approach reproduces backbone spatially-localized solutions. curves with high accuracy. Stefanos Folias Robert Szalai Department of Mathematical Sciences University of Bristol University of Alaska Anchorage [email protected] [email protected]

MS114 MS114 Biological Aggregation Driven by Social and En- Patterns and Waves in a Spatially-Extended Neural vironmental Factors: A Nonlocal Model and Its Field Model Degenerate Cahn-Hilliard Approximation Corticalwaveshavebeenobservedinavarietyofneural Biological aggregations such as insect swarms and fish circuit experiments. These events of activity have impor- schools may arise from a combination of social interactions tant features such as velocity and phase relations which and environmental cues. Nonlocal continuum equations impact the termination and propagation of neural activ- are often used to model aggregations, which manifest as ity. In this talk, we analyze traveling waves occurring in localized solutions. While popular in the literature, the a 1D spatially-distributed Wilson-Cowan firing rate model nonlocal models pose significant analytical and computa- to better understand how the system transitions from trav- tional challenges. Beginning with the nonlocal aggregation eling fronts to pulses as the time constant of inhibition in- model of [Topaz, Bertozzi & Lewis, Bull. Math. Bio., creases (τ>0). The traveling front represents an increase 2006], we derive the minimal well-posed long-wave approx- from down to up-state activity, while the traveling pulse imation, which is a degenerate Cahn-Hilliard equation. Us- reflects a wave of excitation that returns to down-state ac- ing analysis and computation, we study energy minimizers tivity. As τ increases, the system passes through a Hopf and show that they retain many salient features of those of bifurcation and limit cycles emerge. Here, the front leads to the nonlocal model. Furthermore, using the Cahn-Hilliard spatially-homogeneous (bulk) oscillations, while further in- model as a testbed, we investigate how an external poten- creasing τ pushes the oscillation through a period-doubling tial modeling food sources can suppress peak population bifurcation, and the front leads to spatiotemporal patterns. density, which is essential for controlling locust outbreaks. The instability to the bulk oscillation makes for interesting Random potentials tend to increase peak density, whereas dynamics near the transition to a pulse solution, and for a periodic potentials can suppress it. small range of τ-values, the system shows a traveling front that tends to a spatiotemporal pattern or a traveling pulse, Andrew J. Bernoff depending on the stimulus duration. Harvey Mudd College Department of Mathematics Jeremy D. Harris [email protected] Department of Mathematics University of Pittsburgh [email protected] Chad M. Topaz Dept. of Mathematics, Statistics, and Computer Science Macalester College MS114 [email protected] Compositional Evolution of Quasi-bilayers in the Symmetric Two-component Functionalized Cahn- MS114 Hilliard Equation Traveling Waves and Breathers in An Excitatory- Multicomponent mixtures in general support bilayers with Inhibitory Neural Field a diversity of lipid compositions. We study a strong two- component FCH model with a radial symmetric potential Neural fields are spatially-extended nonlinear integrodiffer- which admits a family of quasi-bilayers with various com- ential equations that represent the large-scale dynamics of positional ratios between amphiphile A and B. In the ab- populations of neurons mediated by synaptic interactions sence of pearling bifurcation, the compositional and geo- and other neuronal processes (e.g., firing rate adaptation metric evolution of quasi-bilayers decouples, in the sense and synaptic depression) and support a wide range of spa- that the former evolution takes place in the 1/ time scale tiotemporal dynamics, including traveling waves. We study when the normal velocity of the interface is still zero. More the existence and stability of traveling waves of localized specifically, the composition ratio satisfies a nonlocal equa- activity in a two-population E-I neural field which governs tion accommodating rich dynamics. Depending on the the interaction between excitatory and inhibitory neurons competition between the phase separation and the quench- along a one-dimensional spatial domain. We develop exis- ing of the background, the composition ratio evolves into, tence conditions for activity bumps traveling with constant (1) a homogeneous profile; (2) a phase separation profile speed and derive an Evans function to analyze their spec- where the bilayer consists of pure A regions and pure B re- tral stability. We show that drift bifurcations of stationary gions; or, (3) a quenched profile in a co-dim two manifold. bumps serve as a mechanism for generating traveling bump In case (2) and (3), a rapid spatial variation of the com- solutions in the E-I neural field as parameters are varied. position ratio promotes surface diffusion terms from lower Furthermore, we explore the interrelations between sta- orders. While the evolution of phase separation profiles tionary and traveling types of bumps and breathers (time- mimics Allen-Cahn type coarse graining, novel dynamics periodic oscillatory bumps) by bridging together analytical emerges from the evolution of rapid varying profiles in a and simulation results for stationary/traveling bumps and neighborhood of the quenched manifold: the compositional their bifurcations in a region in parameter space. Inter- profile stays nearby the quenching manifold and evolves estingly, we find evidence for a codimension 2 drift-Hopf into a periodic profile with a large period. bifurcation and show that the codim 2 point serves as an organizing center for the dynamics of these four types of Qiliang Wu, Keith Promislow DS17 Abstracts 241

Michigan State University of Drug Resistance [email protected], [email protected] Recent experimental work has shown how drug treatment can cause competitive release of drug resistant strains MS115 within an infected individual. How this competitive re- Rhinovirus: From Biochemistry and Immunology lease at the within-host scale translates into the spread to Families and Communities of drug resistance through a population, however, is less clear. I will present a mathematical model that integrates Rhinoviruses have been described as the perfectly adapted the processes occurring at multiple scales during the emer- virus, and have striking features ranging from enormous gence and spread of drug resistance. Analysis of the model serotype diversity to low virulence and immunogenicity. shows that, although competitive release can enhance the Models that combine the full complexity of the immune spread of resistance within a host, processes operating at response with the idiosyncracies of transmission, such as the between-host scale can nevertheless curtail the spread strong age and symptom dependence, would be impossible of resistance. to parameterize or analyze. I will present a series of mod- els of rhinovirus, starting within hosts and stepping up to Troy Day families and communities, showing how we can extract the Departments of Mathematics & Biology key aspects of each level of organization to feed into the Queen’s University, Kingston, Canada next level to begin to explain the striking features of rhi- [email protected] novirus. I will apply these results to examine the many connections between rhinovirus and asthma, perhaps the Johanna Hansen most important clinical effect of this virus, and consider Department of Mathematics and Statistics the possible consequences of the newly proposed vaccine. Queen’s University [email protected] Fred Adler University of Utah Dept. of Mathematics, Dept. of Biology MS115 [email protected] Emerging Disease Dynamics in a Model Coupling Within and Between-Host Systems

MS115 Epidemiological models and immunological models have Examining Signatures of Within-Host Malaria Het- been studied largely independently. However, the two bi- erogeneity at an Epidemiological Level ological processes (within- and between-host interactions) occur jointly and models that couple the two processes may Plasmodium falciparum, the most virulent human malaria generate new biological insights. We developed and ana- parasite, matures via asexual reproduction within the hu- lyzed ODE models that link an SI epidemiological model man host, but undergoes sexual reproduction within its and an immunological model for pathogen-cell dynamics. vector host, the Anopheles mosquito. Consequently, the When the two sub-systems are considered in isolation, mosquito stage of the parasite life cycle provides an op- dynamics are standard and simple. That is, either the portunity to create novel parasite genetic information in infection-free equilibrium is stable or a unique positive mosquitoes infected with multiple parasite populations, al- equilibrium is stable depending on the whether the rele- tering parasite diversity at the population level. Despite vant reproduction number is less or greater than 1. How- the important implications for disease transmission and ever, when the two sub-systems are dynamically coupled, malaria control, a quantitative mapping of diversity gen- the full system exhibits more complex dynamics including eration within the vector is lacking. To examine the role backward bifurcations; that is, multiple positive equilibria that vector biology plays in modulating parasite diversity, exist with one of which being stable even when the repro- we develop a multi-scale model that estimates the diver- duction number is less than 1. The biological implications sity as a consequence of different bottlenecks and expansion of such bifurcations are illustrated using an example con- events occurring during the parasite life cycle and couples cerning the spread and control of toxoplasmosis. the diversity development within the mosquito to trans- mission to humans. For the underlying framework, we use Zhilan Feng a stochastic model of within-vector P. falciparum dynam- Department of Mathematics ics and simulate the dynamics of infections with multiple Purdue University distinct parasite populations. We use the results of the [email protected] within-vector portion of our model to inform transmission of the parasites to the human population. Our model quan- titatively maps parasite diversity through the life cycle of MS116 the parasite including bottlenecks between the mosquito Recent Advances in Rigorous Computation of the and the human host. Evans Function

Lauren Childs For a wide class of physical systems of interest, asymptotic Virginia Polytechnic Institute and State University orbital stability of traveling waves is determined by spectral [email protected] stability. For many systems, analytical techniques deter- mine spectral properties in various limits, but are unable to Olivia Prosper treat the entire space of solutions. In these cases, numeri- University of Kentucky cal studies are used to provide strong evidence of spectral [email protected] properties. Our recent work deals with rigorous verification of spectral stability by controlling all error, including ma- chine truncation error, in these numerical computations. MS115 In this talk we describe recent advances in rigorous com- The Role of Multi-Scale Selection in the Emergence putation of the Evans function, an analytic function whose 242 DS17 Abstracts

zeros determine spectral stability of the underlying travel- Institute for Problems in Mechanical Engineering ing wave. [email protected]

Blake Barker Brown University MS117 [email protected] A First Step Toward Quantifying the Climates In- formation Production Over the Last 68,000 Years Kevin Kevin Zumbrun Indiana University We take an information-theoretic approach to analyzing 68 USA thousand years of water isotope data from the WAIS Di- [email protected] vide ice core, the longest continuous and highest-resolution record yet recovered from Antarctica. The water isotopes are primarily a proxy for local temperature at the time MS116 of snow deposition, but also contain information about The Gray-Scott Model: Bistable Regime broader atmospheric circulation patterns. Using a well de- fined depth-age scale, we apply weighted permutation en- Using singularly perturbed nature of the Gray-Scott model, tropy to calculate the Shannon entropy rate of the isotope we apply multi-scale analysis in a systematic way to show measurements. We find that the rate of information pro- the existence and stability of a traveling front and a trav- duction reveals differences in analytical techniques, even eling pulse. While the traveling front is stable, the pulse is when those differences leave no visible traces in the raw unstable. data. The entropy calculations also allow us to identify a number of intervals in the data that may be of direct rel- Vahagn Manukian evance to paleoclimate interpretation. To validate the re- Miami University Hamilton sults, we perform a number of statistical tests, particularly [email protected] whether the information production arises from climatic signals or post-processing of the ice after recovery from the drill site. MS116 Center Manifolds for a Class of Degenerate Evolu- Joshua Garland tion Equations and Existence of Small-Amplitude University of Colorado at Boulder Kinetic Shocks [email protected]

We construct center manifolds for a class of degenerate Tyler Jones evolution equations including the steady Boltzmann equa- University of Colorado at Boulder tion and related kinetic models, establishing in the process INSTAAR existence and behavior of small-amplitude kinetic shock [email protected] and boundary layers. Notably, for Boltzmann’s equation, elements of the center manifold are shown to decay in ve- Elizabeth Bradley locity at near-Maxwellian rate, in accord with the formal University of Colorado Chapman-Enskog picture of near-equilibrium flow as evo- Department of Computer Science lution along the manifold of Maxwellian states, and Grad’s [email protected] 13-moment approximation via Hermite polynomials in ve- locity. Ryan G. James Alin Pogan University of California, Davis Miami University [email protected] Department of Mathematics [email protected] James White University of Colorado at Boulder INSTAAR MS116 [email protected] Complex Bifurcations in Benard-Marangoni Con- vection MS117 We report a stability analysis of an idealized model of Overview of the Boundary Layer, Cyclones, and Benard-Marangoni convection in two spatial dimensions. the Dynamics Involved We show that more complicated bifurcations can appear in this system for a certain nonlinear temperature profile as An introduction to atmosphere physics will be given as a compared to bifurcations in the classical Rayleigh-Benard prelude to the other talks in this mini-symposium. Specifi- and Benard-Marangoni systems with simple linear verti- cally, we’ll introduce the atmospheric boundary layer, and cal temperature profiles. The obtained results lead to our what is means to be stably stratified. Turbulence will be understanding of complex spatial patterns at a free liquid discussed in the context of atmospheric dynamics, and an surface and will be useful for turbulence theory. overview of extra-tropical cyclones will be given.

Ivan Sudakov David Collins University of Dayton Vancouver Island University Department of Physics [email protected] [email protected]

Sergey Vakulenko MS117 Russian Academy of Sciences Dynamics of the Stably Stratified Atmospheric DS17 Abstracts 243

Boundary Layer thick viscous fluid layer filling the STS. This is a surpris- ing evolutionary development as the thermoviscous noise Flow, stratification, and turbulence in the atmospheric engendered by such positioning might result in an ineffec- boundary layer are dynamically coupled: turbulent inten- tive sensor. In order to understand the trade-off between sity and fluxes are influenced by the the eddy-averaged noise and sensitivity at the IHC HB, we developed a model state, which itself depends on the turbulent fluxes. This of the fluid-structure interaction between the viscoelastic interplay is of particular interest in the stably stratified structures and the viscous STS and sulcus fluids. We used boundary layer, which is observationally found to exist in the fluctuation dissipation theorem to compute the thermo- distinct regimes (respectively very and weakly stably strat- viscous noise forces and a stochastic model of the channel ified). This talk will present recent observational and ide- to estimate the gating noise force. Using this model we alized modelling results regarding these regimes and the found the noise induced rms HB tip motion of less than a transitions between them. While the models considered are few nanometers (less than the expected threshold levels) highly idealized from a meteorological perspective, they are and that the channel noise is the dominant noise at the highly nonlinear dynamical systems with a large number of apex while viscous noise is the dominant at the base. We degrees of freedom. Equilibrium and linear stability anal- also computed the minimal detectable signals associated yses will be presented and compared with time-dependent with shear motion of the tectorial membrane along with model behaviour. The relation of these model results to the effect of HB channel gating and adaption on thresh- the observed structures of boundary layer regimes will also olds. be discussed. Karl Grosh,AritraSasmal Amber Holdstworth University of Michigan SEOS, University of Victoria [email protected], [email protected] [email protected]

Adam Monahan MS118 University of Victoria Effect of Electro-Mechanical Coupling on Stochas- [email protected] tic Oscillations in a Model of Hair Cells

Hair cells are mechanoreceptors which transduce mechan- MS117 ical vibrations to electrical signals in peripheral organs of The Dissipation Structure of Extratropical Cy- senses of hearing and balance in vertebrates. Somatic cell clones motility and active motility of the hair bundle, mechani- cally sensitive structure on the hair cell apex, are two main The physical characteristics of extratropical cyclones are mechanisms by which hair cells can amplify mechanical investigated based on non-equilibrium thermodynamics. stimuli. In amphibians and some reptiles active processes Non-equilibrium thermodynamics (developed mostly in the in hair cells result in noisy mechanical oscillation of hair 1970-90) has been widely used in many scientific fields. bundles, which may lead to frequency selective amplifica- Non-equilibrium thermodynamics can reveal the dissipa- tion. The same cells often demonstrate spontaneous elec- tion structure for any thermodynamic and dynamic system. trical oscillation of their somatic potentials, a signature of It is found that dissipation is always present in an extra- yet another amplification mechanism. Functional role of tropical cyclone, and the dissipation center is not always voltage oscillation is not well understood. We use com- coincident with the low pressure center. This is especially putational modeling to study how the interaction of two for true for young cyclones. The different components of distinct unequally noisy oscillators, mechanical and electri- internal entropy production correspond to different dissi- cal, may affect their spontaneous dynamics, battle noise, pation processes. Usually, the thermal dissipation due to and shape sensitivity of hair cells to external mechanical turbulent vertical diffusion and convection lags geograph- signals. ically behind the dynamic dissipation, and this is due to wind stress. At the incipient stage, the dissipation is mainly Alexander Neiman thermal in nature. A concept of temperature shear is in- Ohio University troduced as the result of thermal dissipation. The temper- Department of Physics and Astronomy ature shear provides a useful diagnostic for extratropical [email protected] cyclone identification. We present results from a regional study in the western Pacific. Rami Amro Department of Physics Jiangnan Li University of California Los Angeles Environment and Climate Change Canada [email protected] [email protected]

MS118 MS118 From Passive to Active Mechanics in the Cochlea The Signal and the Noise: Microfluidics and Sens- ing in the Cochlea Active mechanics in the cochlea, known as cochlear ampli- fication, refers to the outer hair cell (OHC) based forcing The nonlinear and finely tuned response of the organ of that enhances mechanical responses to sound. Responses Corti of the mammalian cochlea to external acoustic stimu- to a particular frequency are only enhanced over a limited lation serves to induce a radial fluid flow in the sub-tectorial spatial region, where that frequency peaks in the passive space (STS) and over the inner hair cell (IHC) hair bun- cochlea. Thus, the cochlear amplifier increases the sharp- dle (HB). In turn, this flow gives rise to forces on the HB ness of cochlear frequency tuning, explaining why external that are transduced into neural impulses at the synaptic hearing aids are limited in their ability to compensate for pole of the cell. The free-standing IHC HBs responsible for hearing loss. The mechanism for localized amplification transducing the acoustic signal are immersed in the 2-5 μm was explored with synchronous measurements of basilar 244 DS17 Abstracts

membrane motion and OHC electrical responses. These ping measurements revealed a phase shift of voltage relative to displacement, occurring at the frequency for which me- Wave breaking in deep oceans is a challenge that still de- chanical enhancement began. Consideration of the known fies complete scientific understanding. Sailors know that at OHC voltage-force relationship showed that the phase shift wind speeds of approximately 5m/sec, the random looking introduces the force-velocity phasing needed for power windblown surface begins to develop patches of white foam transfer from OHCs to the cochlear traveling wave. The (whitecaps) near sharply angled wave crests. We idealize phase shift was independent of stimulus level and thus such a sea locally by a family of close to maximum ampli- seems to be based in the cochleas passive mechanics. Ongo- tude Stokes waves and show, using highly accurate simula- ing measurements of cochlear motion and voltage explore tion algorithms based on a conformal map representation, the basis for the phase shift that activates cochlear am- that perturbed Stokes waves develop the universal feature plification, and to determine if it is altered in unhealthy of an overturning plunging jet. We analyze both the cases cochleae. The cochlear amplifier is a key, yet fragile ele- when surface tension is absent and present. In the latter ment of auditory processing. Understanding the reason for case, we show the plunging jet is regularized by capillary its fragility is closely tied with understanding the operation waves which rapidly become nonlinear Crapper waves in of the healthy cochlea. whose trough pockets whitecaps may be spawned.

Elizabeth S. Olson Sergey Dyachenko Columbia University Brown University [email protected] sergey [email protected]

Alan Newell MS118 University of Arizona Department of Mathematics Local and Spatially Extended Frequency Locking: [email protected] Distinguishing Between Additive and Parametric Forcing MS119 The auditory system displays remarkable sensitivity and frequency discrimination, attributes shown to rely on an Time-delay Models of Multi-mode Laser Dynamics amplification process that involves a mechanical as well as a biochemical response. Models that display proximity to Delay differential equations (DDE) model of a passively an oscillatory onset (a.k.a. Hopf bifurcation) exhibit a res- mode-locked semiconductor laser was first proposed as an onant response to distinct frequencies of incoming sound, alternative to semiclassical Maxwell-Bloch equations and and can explain many features of the amplification phe- simplified complex Ginzburg-Landau-type (CGL) Haus nomenology. To understand the dynamics of this reso- master equation of mode-locking as well as single-mode nance, frequency locking is examined in a system near the ODE models. While these DDE models retain all the most Hopf bifurcation and subject to two types of driving forces: important dynamical characteristics of multi-mode semi- additive and parametric. Derivation of a universal ampli- conductor lasers that can be observed directly in Maxwell- tude equation that contains both forcing terms enables a Bloch equations, they can be also studied using the pertur- study of their relative impact on the hair cell response. In bation analysis tools similarly to ODEs or CGL-type equa- the parametric case, although the resonant solutions are tions and using asymptotic and numerical bifurcation anal- 1:1 frequency locked, they show the coexistence of solu- ysis methods. In particular, they proved to be useful for tions obeying a phase shift of π, a feature typical of the qualitative explanation and prediction of the experimen- 2:1 resonance. Different characteristics are predicted for tally observed phenomena for passively mode-locked lasers the transition from unlocked to locked solutions, leading to under the effects of external periodic forcing and noise. In smooth or abrupt dynamics in response to different types the last few years, multi-mode DDE models found applica- of forcing. The theoretical framework provides a more re- tion in spatially extended systems such as broad-area lasers alistic model of the auditory system, which incorporates a under off-axis feedback, frequency-swept sources (Fourier direct modulation of the internal control parameter by an domain mode-locked lasers) and lasers under long optical applied drive. The above seems to have fundamental im- feedback used for generation of temporally localized struc- plications to spatially localized resonances that known to tures. Cavity of a FDML laser contains very long optical arise in the cochlea. fiber that causes strong chromatic dispersion. The effect of chromatic dispersion can be described using a distributed delay term in the laser model, which can lead to the ap- Arik Yochelis pearance of modulational instability and temporal dissi- Jacob Blaustein Institutes for Desert Research, pative solitons. We discuss our recent results and feature Ben-Gurion University challenges. [email protected] Alexander Pimenov Yuval Edri Weierstrass Institute Ben-Gurion University of the Negev [email protected] [email protected] Andrei G. Vladimirov Dolores Bozovic Weierstrass Institute for Applied Analysis and Stochastics UCLA Berlin, Germany [email protected] [email protected]

MS119 MS119 Instability of Steep Ocean Waves and Whitecap- Estimation of Timing Jitter in a Delayed Differen- DS17 Abstracts 245

tial Model of Semiconductor Laser extract useful axes from measurements or simulation data of different non-linear phenomena. Useful hereby refers to We consider a delay differential model of a passively mode the degree of coarsening in which we want to observe our locked semiconductor laser proposed by A. Vladimirov and system, with this degree being specified by appropriately D. Turaev. The system exhibits a stable periodic solution, tuning the kernel scale in the diffusion maps approach. In which models a periodic train of light pulses in the mode addition, we demonstrate that these axes are, provided we locked regime. Timing jitter is the variation of the time have sufficient data, in- dependent of the particular nature interval between the pulses induced by noise. Assuming of the measurement entity. As illus- trative examples, we small additive noise, we derive an asymptotic expression apply our method on spatio-temporal chaotic dy- namics for the diffusion coefficient of the phase diffusion, which apparent in the complex Ginzburg-Landau equation, on quantifies the jitter. Asymptotic formulas are compared to mod- ulated traveling waves in the Kuramoto-Sivashinksy simulation results. equation and on chimera states, states of coexisting coher- ence and incoherence. For the latter, we show that it is Dmitry Rachinskiy possible to extract insightful order parame- ters, allowing Department of Mathematical Sciences further understanding of these intricate dynamics. In ad- University of Texas at Dallas dition, this embedding can be done invariant to the mea- [email protected] surement function, showing similarities to the Koopman operator.

MS119 Ioannis Kevrekidis Generation of Broadband Chaotic Light and Its Dept. of Chemical Engineering Applications in Detection and Communication Princeton University [email protected] Chaotic light has attracted widespread attention in recent years because of its important applications, for instance, in Felix Kemeth encrypted communications, physical random number gen- Technische Universit¨at M¨unchen eration, lidar, and time domain reflectometry. And a semi- [email protected] conductor laser with optical feedback is usually the com- mon choice to generate chaotic light owing to its simple and integratable setup. However, the chaos bandwidth is MS120 limited by the laser relaxation oscillation to a few GHz. In Optimal Parameter Selection for Extended Dy- order to take full advantage of bandwidth, we proposed a namic Mode Decomposition series of methods to generate broadband chaotic light in- cluding optical injection into a laser diode with feedback, Dynamic mode decomposition (DMD) is one of mode de- dual-wavelength injection into a FP-LD with feedback, in- composition methods using the theory of Koopman oper- coherent delayed self-interference of a chaotic laser, fiber ator, which considers latent dynamics of time series data. ring resonator seeded by a chaotic laser, and optical het- Recently, in order to more accurately approximate nonlin- erodyning of chaotic lasers. Using above mentioned meth- ear latent dynamics, an extended DMD has been proposed ods, chaotic light with bandwidth ranging from 14 GHz to [M. Williams et al., 2015]. In this study, we propose a 32 GHz can be readily achieved. Having achieved broad- method for optimal parameter selection in the extended band chaotic light, we applied it to free space ranging and DMD, and show that we can improve the estimation accu- fiber fault location of communication networks. In both racy of Koopman modes. applications, the high spatial resolution of a few centime- ters can be achieved since the wide bandwidth of chaotic Wataru Kurebayashi, Sho Shirasaka light. Furthermore, we expanded the broadband chaotic Tokyo Institute of Technology light to chaos-based communication and secure key distri- [email protected], [email protected] bution. Results indicate that broadband chaotic light en- ables Gbit/s data transmission satisfying the requirement Hiroya Nakao of current high-speed optical communication. Graduate School of Information Science and Engineering, Tokyo Institute of Technology Longsheng Wang, Anbang Wang, Tong Zhao, Yuanyuan [email protected] Guo, Daming Wang Taiyuan University of Technology Institute of Optoelectronic Engineering MS120 [email protected], [email protected], Comparison of Dynamic Mode Decomposition, [email protected], [email protected], Koopman Mode Decomposition, and Vector Prony [email protected] Analysis

Yuncai Wang Dynamic Mode Decomposition (DMD) has since its advent Taiyuan University been applied to experimental data within various research Institute of Optoelectronic Engineering fields such as fluid mechanics, power systems, and thermal [email protected] dynamics of buildings, to name a few. Its connection to spectral analysis of the Koopman operator was pointed out early, and some versions of DMD have been called Koop- MS120 man Mode Decomposition (KMD), an acronym used from An Equal Space, An Equal Time: Data-Driven Em- now on to refer to both. The performance of the algo- beddings of Complex Dynamics rithms is dependent on the spatial dimension, i.e. num- ber of measurement locations, and the number of temporal We present a way to embed nonlinear phenomena based on data snapshots. In this presentation we discuss three KMD their intrinsic variability. In particular we apply diffusion algorithms, and numerically assess their performance for maps, a non- linear dimensionality reduction method, to application to two types of experimental data with varying 246 DS17 Abstracts

temporal length (number of snapshots) and spatial dimen- functions, and also by utilizing multifrequency modes. This sion. In particular, it is shown that the newly proposed, work uses an unsteady boundary-element method to nu- so-called vector Prony analysis, performs well for data with merically explore periodic—but not purely sinusoidal— low spatial dimension, in contrast to standard DMD. gaits of hydrofoils of practical engineering interest and swimming in a perfect fluid. We numerically confirm the Fredrik Raak qualitative features of the experimental studies, and we Kyoto University also identify optimal gaits and wake structures in several [email protected] cases. It is shown that the best kinematic profile may be asymmetric. Finally, we remark on the relation of this work Yoshihiko Susuki to the classical geometric-phase picture of locomotion in a Kyoto University, JST principal bundle, and we allude to some of the challenges in [email protected] practically extending this geometric view to systems which discretely shed vorticity. Igor Mezic University of California, Santa Barbara Michael J. Fairchild [email protected] Department of Mechanical and Aerospace Engineering Princeton University [email protected] Takashi Hikihara Department of Electrical Engineering Kyoto University Clarence Rowley [email protected] Princeton University Department of Mechanical and Aerospace Engineering [email protected] MS120 Extended Dynamic Mode Decomposition for Sys- MS121 tems with Complex Dynamics The Geometry of Self-Propulsion in (and on) Fric- tional Fluids Extended Dynamic Mode Decomposition (EDMD) is a method for approximating the Koopman operator directly Certain animals and robots contend with flowable terres- from data sampled from a dynamical system. It has proven trial environments like sand. We have developed a granular successful for modeling some nonlinear systems with simple resistive force theory (RFT) that accurately describes lo- dynamics (e.g., fixed points and limit cycles), but has not comotion in dry granular media (Zhang & Goldman, PoF, been validated for use on complicated systems like turbu- 2014) in the frictional fluid regime; however, this theory lent fluid flows. In this work, we study the performance of gives limited insight into the character of locomotion. Geo- EDMD on model problems with more complex dynamics metric mechanics (e.g. work of Shapere, Wilczek, Marsden, (e.g., ergodic, mixing, chaotic). For ergodic systems with Kelly, etc) is a powerful framework relating position and pure point spectrum, EDMD successfully finds the eigen- orientation changes of a body to internal shape changes. values and eigenfunctions, and for mixing systems with no Advances by Hatton and Choset (Eur. Phys. J. 2015) now eigenfunctions, EDMD correctly predicts that there are no enable application to large self-deformations; in collabora- eigenvalues other than 1, but does not identify the continu- tion we demonstrated the power of the combination of ge- ous spectrum. For systems with both a structured compo- ometric mechanics and RFT through study of a three-link nent (e.g., point spectrum) and a random component (e.g., robot (two active, controllable DoF) swimming in granular continuous spectrum), the method correctly separates the media (Hatton et al, PRL, 2013). I will present our con- structured and random components. tinued collaborative work extending these tools to describe the locomotion of animals and robots with more than two Clarence Rowley DoF. I will show that the cyclic self-deformations of sand- Princeton University fish lizards, shovel nosed snakes, sidewinder rattlesnakes, Department of Mechanical and Aerospace Engineering salamanders and mudskipper fish (and corresponding robo- [email protected] physical models) that make continuous or intermittent con- tacts with granular substrates can be represented as a lin- Vivian Steyert ear combination of two shape basis functions — the shape Princeton University configuration space can be described with two variables. [email protected] With these approximations we gain insight into optimal self-deformation of high-DoF locomotors in granular me- dia. MS121 Efficient Locomotive Gaits for Swimming in Perfect Daniel Goldman Fluids Georgia Tech School of Physics The literature on oscillatory fishlike swimming focuses [email protected] overwhelmingly on sinusoidal pitching or heaving gaits, or a combination thereof. Perhaps this is because the for- ward swimming of many marine animals, such as thunni- MS121 form swimmers, is most often attributable to purely sinu- 3D Locomotion and Efficiency of a Yaw-Pitch soidal kinematics. However, engineered systems need not Three-Link Robot in a Low Reynolds-Number mimic biological organisms, and it is natural to ask if non- Fluid sinusoidal gaits may perform better in some respects, for example as measured by thrust or propulsive efficiency. In- The efficiency and maneuverability of mircoorganisms pro- deed, recent experiments have achieved performance gains vide an enticing model for the design of biomimetic robots. through the use of kinematics encoded by Jacobi-elliptic These observations serve as guides for the mechanical de- DS17 Abstracts 247

sign and control of microswimmers, wherein changes in the Mechanical Engineering and Engineering Science internally-controlled degrees of freedom result in changes University of North Carolina at Charlotte in net position and orientation of different robot designs. scott@kellyfish.net This work develops a novel geometric model of a non-planar three-link swimmer that executes three-dimensional ma- neuvers in a low Reynolds-number fluid. Whereas prior MS122 work on geometric swimming has considered mechanical Synchronization in Experiments with Coupled Me- designs that allow only planar motion, the model consid- chanical Oscillators ered in this work allows for out-of-plane motions using yaw and pitch control of two internal joints. We leverage tools Abstract Not Available At Time Of Publication. from geometric mechanics to derive a mapping from the underlying shape manifold to the swimmer’s local veloci- Karen Blaha ties and analyze it’s locomotion using this mapping. We University of New Mexico present initial results that synthesize motion primitives for [email protected] the swimmer, which make it move along canonical direc- tions relative to an inertial frame. Concatenating motion primitives can serve as a convenient approach for robot MS122 motion planning because primitives are precomputed of- Chimeras, Cluster States, and Symmetries: Exper- fline and do not require online re-computation or solving a iments on the Smallest Chimera trajectory optimization problem, every time a new target position is specified. Abstract Not Available At Time Of Publication.

Jaskaran S. Grover,TonyDear Joseph Hart Robotics Institute University of Maryland Carnegie Mellon University [email protected] [email protected], [email protected]

Matthew Travers MS122 Carnegie Mellon University Algorithms and Experiments for the Approximate [email protected] Balanced Coloring Problem

Howie Choset Abstract Not Available At Time Of Publication. Robotics Institute David Philips Carnegie Mellon University Operation Research [email protected] United States Naval Academy, Annapolis, MD, USA [email protected] Scott D. Kelly Mechanical Engineering and Engineering Science University of North Carolina at Charlotte MS122 scott@kellyfish.net Approximate Cluster Synchronization in Networks with Symmetries

MS121 Abstract Not Available At Time Of Publication. Entrainment and Cooperative Locomotion in Non- holonomic Systems and Swimming Systems Francesco Sorrentino University of New Mexico The dynamics of a solid body moving through a fluid can Department of Mechanical Engineering be altered substantially by the presence of another body [email protected] nearby. Under some circumstances, self-propelling bod- ies in proximity to one another can realize collective gains in efficiency as a result of their interactions. This phe- MS123 nomenon is well documented in the context of birds flying Interpreting Huybers’ Glacial Cycles Model As a in formation to reduce drag resulting from vortex shedding, Nonsmooth Dynamical System and more complex wake-body interactions play a role in determining the propulsive efficiency of schools of fishlike One of the most interesting open questions in Paleoclimate swimmers, but even irrotational flows provide the means science is the mid-Pleistocene Transition (MPT) problem, for aggregated bodies to influence one another through which surrounds the unknown cause that drove the change added-mass effects. Wheeled robots can influence one an- in the dominant period of deglaciation from 41 kyr to 100 other in similar ways when they operate atop a common kyr. Huybers’ glacial cycles model was first suggested by platform that exhibits sufficient compliance to transmit Peter Huybers’ and Carl Wunsch in 2005 to explain the vibrational energy from one robot to another, leading to glacial variability of the earth’s climate, in hopes of shed- variations of the kind of synchronization observed among ding light on what prompted the sudden shift in the peri- pendulum clocks sharing such a platform. This talk will odicity of the mid-Pleistocene. In this talk, I will present describe nonlinear phenomena arising in the coupled dy- how to construct a nonsmooth dynamical system (in par- namics of wheeled robots that exploit inertial effects to ticular a Filippov system) from Huybers’ model by creating propel themselves and will develop the analogy between a manifold from the quotient space inspired by the thresh- the coupled locomotion of such robots and the coupled lo- old of the model. Then I will discuss a way of reducing comotion of bodies deforming in fluids. the system to a circle relation and possible implications of this approach. Through imposing well-studied structures Scott D. Kelly on Huybers’ model, we hope to gain further insight into 248 DS17 Abstracts

the mid-Pleistocene Transition problem. cal for high-dimensional systems. The semiparametric ap- proach loosens the structure of a parametric model by fit- Somyi Baek ting a data-driven nonparametric model for the parame- University of Minnesota ters. Given a parametric dynamical model and a noisy data [email protected] set of historical observations, an adaptive Kalman filter is used to extract a time-series of the parameter values. A nonparametric forecasting model for the parameters is built MS123 by projecting the discrete shift map onto a data-driven ba- Conceptual Climate Models with Global Feedback sis of smooth functions. Ensemble forecasting algorithms Mechanisms extend naturally to the semiparametric model which can ef- fectively compensate for model error, with forecasting skill In contrast to highly complex general circulation models of approaching that of the perfect model. Semiparametric the Earths climate, conceptual climate models concern the forecasting is a generalization of statistical semiparamet- behaviour of only a few variables. The goal is to obtain ric models to time-dependent distributions evolving under a model that is amenable to mathematical analysis, while dynamical systems. still giving an adequate description of the phenomenon un- der consideration. The modeling of feedback mechanisms Tyrus Berry, John Harlim on a global scale is of particular relevance in this context. Pennsylvania State University This talk will give a brief overview of different types of con- [email protected], [email protected] ceptual climate models and the dynamical systems tools available for their analysis. MS124 Bernd Krauskopf Uncertainty Quantification for Generalized University of Auckland Langevin Dynamics Department of Mathematics [email protected] We present efficient finite difference estimators for goal- oriented sensitivity indices with applications to the gener- alized Langevin equation (GLE). In particular, we apply MS123 these estimators to analyze an extended variable formula- Understanding the Variability of the Indian Mon- tion of the GLE where other well known sensitivity analy- soons - Combining Data with Model sis techniques such as the likelihood ratio method are not applicable to key parameters of interest. These easily im- The Indian monsoons show significant year-to-year vari- plemented estimators are formed by coupling the nominal ability, as well as variability within a given rainfall season. and perturbed dynamics appearing in the finite difference It is unknown how much of this variability is driven by through a common driving noise, or common random path. climatic conditions external to the Indian subcontinent, or internal feedback mechanisms. While many studies have shown correlations between global climate states and the Eric J. Hall Indian summer monsoon rainfall, the dynamical connec- University of Massachusetts tions are not well understood. In our study, we develop a [email protected] highly simplified box model of the monsoon system, driven by real boundary conditions derived from satellite data. Markos A. Katsoulakis Daily values of near-surface temperature, winds, and hu- University of Massachusetts, Amherst midity measurements are used to compute heat and mois- Dept of Mathematics and Statistics ture fluxes into the box situated over India and Bangladesh. [email protected] Precipitation is then computed from vertical air flux, evap- oration of groundwater, and parameterized cloud forma- tions. Processes within the box are modeled using stan- Luc Rey-Bellet dard physics currently employed in state-of-the-art climate Department of Mathematics & Statistics models. The dynamical variables are heat, moisture, and University of Massachusetts groundwater content. We hope that this approach will give [email protected] us some insight about the role and importance of internal feedbacks and external climate states on the monsoon rain- fall. MS124 Accounting for Model Errors from Unresolved Raj Saha Scales by Stochastic Parametrization in Ensemble University of North Carolina Kalman Filters [email protected] We investigate a discrete-time stochastic parametrization method to account for model errors due to unresolved MS124 scales in the context of the ensemble Kalman filter with Data-Driven Correction of Model Error for Fore- perturbed observations. The parametrization quantifies casting the model errors and produces an improved non-Markovian forecast model, which generates high quality forecast en- Semiparametric forecasting is introduced as a method of sembles and improves filter performance. We compare this addressing model errors arising from unresolved physi- method with the standard method of dealing with model cal phenomena. While traditional parametric models are errors through covariance inflation and localization, and il- able to learn high-dimensional systems from small data lustrate the comparison with numerical simulations on the sets, their rigid parametric structure makes them vulner- two-layer Lorenz 96 system. Results show that when the able to model error. On the other hand, nonparamet- ensemble size is sufficient, the parametrization is more ef- ric models have a very flexible structure, but they suf- fective in accounting for such model errors than inflation fer from the curse-of-dimensionality and are not practi- and localization; if the ensemble size is small, inflation and DS17 Abstracts 249

localization are needed, but the parametrization provides [email protected] a further improvement in performance. The general impli- cations of the results is discussed. MS125 Fei Lu Yeast Cultures Have Large Coupling Strengths: Department of Mathematics, UC Berkeley Random Perturbations on Cell-cell Coupling Dy- Mathematics group, Lawrence Berkeley National namics Laboratory fl[email protected] We study the effects of random perturbations on collec- tive dynamics of a large ensemble of interacting cells in a Alexandre Chorin model of the cell division cycle. We consider a parameter Department of Mathematics region for which the unperturbed model possesses asymp- University of California at Berkeley totically stable two-cluster periodic solutions. Two biolog- [email protected] ically motivated sources of noise are introduced into the cell cycle system and we compare them with the model that has Gaussian white noise perturbations. In simula- Xuemin Tu tions, we explore how an ordered two-cluster periodic state University of Kansas of cells becomes disordered by the increase of noise and a [email protected] uniform distribution emerges for large noise. As a corol- lary to the results, we can estimate the coupling strength of the cell cycle in yeast autonomous oscillation experiments MS124 where clustering is observed. This work was done with Estimating Parameters with Linear Response Alexander Neiman and Todd Young at Ohio University. Statistics Xue Gong I will present a new parameter estimation method for Itˆo Department of Mathematics and Computer Science diffusions such that the resulting model predicts the equi- Augustana College librium statistics as well as the sensitivities of the underly- [email protected] ing system to external disturbances. Our formulation does not require the knowledge of the underlying system, how- Gregory Moses ever we assume that the linear response statistics can be Ohio University computed via the fluctuation-dissipation theory. In this [email protected] talk, I will demonstrate the consistency of our method on a Langevin dynamics. Alexander Neiman Ohio University John Harlim Department of Physics and Astronomy Pennsylvania State University [email protected] [email protected] Todd Young Xiantao Li Ohio University Department of Mathematics Department of Mathematics Pennsylvania State University [email protected] [email protected]

He Zhang MS125 Department of Mathematics Hope Bifurcation in Random Dynamical Systems The Pennsylvania State University [email protected] Despite the obvious relevance to applications, a bifurcation theory for random dynamical systems is still only in it’s in- fancy. We discuss some progress on the understanding of MS125 the dynamics of a random dynamical systems near a (deter- ministic) Hopf bifurcation that illustrate recent advances Stochastic Dynamics that are Central to Biological and remaining challenges. This presentation is based on Mechanisms: Sleep-wake Transitions joint work with Maximilian Engel (Imperial College Lon- don), Martin Rasmussen (Imperial College London) and The study of sleep-wake transitions is an excellent arena Doan Thai Son (Hokaido University). for the methods of stochastic analysis. Sleep and the tran- sitions between sleep and wake are variable within each Jeroen Lamb individual and the frequency of transitions and the length Department of Mathematics of bouts change across the lifespan. In addition, there is Imperial College London enormous variability between individuals. Sleep-wake tran- [email protected] sitions and the lengths of bouts are easy to measure, how- ever, they depend on underlying neural stochastic mecha- nisms that are not well understood. In this talk, we will MS125 describe various approaches for analytical, statistical and Transitions in a Genetic Transcriptional Regula- numerical calculations about the nature of sleep-wake tran- tory System Under L´evy Motion sitions and the underlying biological mechanisms. Based on the stochastic differential equation model of a sin- Janet Best gle genetic regulatory system, we examine the dynamical The Ohio State University effects of noisy fluctuations from the synthesis reaction, on Department of Mathematics the evolution of the transcription factor activator in terms 250 DS17 Abstracts

of its concentration. The fluctuations can be modeled by will be discussed. Brownian motion and a pure jump L´evy motion, respec- tively. Two deterministic quantities, the mean first exit Makoto Iima, Takayuki Yamaguchi time (MFET) and the first escape probability (FEP), are Hiroshima University used to investigate the transition from the low to high con- [email protected], [email protected] centration states.

Yayun Zheng MS126 School of Mathematics and Statistics, Dynamics of Localized Structures in Dissipative Huazhong University of Science and Technology Systems [email protected] We highlight several key features of structures and dynam- ics of spatially localized patterns arising in dissipative sys- MS126 tems. The model systems include chemical reactions, fluid mechanics, nonlinear optics, gas discharge, plant ecology The Dynamics of Interacting Pulses in an Extended and adaptive behaviors of amoeba. These patterns are Klausmeier Model much simpler than single living cell, however they seem to inherit several characteristic living-state features, De- The extended Klausmeier, or Klausmeier-Gray-Scott, spite the diversity, there are common mathematical struc- equation models the ecosystem dynamics of vegetation tures and driving mechanisms leading to rich structures patterns in semi-arid regions. The equation is especially and their associated dynamics including snaking bifurca- used to study the desertification process, i.e. the process tion, self-replication, self-excitation, and collision dynam- through which a homogeneously vegetated terrain eventu- ics. The interplay between intrinsic and extrinsic instabil- ally may degenerate into bare soil (under changing clima- ities is a key to understand those dynamics and one of the tological circumstances). In the model, decreasing yearly powerful approach is to clarify the global geometric inter- rainfall first initiates the formation of (Turing) patterns relation among all relevant solution branches with approxi- that evolve into localized vegetation patches surrounded by mate unfolding parameters. We present several illustrative bare soil – patterns that are observed throughout the globe. examples to show the effectiveness of such geometric ap- The next stage of the desertification process is governed by proach as well as classical singular perturbation methods. the semi-strong interaction of these (far-from-equilibrium) vegetation patches – that have the character of homo- clinic pulses in one spatial dimension. In this terminol- Yasumasa Nishiura ogy – and the the 1D setting – desertification is caused WPI-AIMR, Tohoku University by the interaction and mutual annihilation of pulses (and Japan the (re-)emergence of pulses if the external circumstances [email protected] improve). A central topic of this talk will be the question whether the desertification process has a catastrophic or a more regular nature. It will be argued that this is strongly MS126 related to the (ir)regularity of the multi-pulse pattern. Dynamics of Traveling Spots with Oscillatory Tails in Heterogeneous Media Arjen Doelman Mathematisch Instituut Interactions between traveling spots with oscillatory tails [email protected] and heterogeneities for a generalized three-component FitzHugh-Nagumo equations are investigated in two- Robbin Bastiaansen dimensional space. We in particular focus on the case that Mathematical Institute heterogeneity is given in the form of disk shape. Various Leiden University types of heterogeneity-induced ordered patterns (HIOPs) [email protected] appeared. The important aspect of these localized patterns is that both of them have oscillatory tails and they inter- act through their tails when a traveling spot approaches to MS126 an HIOP. As a result, the interaction manner of them is Interactions of Spatially Localized Structures in very different from that of non-oscillatory case. At the first Bioconvection of Photosensitive Microorganism step we address the global bifurcation structure of HIOPs because many different stable HIOPs coexist at the same Euglena gracilis is a unicellular flagellated photosynthetic parameter. At the second step collisions between traveling alga. The suspension of Euglena has behavioral responses spot and HIOP are investigated. Finally four-dimensional to light environment, and macroscopic localized bioconvec- reduced ODEs is derived and mathematical structure of tion patterns are generated when illuminated from below. the interactions are elucidated. One of the fundamental structures is bioconvection unit, a pair of convection cells sandwiching a high cell density Takeshi Watanabe region [”Localized Bioconvection Patterns and Their Ini- School of Engineering, The University of Tokyo tial State Dependency in Euglena gracilis Suspensions in [email protected] an Annular Container”, E. Shoji, et al. J. Phys. Soc. Jpn. 83, 043001 (2014)]. Experimental studies show var- Yasumasa Nishiura ious types of interaction in the localized convection cells; WPI-AIMR, Tohoku University a quasi-steady bound states and collisions are typical ex- Japan amples. After summarizing individual behavior, we show [email protected] numerical simulations of the interactions based on a hydro- dynamic model proposed by one of the authors, which is based on of the experimental measurements and dilute as- MS127 sumptions. Stability of equally-placed bioconvection units Coupling Single Neuronal Activity to Complex DS17 Abstracts 251

Brain Functions: Large-Scale, High-Resolution Ap- Unfortunately, the function of relevant neural circuits in proaches to Dissect and to Modify Neuronal Net- humans is uncharted at fine temporal scales, limiting the work Activities understanding of changes underlying disruption associated with diseases. In this study, we localize neural circuits In order to understand how behavioral and network level critically involved in human DM on a millisecond scale. functions emerge from the cooperation of neuronal pop- Ten subjects, implanted with depth electrodes for clinical ulations, a representative fraction of neurons must be si- purposes, performed a gambling task while we recorded lo- multaneously monitored at the single-cell level. We de- cal field potential activity from deep and peripheral brain veloped large-scale electrophysiological recording methods structures. The gambling task consisted of logic-driven tri- with precise manipulation approaches, which can give birth als and trials that were ambiguous - decisions likely were to on demand, closed-loop therapeutic interventions for based on internal biases. We first construct dynamical many drug resistant neuropsychiatric disorders. I will in- models of betting behavior for each subject that allows troduce our recent results we gained to understand the de- both logic and biases to integrate in DM. The models cap- velopment of hypersynchronous events in the hippocam- ture variability amongst subjects ranging from irrational pus. Our translational efforts employing closed-loop tran- to logical. Analysis of the neural data suggests key struc- scranial electrical stimulation on human patients to estab- tures (e.g. cuneus, amygdala, orbitofrontal cortex) encode lish an acute intervention approach that promptly disrupt different components of the DM system such as risk of re- pathologic network oscillations will be also highlighted in ward and provides new insight into how humans link their my talk. internal biases with logic to make decisions.

Antal Berenyi SrideviV.Sarma, Pierre Sacre University of Szeged Johns Hopkins University [email protected] [email protected], [email protected]

John Gale MS127 Emory University Control, Optimization and the Dynamics of Func- [email protected] tion in Neuronal Networks Jorge Martinez-Gonzalez This talk will explore how the control properties of neu- Cleveland Clinic ronal networks, mediated by their dynamics, may confer [email protected] functional advantages in terms of information processing capacity. Analytical frameworks for linking control prop- erties and function at two spatial scales will be highlighted. MS127 Examined first will be the issue of how the dynamics of Moving to the Network Level in Brain Activity early sensory networks determine their sensitivity to stim- Control: Implications for Cognition and Develop- ulus orientation and novelty, towards enabling improved ment decoding of stimulus identity. This analysis allows for pre- dictions regarding the neural responses elicited by different While the successful control of brain dynamics remains a stimulus types, under the premise that such responses are great challenge, new advancements in both neural technolo- optimal for control-theoretic decoding functions. Evidence gies and relevant theories make it an opportune moment to of the predicted responses in the olfactory system will be revisit this topic. We review key developments in the con- shown. The concept of optimality will then be further ex- trol of single or a few neurons, as well as in specific domains plored in the context of information processing objectives such as Parkinsons disease. These successes naturally bring at broader spatial scales. Analysis will be presented that to light limitations and key questions in the field, suggest- shows how brain regions interacting via local, optimal rules ing potential areas for growth and work. We discuss ef- may give rise to predictable, yet nontrivial attractor land- forts to study the brain from a systems level, which utilize scapes at the large-network level. Evidence of such at- developments in the field of connectomics and in network tractors, or microstates, will be presented from large scale control theory. Linear control theory links brain regions to recordings of human neural activity in resting awake and their canonically identified cognitive tasks, while connect- pharmacologically unconscious conditions. The alteration ing to individual differences in cognitive performance, as to the attractor structure, and thereby the network control well as brain development in the maturation of a child into properties, in the unconscious condition will be especially adulthood. highlighted. Evelyn Tang ShiNung Ching University of Pennsylvania Washington University in St. Louis [email protected] [email protected]

MS128 MS127 The Motion of Objects Floating on the Ocean Sur- Fragility in the Human Decision Making System: face When Irrationality Hijacks Logic We propose a new Maxey–Riley set for the motion of Decision-making (DM) links cognition to behavior and is spheres floating at the ocean–atmosphere interface. The a key driver of personality, fundamental for survival, and new set predicts long-term accumulation in the center of essential for our ability to learn and adapt. Humans make the subtropical gyres. Observed distributions of undrogued logical decisions (e.g. maximize expected reward), but this drifters and plastic debris support the prediction. rationality is influenced by internal biases such as emotions. Psychiatric patients who have dysfunctional cognitive and Francisco J. Beron-Vera emotional circuitry frequently make irrational decisions. RSMAS, University of Miami 252 DS17 Abstracts

[email protected] Technische Universit¨at M¨unchen, Germany [email protected] M. Josefina Olascoaga University of Miami Johannes Keller [email protected] TU Munich [email protected] Rick Lumpkin NOAA, AOML [email protected] MS128 Objective Eulerian Coherent Structures MS128 We present a variational theory of Objective Eulerian A Comparison of Lagrangian Methods for Coherent Coherent Structure (OECS) in two-dimensional non- Structure Detection autonomous dynamical systems (Serra, M. and Haller, G., Coherent Lagrangian (i.e., material) structures are ubiq- Chaos 26(5), 2016 ). OECSs uncover the instantaneous uitous in unsteady fluid flows, often observable indirectly skeleton of the overall dynamical system, acting as theoret- from tracer patterns they create, e.g., in the atmosphere ical centerpieces of short-time trajectory patterns. OECSs and the ocean. Despite these observations, a direct iden- are computable at any time instant without any assump- tification of these structures from the flow velocity field tion on time scales, and hence are promising tools for (without reliance on seeding passive tracers) has remained flow-control and real-time decision-making problems. We a challenge. Several heuristic and mathematical detection also show that OECSs are null-geodesics of appropriate methods have been developed over the years, each promis- Lorentzian metrics. Exploiting the geometry of geodesic ing to extract materially coherent domains of arbitrary un- flows,wederiveafullyautomatedprocedureforthecom- steady velocity fields over a finite time interval of interest. putation of OECSs (Serra, M. and Haller, G., Proc. of the Here we review a number of these methods and compare Royal Soc. A (2017) 473 20160807 ). As an illustration, their performance systematically on three benchmark ve- we apply our results to ocean velocity data. locity data sets. Based on this comparison, we discuss the strengths and weaknesses of each method, and recommend Mattia Serra minimal self-consistency requirements that Lagrangian co- ETH Zurich, Switzerland herence detection tools should satisfy. [email protected]

Alireza Hadjighasem George Haller McGill University ETH, Zurich, Switzerland [email protected] [email protected]

Mohammad Farazmand School of Physics MS129 Georgia Institute of Technology A Dynamical Systems Approach to the Pleistocene [email protected] Climate - Part II of II Daniel Blazevski In this presentation, we continue the analysis of the Pleis- Insight Data Science tocene climate model proposed by Maasch and Saltzman, [email protected] extending the results obtained for the liming case (q = ∞) discussed in the previous presentation to the case of finite Gary Froyland q.Forq 1, the Maasch-Saltzman model is a fastslow UNSW Australia system. We show that there exists a family of invariant [email protected] slow manifolds, so the long-term dynamics are essentially two-dimensional. For the general case q>1, we show that George Haller there exists a critical value qc such that the long-term dy- ETH, Zurich, Switzerland namics evolve on a two-dimensional center manifold, while [email protected] for 1

Hans Engler MS128 Georgetown University Mathematical Relations Between Geometric and [email protected] Probabilistic Coherent Structure Detection Meth- ods Hans G. Kaper Georgetown University In this talk, we first review three prominent objective de- Argonne National Laboratory tection methods for coherent Lagrangian vortices. These [email protected] are—in chronological order—the probabilistic transfer op- erator approach developed by Froyland and co-workers, the geodesic approach developed by Haller and co-workers, Tasso J. Kaper and recent Laplace operator-based approaches developed Boston University by Froyland as well as by the authors. We present either Department of Mathematics theorems or strong numerical evidence for close mathemat- [email protected] ical connections between these methods. Theodore Vo Daniel Karrasch Boston University DS17 Abstracts 253

[email protected] [email protected]

MS129 MS129 A Dynamical Systems Approach to the Pleistocene Interconnected Climate Variability in the Pacific Climate - Part I of II and Indian Oceans

In 1990, K. Maasch and B. Saltzman introduced a three- The Pacific Ocean is home to one of the dominant large- dimensional dynamical system to explain the glacial cy- scale climate modes arising from a coupled instability of cles of the Pleistocene Epoch. The model consists of three the ocean/atmosphere system: the El Ni˜no-Southern Os- ODEs with four parameters, and emphasizes the role of cillation (ENSO). ENSO events occur irregular on interan- atmospheric CO2 in the generation and persistence of pe- nual timescales and are characterized by large amplitude riodic orbits (limit cycles). In this presentation, we review changes of Sea Surface Temperatures (SSTs) in the Central the physics underlying the Maasch-Saltzman model and and Eastern equatorial Pacific. These SST changes induce focus on a limiting case, obtained by letting one of the pa- large shifts of the atmospheric circulation, thereby causing rameters (q, a ratio of two characteristic times) tend to anomalous droughts and flooding events in many regions infinity. The resulting two-dimensional model has many of of the Earth. While linear ENSO theory has matured over the same dynamical features as the full Maasch-Saltzman recent decades, many aspects of ENSOs complexity are still model. In the special case of Z2 symmetry, a Bogdanov– puzzling. These include for instance ENSOs temporal evo- Takens (BT) point serves as an organizing center for the lution irregularity, its spatial complexity, and its distinct local and global dynamics. In the absence of symmetry, nonlinear climate impacts (such as precipitation), which the BT point splits into two BT points, with different local depend crucially both on the ENSO phase (El Ni˜no or La dynamics in their neighborhoods. Ni˜na) and the annual cycle phase. The Indian Ocean and the Atlantic exhibit similar modes of intrinsic climate vari- Hans Engler ability, albeit considerably smaller both in spatial scale and Georgetown University amplitude. One of these climate modes is the Indian Ocean [email protected] Dipole (IOD). Here we are presenting a simple conceptual model framework that captures the observed interactions Hans G. Kaper between ENSO and the IOD. Using this model we are able Georgetown University to explain a variety of spectral and cross-correlation char- Argonne National Laboratory acteristics in observable climate variables, with important [email protected] implications for the interpretation of seasonally modulated climate variability.

Tasso J. Kaper Malte Stuecker Boston University University of Hawaii at Manoa Department of Mathematics [email protected] [email protected]

Theodore Vo MS130 Boston University Chaotic Ollations of Smple Mchanical Sstems [email protected] We describe two simple experimental systems with Lorenz like chaotic dynamics: a MalkusLorenz water wheel and MS129 an electromechanical chaotic oscillator. The MalkusLorenz Palaeoclimate Dynamics Modelled with Delay Dif- water wheel dynamics are found to be in good, but not per- ferential Equations fect, agreement with predictions from the Lorenz model. We address the issue of multi-parameter estimation from The Quaternary period is characterised by oscillations be- scalar outputs of this chaotic system. The electromechani- tween glacial and interglacial states along with unexplained cal oscillator produces chaotic oscillations with a topologi- jumps in temperature on shorter timescales (Dansgaard- cal structure similar the Lorenz butterfly or Rsslers folded- Oeschgar events). External orbital forcing only explains band oscillator, depending on the configuration. An inter- some of the behaviour in the climate record, so it is nec- esting feature of this oscillator is that its model admits an essary to study the associated internal processes of the cli- exact analytic solution. mate variables involved. We study where delays can play a role in these internal processes. Delayed feedback mecha- Lucas Illing nisms are typically described by delay differential equations Reed College (DDEs), i.e. differential equations which depend on past [email protected] states of the system. These can be useful in parameter- ising small-scale processes into just an effective delay of the feedback. In climate models, delays can approximate MS130 the transport of mass or energy across large distances. We Coins Falling in Water explore possible delays in the climate system relative to the timescales of the Quaternary period. In particular we Objects falling freely under gravity through a fluid medium focus on delays related to deep ocean transport in the At- may follow various periodic or chaotic paths depending on lantic. A few different conceptual climate DDE models are their geometry, the ratio of the disc to fluid density, and studied, and their behaviour is compared to the climate the fluid viscosity. Understanding the origin and nature record. of these nonstraight paths is relevant to many branches of science and engineering. Examples range from seed dis- Courtney Quinn persal and gliding animals to unpowered robotic vehicles. University of Exeter In this talk, we present a series of experimental studies of 254 DS17 Abstracts

the falling modes of coins of various shapes and how these tribute to the better understanding of the role of LFY and falling modes influence the probability distribution of the AP1 in the flowering of Arabidopsis thaliana. coins landing sites. Maia Angelova Lionel Vincent Department of Mathematics and Information Sciences Aerospace and Mech. Engineering Northumbria University University of Southern California [email protected] [email protected] Emrah Haspolat Try Lam Deaprtment of Mathematics, Physics and Electrical NASA Jet Propulsion Laboratory Engineerin [email protected] Northumbria University [email protected] Eva Kanso University of Southern California Benoit Huard [email protected] Northumbria University Department of Mathematics, Physics & Electrical Engineering MS130 [email protected] Pattern Formation of Swarms of Artemia Francis- cana MS131 Swarming is a ubiquitous self-organization phenomenon Delays and Nonlinearities in Biological Clocks which occurs in many biological systems such as groups of the flocking of birds and insects, the schooling of fish, As an introduction to this mini-symposium on gene regula- and the collection of bacteria. This sort of behavior is tory networks we provide a brief history of models for cellu- an emergent phenomenon as most of this behavior aris- lar oscillators, beginning with the three-variable Goodwin ing without any sort of centralized control or leadership. oscillator with negative feedback. We then consider a very Swarming depends primarily on local interactions between general model for a cyclic sequence of J reactions, where individuals in the swarm. We will discuss the patterns that reacting species j is the source for the generation of species can be observed by the swarming of brine shrimp, specifi- j + 1, until the very last species J, which provides negative cally the Great Salt Lake strain of Artemia franciscana, at feedback to the first species j = 1. These species may rep- high concentration. These patterns can be easily observed resent, for example, the concentration of mRNA, which is with simple tabletop experiments; however, the cause of translated into a sequence of proteins, and a final product these patterns are unknown. We experimentally test the that acts to inhibit the transcription of mRNA. From our effects on certain physical parameters such as concentra- analysis we are able to generalize and expand on results ob- tion, temperature, salinity, luminosity and the relationship tained from a variety of simpler models. For example, we of extended-area to depth on the patterns that arise. We show that rather than the details of specific reaction steps, then develop a model for the shrimps behavior which will it is the net nonlinearity and the net delay of the system also yield the same sort of patterns. We hope to model the that determine the onset and properties of oscillatory so- basic length and times scales of the patterns, the patterns lutions due to a Hopf bifurcation. Or, a linear sequence of selected, the stability of those patterns, and the transitions reactions with non-uniform coefficients can be captured by that occur between patterns. an appropriately chosen distributed delay, hence, general- izing what is often referred to as the linear-chain trick. Andrea J. Welsh,KrishmaSingal,FlavioH.Fenton Georgia Institute of Technology Thomas W. Carr [email protected], [email protected], Southern Methodist University fl[email protected] Department of Mathematics [email protected]

MS131 Deterministic and Stochastic Stability of Arabidop- MS131 sis Flowering Model Characterization of Transcriptional Delays under Time-Varying Temperatures Experimental studies of the flowering of Arabidopsis Thaliana have shown that a large complex gene regulatory Delays play a critical role in genetic networks through their network is responsible for its regulation. In this work, we nonlinear effects on cell signaling. For example, it is often consider a deterministic and stochastic delayed non-linear delays that allow for oscillations in a genetic network given dynamic model of Arabidopsis flowering time based on the its narrow parameter range. One of the most prevalent deterministic model introduced in Valentim et al 2015. We environmental factors that affect timing and regulation of investigate the conditions for deterministic and stochastic gene expression is temperature. However, it is not known stability of the full model. By using a quasi-steady state how delays change with time-varying temperatures. We approximation, the system is reduced by focusing on a sub- analytically derive the time dependence of delay distribu- network composed of the transcription factor Suppressor of tions in response to time-varying temperature changes. We Overexpression of Constants 1 (SOC1) and two important find that the resulting time-varying delay is nonlinearly floral meristem identity genes, Leafy (LFY) and Apetala1 dependent on parameters of the time-varying temperature (AP1). We also consider three motifs from the reduced such as amplitude and frequency; therefore, applying an network, based on LFY and AP1, which lead to simpli- Arrhenius scaling may result in erroneous conclusions. We fied models of the GRN. The steady state regimes and the apply the results to a model of a synthetic gene oscillator effect of stochasticity are investigated analytically and nu- with dynamics described by a delay differential equation. merically for the full and reduced models. The results con- This model describes a genetic oscillator that maintains DS17 Abstracts 255

the same period of oscillation for a wide range of temper- been shown to adequately describe optical Kerr frequency atures. We show that entrainment of the synthetic gene combs. Frequency combs are widely studied as they can be oscillator follows from the same mechanism that results in used to measure light frequencies and time intervals with temperature compensation. great accuracy in a compact integrated platform [Science 332, 555 (2011), Phys. Rev. Lett. 107, 063901 (2011), Marcella M. Gomez Nat. Photon. 6, 480 (2012)]. As such, our work provides Department of Electrical Engineering and Computer new insights into the generation and stability of these fre- Science quency combs. University of California, Berkeley [email protected] Lendert Gelens KU Leuven, Belgium Richard Murray [email protected] Caltech Control and Dynamical Systems Pedro Parra-Rivas [email protected] Applied Physics Research Group Vrije Universiteit Brussel, Belgium Matthew Bennett [email protected] Rice University [email protected] Damia Gomila IFISC (CSIC-UIB) Instituto de Fisica Interdisciplinar y Sistemas Complejos MS131 damia@ifisc.uib.es WhichDelaysMatterinGeneExpression? Fran¸cois Leo Many gene expression models either include separate tran- OPERA-Photonique, Universit´e libre de Bruxelles (ULB) scription and translation delays, or simply a total (tran- 50 av. F.D. Roosevelt, CP194/5, B-1050 Bruxelles, scription + translation) expression delay for each protein. Belgium However, some models also include delays representing pro- [email protected] moter and ribosome binding site (RBS) clearance. Do these clearance delays matter? Intuitively, one would think that delays of a few seconds matter little when the total gene St´ephane Coen expression time is of the order of one minute (prokary- Department of Physics, University of Auckland, otes) to several minutes or longer (eukaryotes). On the New Zealand other hand, including a delay for promoter clearance, for [email protected] example, guarantees a minimum temporal spacing between transcription initiation events. Are there observable con- Edgar Knobloch sequences associated with promoter clearance, or with the University of California, Berkeley - analogous process of RBS clearance, in delay-differential Nonlinearity, Institute of Physics equation models of gene expression? Do these clearance [email protected] times affect the statistics of protein expression bursts in delay-stochastic models? And does it matter if we repre- sent these processes with a fixed delay, or is a distributed MS132 delay necessary? These questions will be explored in the Patterns vs. Localized Structures context of a simple model for the expression of a single, constitutively expressed gene. We analyze the formation of patterns and localized struc- Marc R. Roussel tures in the Lugiato-Lefever equation. For some parameter University of Lethbridge, Canada regions this model presents a subcritical Turing instability Department of Chemistry and Biochemistry leading to the formation of periodic patters and localized [email protected] structures displaying homoclinic snaking. In this regime localized solutions have clearly different properties than the patterns. For other parameter values, however, the MS132 system goes through a Quadruple Zero (QZ) point where Localized Structures in Nonlinear Optical Res- the critical wavenumber of the pattern forming instabil- onators ity goes to zero. In this case we show how patterns and localized states are no longer that different. In fact, the bi- We analyze the dynamics and stability of localized struc- furcation diagram of localized states changes dramatically tures in the mean-field Lugiato-Lefever equation describing going through that point, resembling more the typical bi- driven nonlinear optical resonators. We show how different furcation structure of periodic patterns. We will also dis- types of localized structures can form depending on system cuss the role of the interaction between localized structure parameters, such as pump power, cavity detuning, and the in this scenario. regime of chromatic dispersion (normal vs. anomalous). We discuss that these structures are organized in so-called Damia Gomila homoclinic or collapsed snaking diagrams, and that they IFISC (CSIC-UIB) can undergo a variety of dynamical instabilities, such as Instituto de Fisica Interdisciplinar y Sistemas Complejos oscillatory and chaotic dynamics [Phys. Rev. A 54, 5707 damia@ifisc.uib.es (1996), Opt. Expr. 21, 9180 (2013), Phys. Rev. A 89, 043813 (2014), Phys. Rev. A 89, 063814 (2014), Opt. Pedro Parra-Rivas Expr. 23, 7713 (2015), Phys. Rev. A 93, 063839 (2016), Applied Physics Research Group Opt. Lett. 41, 2402 (2016)]. These results are timely as Vrije Universiteit Brussel, Belgium localized light pulses in the Lugiato-Lefever equation have [email protected] 256 DS17 Abstracts

Lendert Gelens strates the importance of the presence of oscillations in KU Leuven, Belgium neuronal systems, and the role they play in information [email protected] processing.

Edgar Knobloch Helmut Schmidt University of California at Berkeley Centre de Recerca Matematica, Barcelona, Spain Dept of Physics [email protected] [email protected] Daniele Avitabile School of Mathematical Sciences MS132 University of Nottingham Forced Snaking [email protected]

We study spatial localization in the real subcritical Ernest Montbrio Ginzburg-Landau equation Universitat Pompeu Fabra Barcelona, Spain 2π 2 4 ut = m0u + m1 cos x u + uxx + d|u| u −|u| u [email protected]  Alex Roxin with spatially periodic forcing. When d>0andm1 =0 this equation exhibits bistability between the trivial state Centre de Recerca Matematica Bellaterra (Barcelona), Spain u = 0 and a homogeneous nontrivial state u = u0 with stationary localized structures which accumulate at the [email protected] 2 Maxwell point m0 = −3d /16. When spatial forcing is included its wavelength is imprinted on u0 creating condi- tions favorable to front pinning and hence spatial localiza- MS133 tion. We use numerical continuation to show that under A Bifurcation of the Kuramoto Model on Networks appropriate conditions such forcing generates a sequence of localized states organized within a snakes-and-ladders For the mean-field limit of a system of globally coupled structure centered on the Maxwell point, and refer to this phase oscillators defined on networks, a bifurcation from phenomenon as forced snaking. We determine the stability the incoherent state to the partially locked state at the properties of these states and show that longer lengthscale critical coupling strength is investigated based on the gen- forcing leads to stationary trains consisting of a finite num- eralized spectral theory. This reveals that a network topol- ber of strongly localized, weakly interacting pulses exhibit- ogy affects the dynamics through the eigenvalue problem ing foliated snaking. [Ponedel, B. and Knobloch, E., Eur. of a certain Fredholm integral operator which defines the Phys. J. Special Topics, 2016 (to be published).] structure of a network.

Benjamin C. Ponedel Hayato Chiba Department of Physics, UC Berkeley Institute of Mathematics for Industry [email protected] Kyushu University [email protected] Edgar Knobloch University of California at Berkeley Dept of Physics MS133 [email protected] Bumps in Small-World Networks

MS132 We consider a network of coupled excitatory and in- hibitory theta neurons which is capable of supporting sta- Spatially Localized Patterns in Networks of Spiking ble spatially-localised “bump” solutions. We randomly add Neurons long-range and simultaneously remove short-range connec- We study spatially localized solutions in a neural tions within the network to form a small-world network field model that describes the macroscopic dynamics of and investigate the effects of this rewiring on the existence quadratic integrate-and-fire neurons. These localized so- and stability of the bump solution. Two limits are consid- lutions are the physical representation of working memory ered in which continuum equations can be derived; bump and short-term memory. They bifurcate from a branch solutions are fixed points of these equations. We can thus of stationary homogeneous solutions, in the proximity of use standard numerical bifurcation analysis to determine a saddle-node bifurcation of the spatially clamped model. the stability of these bumps and to follow them as parame- The solution branch exhibits damped snaking for homoge- ters (such as rewiring probabilities) are varied. Under some neous (translationally invariant) synaptic connections. If rewiring schemes bumps are quite robust, whereas in other synaptic connections are heterogeneous (specifically, if they schemes they can become unstable via Hopf bifurcation or are sinusoidally modulated in space) then localized solu- even be destroyed in saddle-node bifurcations. tion branches exhibit a typical snakes-and-ladders struc- ture, previously reported by Avitabile and Schmidt for an Carlo R. Laing Amari-type neural field model. In addition, we study the Massey University impact of external oscillatory forcing on these localized so- Auckland lutions, which represents oscillations that are ubiquitous in [email protected] neuronal systems. The resulting spatiotemporal dynamics is organized by limit points and period doubling bifurca- tions, which can lead to the annihilation of localized pat- MS133 terns at different frequencies and amplitudes. This demon- The Mean Field Limit of the Kuramoto Model on DS17 Abstracts 257

Convergent Graph Sequences an exact mean-field description for instantaneous pulsatile interactions. I will show that the inclusion of space and Mathematical analysis of many interesting dynamical ef- a more realistic synapse model leads to a similar redused fects in systems of coupled oscillators relies on the mean model that has many of the features of a standard neural field equation formally derived in the limit, as the num- field model coupled to a further dynamical equation that ber of oscillators goes to infinity. Examples include the describes the evolution of network synchrony. Both Tur- onset of synchronization in the Kuramoto model with dis- ing instability analysis and numerical continuation software tributed intrinsic frequencies and emergence and bifurca- were used to explore the existence and stability of spatio- tions of chimera states. In this talk, we discuss the deriva- temporal patterns in the system. tion and rigorous justification of the mean field limit for the Kuramoto model on convergent families of certain de- Aine Byrne terministic and random graphs, including Erd˝os-R´enyi and University of Nottingham small-world graphs. University of Nottingham [email protected] Georgi S. Medvedev Drexel University Stephen Coombes [email protected] University of Nottingham [email protected] MS133 Daniele Avitabile Bifurcations Mediating Appearance of Chimera School of Mathematical Sciences States University of Nottingham Coherence-incoherence patterns, better known as ’chimera [email protected] states’, are complex dynamical regimes observed in spa- tially homogeneous systems of nonlocally coupled oscilla- MS134 tors. In many cases they can be adequately represented by relatively simple solutions, e.g. rotating or traveling waves, Weakly Coupled Oscillators in a Slowly Varying of a certain partial integro-differential equation. Bifurca- World tion analysis and continuation of these solutions is usually We extend the theory of weakly coupled oscillators to incor- a difficult problem, because of the critical continuous spec- porate slowly varying inputs and parameters. We employ trum of the corresponding linearized operator. In this talk, a combination of regular perturbation and an adiabatic I will discuss mathematical challenges concerned with the approximation to derive equations for the phase-difference consideration of chimera states and show their possible so- between a pair of oscillators. We apply this to the sim- lution in a few special cases motivated by recent research ple Hopf oscillator and then to a biophysical model. The in the field. latter represents the behavior of a neuron that is subject Oleh Omel’chenko to slow modulation of a muscarinic current such as would Weierstrass Institute occur during transient attention through cholinergic activa- [email protected] tion. Our method extends and simplifies the recent work of Kurebayashi 2013 to include coupling. We apply the method to an all-to-all network and show that there is a MS134 waxing and waning of synchrony of modulated neurons. Introduction, Exact Results for Globally-Coupled Youngmin Park Theta Neurons, and Extensions University of Pittsburgh I will provide an introduction to the Ott-Antonsen ansatz [email protected] method of obtaining the asymptotic collective behavior of globally-coupled oscillators and its application to networks Bard Ermentrout of theta neurons, and review a number of extensions of this University of Pittsburgh work. Department of Mathematics [email protected] Ernest Barreto George Mason University Krasnow Institute MS134 [email protected] Exact Low-Dimensional Mean-Field Dynamics of Neuronal Networks

MS134 An urgent challenge in neuroscience is to properly de- Next Generation Neural Field Modelling scribe the collective dynamical behavior of networks of spiking neurons. Recently, an exact correspondence has Neural field models are actively used to describe wave prop- been established between the macroscopic firing rate of agation and bump attractors at a tissue level in the brain. the network and its underlying microscopic state. Mont- Although motivated by biology, these models are entirely bri´o and co-workers [PRX 5, 2015] showed for a network phenomenological in nature. They are built on the assump- of quadratic integrate-and-fire (QIF) neurons that an ef- tion that the neural tissue is synchronous, and hence, can- fective coupling between firing rate and mean membrane not account for changes in the underlying synchrony pat- potential prescribes the networks evolution. This low- terns. It is customary to use spiking neural network models dimensional formulation, however, requires an extension of when examining within population synchronisation. Un- the Ott-Antonsen (OA) ansatz for parameter-dependent fortunately, these high dimensional models are notoriously phase oscillator networks. This is mandatory to describe hard to gain insight from. In this talk I will consider the θ- the macroscopic states of the QIF neurons. We provide a neuron model, which has recently been shown to admit to proof that the dynamics of such parameter-dependent sys- 258 DS17 Abstracts

tems reduces to the attracting OA manifold. By that the show little information flux. Taken together, these results OA ansatz becomes applicable to physiologically realistic provide new quantitative bounds on information transfer in oscillator networks. social networks, useful for better understanding the spread of ideas and influence in human populations. Bastian Pietras Vrije Universiteit Amsterdam James Bagrow Lancaster University Department of Mathematics & Statistics [email protected] University of Vermont [email protected]

MS135 Lewis Mitchell Extracting Information Flow Between Animals in University of Adelaide the Wild [email protected]

Social animals exhibit collective behavior wherein they ne- gotiate to reach an agreement, for example to coordinate MS135 motion and synchronize migration. Bats are unique among Inferring Influence and Leadership in Mobile Ani- such social animals in that they use active sensory echolo- mal Groups cation or “bio-sonar’, that is, they emit ultrasonic waves and sense echoes to detect and navigate surroundings. In Most models of collective moment (e.g., flocking, schooling) real bat swarms, understanding navigational leadership assume that individuals follow identical behavioral rules. roles is a challenge, and quantitative assessment of lead- However, in reality individuals almost certainly vary, for ership appears to be an entirely untouched area of study. example in rank, internal state or behavioral type. Recent Given the broadcast nature of bio-sonar, the active sensing advances in GPS and automated tracking technologies are of one individual can directly influence the motion of oth- yielding fine-scale trajectories for each individual within ers, making the directionality of leadership unclear. Here, mobile animals groups. Here we apply information theo- we seek to understand navigational leadership in bats from retic measures, specifically optimized causation entropy, to direct observation of bat swarms in flight. Small groups sets of trajectories of collectively moving animals to infer of bats were continuously tracked in a mountain cave in leadership from differential information flow, in a variety Shandong Province, China, from which 3D path points are of taxa, including storks, bison and baboons. Using the extracted and converted to 1D curvature time series. We inferred influence networks, we explore whether an indi- explore two recent tools from dynamical systems to ex- viduals position in the influence network is related to in- tract causal relationships between these time series, namely dividual traits, such as size, age, experience, dominance or transfer entropy and convergent cross mapping. These spatial position in the group. In contrast to previous re- tools identify the direction of information transfer between sults, which found democratic decision making in animal a pair of bats flying together, which allows us to answer the groups, we find evidence for leadership according to social question of whether individuals fly independently of each rank, by applying our less subjective measures to the same other or interact to plan flight paths. We find that the data. identified causal relationships between bats is sensitive to the selected mathematical tools and parameters. Andrew Berdahl Santa Fe Institute Nicole Abaid, Subhradeep Roy [email protected] Virginia Polytechnic Institute and State University [email protected], [email protected] Joshua Garland University of Colorado at Boulder [email protected] MS135 Information and Prediction Limits in Online Social Jie Sun Activity Mathematics Clarkson University Massive datasets of human activity are now available, rev- [email protected] olutionizing research on human dynamics and computa- tional social science. We study the online posts of thou- Erik Bollt sands of Twitter users and their followers. Information Clarkson University flows across the Twitter network by these posts, but it is [email protected] unclear how much information flows, how much influence do users have on one another, and if we can accurately mea- sure these effects. Treating each Twitter user’s text stream MS135 as a symbolic time series, the entropy rate measures how Network Flux based on Directed Links in Spatially much information about a future word choice is available in Embedded Climate Networks the past history. We apply theorems from data compres- sion to estimate a “correlated’ entropy that accounts for Abstract Not Available At Time Of Publication. both temporal ordering and long-range correlations in the data. The correlated entropy rate estimates the inherent Ugur Ozturk uncertainty about someone’s future word choice. Crucially, Potsdam Institute for Climate Impact Research this technique can also capture social information trans- Institute of Earth and Environmental Science fer between pairs of users (denoted egos and alters), via a [email protected] cross-entropy that estimates how much information about the ego’s future word choice is present in the alter’s past words. Some alters contain nearly as much information MS136 about the ego as the ego itself, but other ego-alter pairs Cell Based Modelling of Tissue Size Control in the DS17 Abstracts 259

Drosophila Embryonic Epidermis [email protected]

Embryogenesis is an extraordinarily robust process, ex- hibiting the ability to control tissue size and repair pattern- MS136 ing defects in the face of environmental and genetic per- Optimal Chemotherapy Scheduling Based on a Pair turbations. Understanding developmental control mecha- of Collaterally Sensitive Drugs nisms requires studying multiple concurrent processes that make up morphogenesis, including the spatial patterning Despite major strides in the treatment of cancer, the de- of cell fates and apoptosis, as well as cell intercalations. velopment of drug resistance remains a major hurdle. To Here, we develop a computational model that aims to un- address this issue, researchers have proposed sequential derstand aspects of the robust pattern repair mechanisms drug therapies with which the resistance developed by a of the Drosophila embryonic epidermal tissues. Size con- previous drug can be relieved by the next one, a concept trol in this system has previously been shown to rely on called collateral sensitivity. Inspired by such studies, we the regulation of apoptosis rather than proliferation; how- develop multiscale dynamical models and study the effect ever, to date little work has been carried out to understand of sequential chemotherapy utilising a pair of drugs (A- the role of cellular mechanics in this process. We employ type, B-type) switched in turn within the therapy sched- a vertex model of an embryonic segment to test hypothe- ule. An important assumption in our model is that a tu- ses about the emergence of this size control. Comparing mor is comprised by two types of cell groups, A-resistant the model to previously published data across wild type and B-resistant, each of which is resistant to the indicated and genetic perturbations, we show that passive mechan- drug and sensitive to the other. Based on a population ical forces suffice to explain the observed size control in a based ODE system, we determined that the optimal treat- segment of the epidermis. However, observed asymmetries ment strategy consists of two stages: (i) the initial stage in cell death frequencies across the segment are demon- in which a chosen better drug is utilised until a specific strated to require patterning of cellular properties in the time point, T, and afterward; (ii) a combination of the two model. Finally, we propose multiple experiments that al- drugs switched in turn with a definite ratio in duration, low to distinguish distinct forms of mechanical regulation k. Of note, we prove that the initial period, in which the in the model. first drug is administered, T, is shorter than the period in which remains effective, contrary to clinical intuition. Be- Jochen Kursawe yond our analytic results, we explore an individual based University of Oxford stochastic model and present the distribution of extinction [email protected] times for the classes of solutions found. Taken together, our results suggest opportunities to improve chemotherapy scheduling in clinical oncology. MS136 The Dynamics of Colonic Crypt Dysplasia Nara Yoon Case Western Reserve University Colorectal cancer is the third most common cancer in the [email protected] UK and USA, with over 170,000 new cases reported each year. These cancers are thought to originate in the crypts Robert Vander Velde that reside along the surface of the intestines. The crypts University of South Florida are small test-tube like depressions of epithelial cells, con- [email protected] taining both intestinal stem cells and their progeny, the proliferation of which cause a total renewal of the intesti- Andriy Marusyk nal lining approximately every 5 days. H Lee Moffitt Cancer Center and Research Institute This talk will focus on how nonlinear multiscale mathemat- andriy.marusyk@moffitt.org ical models allow us to better understand the dysplastic crypt dynamics that lead to the development of colorectal Jacob G. Scott cancers. The concept of multiscale modelling in this case Cleveland Clinic Lerner Research Institute relates to the inclusion of cell-level mechanical behaviours, [email protected] such as cell-cell adhesion and cell division, as well as sub- cellular processes, such as individual cell-cycle models and gene regulatory networks, to create a more complete model MS136 of the crypt system. In particular, we will show how the Multiscale Agent-Based Models Predict Emergent inclusion of physiologically realistic contact inhibition re- Dynamics of Heterogeneous Tumor Growth lated behaviours, including cell apoptosis, affects the col- lective cell dynamics, and ultimately how mutations at the Computational modeling is an essential tool for integrating microscopic level lead to different dysplastic macroscopic and understanding complex biological systems. With in- dynamics. creasingly high-resolution, high-throughput, and dynamic experimental data, we are able to develop better informed Daniel Ward models to interrogate the complex, heterogeneous, and University of Bristol multiscale nature of cellular microenvironments. We em- [email protected] ploy agent-based models (ABMs) as an intuitive, modu- lar, and flexible framework to study emergence in hetero- Lucia Marucci geneous cell populations. ABMs utilize dynamic cellular Telethon Institute of Genetics and Medicine, Naples, Italy agents that can change state and location over time. Cell [email protected] states are governed by intracellular agent rules, interactions with neighboring agents, and the local microenvironment. Martin Homer Using this platform, we investigate the impact of intra- University of Bristol cellular network dynamics on intercellular signaling in the Department of Engineering Mathematics context of cancer. Preliminary results demonstrate how 260 DS17 Abstracts

tumor growth and shape are sensitive to parameters gov- Department of Mathematics erning glucose uptake and consumption, suggesting that University of Kansas our model is able to recover the importance of metabolic [email protected] deregulation, a known hallmark of cancer. Our multi-scale, multi-class simulations help interrogate how the whole is Aaron Hoffman greater than the sum of its parts by predicting emergent Franklin W. Olin College of Engineering cell population dynamics from simple cellular rules. We be- aaron.hoff[email protected] lieve such insights are critical for identifying effective drug targets and designing personalized, dynamic therapeutic Leonardo Morelli strategies. University of Leiden Jessica Yu [email protected] Chemical & Biological Engineering Northwestern University MS137 [email protected] Phase Separation from Directional Quenching and Neda Bagheri Unbalanced Patterns in Cahn-Hilliard and Allen- Chemical and Biological Engineering Cahn Equations Northwestern University [email protected] In this talk I describe the effect of directional quenching on patterns formed in simple bistable systems such as the Allen-Cahn and the Cahn-Hilliard equation on the plane. MS137 Directional quenching is considered as an externally trig- gered change in system parameters, changing the system Oblique Stripes in a Triggered Swift-Hohenberg from monostable to bistable across an interface; numeri- Equation cally and experimentally, one can see patterns forming in Externally mediated, or triggered, spatial patterns have the bistable region, in particular as the trigger progresses become a topic of recent interest in many fields, where and increases the bistable region. I will discuss existence researchers wish to harness natural pattern forming pro- and non-existence results of single interfaces and striped cesses to form novel and functional materials. Such trigger- patterns. Joint work with Arnd Scheel. ing mechanisms can be modeled in the prototypical Swift- Hohenberg equation, posed on the plane, with a planar in- Rafael Monteiro da Silva homogeneity which travels horizontally across the domain, School of Mathematics converting the trivial homogeneous equilibrium into an un- University of Minnesota stable state. We prove the existence of pattern-forming [email protected] modulated traveling waves in this system which leave be- hind oblique, or slanted, stripes of long transverse wave- MS137 length. We employ a functional analytic approach which regularizes the singular limit of small transverse wavenum- A Dynamical Approach to Elliptic Equations on ber to obtain such oblique solutions as a continuous per- Bounded Domains turbation of one-dimensional trigger fronts, which leave be- hind stripes parallel to the interface of the inhomogeneity. We develop the method of spacial dynamics, and obtain a system of operator-valued ODEs equivalent to elliptic prob- Ryan Goh lems on bounded domains. Let Ω ⊂ Rn be a bounded do- Mathematics and Statistics main, and suppose u satisfies the equation −Δu+Vu= λu Boston University on Ω. When the domain is deformed through a one- [email protected] parameter family Ωt, the Cauchy data of u on ∂Ωt satisfies a Hamiltonian evolution equation. Moreover, this equation Arnd Scheel admits an exponential dichotomy with the unstable sub- University of Minnesota space at time t corresponding to the Cauchy data of weak [email protected] solutions to the PDE on Ωt as Ω is deformed smoothly to a point. (Joint work with Margaret Beck, Graham Cox, Chris Jones and Yuri Latushkin) MS137 Waves and Obstacles in Square Lattices Alim Sukhtayev Department of Mathematics We study dynamical systems posed on a square lattice, Indiana University Bloomington with a special focus on the behaviour of basic objects such [email protected] as travelling corners and travelling waves under (poten- tially large) perturbations of the wave and the underlying spatial lattice. Such travelling structures satisfy functional MS138 differential equations of mixed type, which can be seen as Coarse Graining under Prevalent Embeddings generalizations of delay equations in the sense that both delays and advances in the arguments are allowed. This talk considers high dimensional drift-diffusion sys- tems, whose essential dynamics is constricted to a smooth, Hermen Jan Hupkes low-dimensional, but unknown manifold. Using the well- University of Leiden known Whitney embedding theorem, I describe how the Mathematical Institute analysis of prevalent, i.e. almost arbitrary embeddings of [email protected] the dynamics into a low-dimensinal euclidean space is suf- ficient to not only find the manifold, but to construct re- Erik Van Vleck duced drift and diffusion terms to coarse-grain the original DS17 Abstracts 261

dynamics. when the microscopic dynamics are replaced with corre- sponding correlated data in the form of time series. We Andreas Bittracher present applications of the method to coarse-graining poly- Center for Mathematics mers, and to defining empirical potentials from quantum TU Munich mechanical data in materials modeling. [email protected] Petr Plechac University of Delaware MS138 Department of Mathematical Sciences Diffusion Forecast: A Nonparametric Probabilistic [email protected] Modeling of Dynamical Systems

I will discuss a nonparametric modeling approach for fore- MS139 casting stochastic dynamical systems on smooth manifolds Shannon Entropy Estimation Based on Chaotic At- embedded in Euclidean space. In the limit of large data, tractor Reconstruction for a Semiconductor Laser this approach converges to a Galerkin projection of the semigroup solution of the backward Kolmogorov equation State-space reconstruction of chaotic attractors is investi- of the underlying dynamics on a basis adapted to the invari- gated for deducing the randomness generated by an opti- ant measure. This approach, which we called the diffusion cally injected semiconductor laser. The attractors are re- forecast, allows one to evolve the probability distribution of constructed using the intensity time series from both sim- non-trivial dynamical systems with an equation-free mod- ulations and experiments. The time-dependent exponent eling. is adopted for estimating the divergence between nearby trajectories, thereby verifying the effective noise amplifi- John Harlim cation by chaotic dynamics. After discretization for ob- Pennsylvania State University taining the output bits, the Shannon entropies continually [email protected] generated by the laser are estimated. Here, the effect of postprocessing on the randomness of the output bits is in- MS138 vestigated by evaluating the Shannon entropies. The role of a delayed exclusive-or operation is investigated, where Complex Dynamics in Multiscale Systems the time-dependent exponents as well as the Shannon en- Understanding the evolution of complex multiscale systems tropies are affected. The results fundamentally confirm the is crucial from a fundamental but also the applications’ provision of non-deterministic randomness by the chaotic point of view. For instance, many engineering systems are lasers. Practical tests for randomness at a combined out- complex and multiscale and understanding their dynam- put bit rate of 200 Gbps are successfully conducted. ics has the potential to predict a specific system’s behav- Xiao-Zhou Li, Jun-Ping Zhuang, Song-Sui Li ior, engineer its design and build-in response to arrive at a City University of Hong Kong highly optimal and robust system. We combine elements [email protected], [email protected], from homogenization theory, nonlinear science, statistical [email protected] physics, critical phenomena and information theory to de- velop a number of novel and generic methodologies that en- able us to undertake the rigorous and systematic study of Sze-Chun Chan the emergence of complex behavior in multiscale systems. City University of Hon The methodologies are exemplified with paradigmatic pro- Department of Electronic Engineering totypes from different classes of complex systems such as [email protected] interface dynamics in disordered media, convectively un- stable open flows and stochastic multiscale phenomena in MS139 noisy spatially extended systems and diffusion in multi- scale potentials. [Joint work with Marc Pradas (Open), Progress in Random Number Generation with Markus Schmuck (Heriot Watt), Dmitri Tseluiko (Lough- Chaotic Lasers borough), Grigorios A. Pavliotis (Imperial) and Andrew Duncan (Sussex)] Fast physical random number generators based on chaotic lasers have been investigated intensively for a decade. The Serafim Kalliadasis generation rate of random number generators based on Department of Chemical Engineering chaotic laser dynamics has been progressing over Terabit Imperial College London per second (Tb/s). In addition, photonic integrated cir- [email protected] cuits (PICs) are promising devices to minimize the size of random number generators for practical implementation. We present the recent progress of the speed and the minia- MS138 turization of physical random number generators based on Information-Theoretic Methods for Coarse- chaotic lasers. We experimentally demonstrate ultra-fast Graining Stochastic Dynamics physical random number generation with the bandwidth- enhanced chaos obtained from three-cascaded semiconduc- We discuss information-theoretic metrics for obtaining op- tor lasers. We generate physical random numbers from the timized coarse-grained molecular models for both equilib- bandwidth-enhanced chaos by using a multi-bit extraction rium and non-equilibrium molecular dynamics. The pre- method. We achieve the maximum generation rate of 1.2 sented approach compares microscopic behavior of molecu- Tb/s for the physical random bit sequences whose random- lar systems to parametric or non-parametric coarse-grained ness is verified by the standard statistical tests of random- systems using the relative entropy between distributions ness. We also fabricate PICs with short external cavity on the path space. The use of goal-oriented information- lengths to perform the miniaturization of physical random theoretic inequalities allows us to bound the error for quan- number generators. We use PICs with different external tities of interest. The methods become entirely data-driven cavity lengths (1-10 mm) to generate random bit sequences. 262 DS17 Abstracts

These PICs are promising miniature entropy sources for the vices. integration of random number generators for applications in information security and numerical computations. Jun Zhang University of Science and Technology of China Atsushi Uchida, Kazusa Ugajin, Yuta Terashima, Keigo Hefei National Laboratory for Physical Sciences at Yoshiya Microscal Department of Information and Computer Sciences [email protected] Saitama University [email protected], [email protected] u.ac.jp, [email protected], MS140 [email protected] Delay Embeddings and the Spectra of Koopman Operators

MS139 The Koopman operator induced by a dynamical system Real-Time Fast Physical Random Number Gener- is inherently linear and provides an alternate method of ators studying many properties of the system, including attrac- tor reconstruction and forecasting. Koopman eigenfunc- Physical random number generators (RNGs) have great tions represent the non-mixing component of the dynam- applications in cryptography and secure communication. ics. They factor the dynamics, which can be chaotic, into Due to the high bandwidth and large amplitude fluctua- quasiperiodic rotation on tori. We will describe a method tion, chaotic laser is recently viewed as an ideal entropy in which these eigenfunctions can be obtained by apply- source to solve the problem of limited low rates confronted ing an integral operator, which behaves as a projection by conventional RNGs. In this report, we focus on the operator. The underlying idea is a data-driven method of real-time implementation of physical RNGs based on laser using an integral operator approximation of the Laplace- and electrical chaos. Specifically, we will present our works Beltrami operator. We will show that incorporating a large from two aspects: one is how to generate the ultra-broad number of delay coordinates in constructing the kernel of or white chaos using semiconductor laser diodes or circuits, the operator results in the creation of a projection into the and the other is the real-time extraction of ultrafast ran- discrete spectrum subspace, in the limit of infinitely many dom numbers involving electrical and all-optical methods. delay coordinates. Based on these researches, several types of RNG proto- types or modules are also be presented, whose maximum Suddhasattwa Das, Dimitrios Giannakis real-time rate can reach a level of 10 Gb/s. Courant Institute of Mathematical Sciences New York University Yuncai Wang [email protected], [email protected] Taiyuan University Institute of Optoelectronic Engineering [email protected] MS140 Nonparametric Reconstruction from Noisy Time Pu Li, Anbang Wang, Jianguo Zhang, Xianglian Liu, Series: The Kalman-Takens Method Yanqiang Guo, Luxiao Sang Taiyuan University of Technology Modern methods of data assimilation rely on a key assump- Institute of Optoelectronic Engineering tion: that a mathematical model of the physical system is [email protected], [email protected], ik- known. In absence of a model, outright implementation of [email protected], [email protected], guoyan- standard techniques such as ensemble Kalman filtering is [email protected], [email protected] impossible. In this talk, I will present a novel approach to filtering noisy time series when a model is not known. This nonparametric approach to filtering merges the key MS139 ideas behind Takens’ method of attractor reconstruction Quantum Random Number Generator: Speed and and the Kalman filter. The resulting Kalman-Takens fil- Security ter replaces the model with dynamics reconstructed from delay coordinates, while using the Kalman update formu- Quantum random number generator (QRNG) is an impor- lation to assimilate new observations. I will show that this tant approach to generate true random numbers that are model-free approach to filtering is able to handle significant unpredictable, irreproducible, and unbiased. Speed is one observational and dynamical noise in identifying underly- of the key QRNG parameters. In this talk, we will first in- ing dynamics. troduce the typical QRNG schemes such as beam splitters, photon arrival times, vacuum fluctuations, and laser phase Franz Hamilton fluctuations. Then we will present our recent progresses Center for Quantitative Sciences in Biomedicine on QRNG implementations. In QRNGs, the randomness North Carolina State University relies on the accurate characterization of its devices. How- [email protected] ever, device imperfections and inaccurate characterizations can result in wrong entropy estimation and bias in prac- tice, which highly affects the genuine randomness genera- MS140 tion. Very recently the concept of measurement-device- Topological Symmetries: Quasiperiodicity and Its independent (MDI) QRNG has been proposed, and we Application to Filtering and Classification Prob- have experimentally implemented a MDI-QRNG for the lems first time based on time-bin encoding to achieve certified quantum randomness even when the measurement devices Chirped sinusoids and interferometric phase plots are func- are uncharacterized and untrusted. Our implementation tions that are not periodic, but are the composition of a with an all-fiber setup provides an approach to construct smooth function and a periodic function. These functions a practical MDI-QRNG with trusted but error-prone de- factor into a pair of maps: from their domain to a circle, DS17 Abstracts 263

and from a circle to their codomain. One can easily imag- Vortices in Astrophysical Plasmas ine replacing the circle with other phase spaces to obtain a general quasiperiodic function. Under appropriate re- Previous works have demonstrated that the technique of strictions, each quasiperiodic function has a unique univer- hyperbolic Lagrangian coherent structures is useful for an- sal factorization. Quasiperiodic functions can therefore be alyzing the dynamics of velocity and magnetic fields in nu- classified based on their phase space and the phase function merical simulations and satellite observations of astrophys- mapping into it. This talk will present a two-stage topolog- ical plasmas such as solar photospheric turbulence. An ac- ical algorithm for recovering an estimate of a quasiperiodic curate detection of the boundary of coherent structures in function from a set of noisy measurements. By respecting velocity and magnetic fields of a plasma is fundamental for an estimate of the phase space and phase function, the al- the study of astrophysical turbulence, for example, the de- gorithm avoids creating distortion even when it uses a large termination of the front and rear boundary layers of mag- number of samples for the estimate of the function. netic flux ropes in the solar wind is essential for locating the preferential sites for the genesis of interplanetary inter- Michael Robinson mittent turbulence. In this work we apply the approach of Department of Mathematics and Statistics rotational Lagrangian coherent structures to detect kine- American University matic vortices in 3D magnetohydrodynamic simulations of [email protected] an astrophysical dynamo. The objective kinematic vor- tices are given by tubular level surfaces of the Lagrangian Averaged Vorticity Deviation (LAVD), the trajectory in- MS140 tegral of the normed difference of the vorticity from its spatial mean. In addition, we adapt the LAVD technique Stabilizing Embedology: When Do Delay- to use it on magnetic fields and propose the method of In- Coordinate Maps Preserve Geometry? tegrated Averaged Current Deviation (IACD) to determine precisely the boundary of magnetic vortices. The relevance The efficacy of delay-coordinate mapping has long been of objective kinematic and magnetic vortices in solar plas- supported by Takens’ embedding theorem, which guaran- masisdiscussed. tees that delay-coordinate maps use the time-series output to provide a reconstruction of the hidden state space that is Erico L. Rempel a one-to-one embedding of the system’s attractor. While Institute of Aeronautial Technology, Brazil this topological guarantee ensures that distinct points in erico [email protected] the reconstruction correspond to distinct points in the orig- inal state space, it does not characterize the quality of Abraham C. Chian this embedding or illuminate how the specific problem de- National Institute for Space Research (INPE), P.O. Box tails affect the reconstruction. We extend Takens’ result 515, by establishing conditions under which delay-coordinate Sao Jose dos Campos, SP, Brazil mapping is guaranteed to provide a stable embedding of [email protected] a system’s attractor. Beyond only preserving the attractor topology, a stable embedding preserves the attractor geom- Francisco J. Beron-Vera etry by ensuring that distances between points in the state RSMAS, University of Miami space are approximately preserved. In particular, we find [email protected] that delay-coordinate mapping stably embeds an attractor of a dynamical system if the stable rank of the system is large enough to be proportional to the dimension of the Sandor Szanyi attractor. The stable rank reflects the relation between 5Institute for Mechanical Systems, Department of the sampling interval and the number of delays in delay- Mechanical coordinate mapping. Our theoretical findings give guid- [email protected] ance to choosing system parameters, echoing the trade-off between irrelevancy and redundancy that has been heuris- George Haller tically investigated in the literature. ETH, Zurich, Switzerland [email protected] Armin Eftekhari Alan Turing Institute Tiago F. P. Gomes [email protected] National Institute for Space Research (INPE), Brazil [email protected] Han Lun Yap Georgia Institute of Technology Suzana S. A. Silva [email protected] Institute of Aeronautial Technology, Brazil [email protected] MichaelB.Wakin Dept. of Electrical Engineering and Computer Science Colorado School of Mines MS141 [email protected] Application of Dynamical Systems Approach to Lagrangian Separation in Laminar and Turbulent Chris Rozell Flows Georgia Institute of Technology In fluid mechanics, the separation phenomenon is as com- [email protected] mon as unpredictable. It causes the aerodynamic stall of aircraft wings, reduces the performance of turbomachines and limits the efficiency of hydraulic turbines. Although MS141 it has long been known, to date we can neither detect Objective Detection of Kinematic and Magnetic nor predict it with a unique and universal way. While in 264 DS17 Abstracts

two-dimensional steady flows Prandtl stated that the sep- flow in a rotating and stratified setting. The model allows aration point coincides with the location on a solid wall for submesoscale processes such as internal waves and sur- where fluid friction vanishes, it turns out that in presence face forcing by the wind. Here we discuss some preliminary of three-dimensionality or unsteadiness this criterion is not results on identifying transport pathways in this environ- valid. When separation occurs, the fluid flowing along the ment. The results generally are encouraging, but it is crit- wall suddenly detaches from the boundary. Based on this ical that the vertical velocity be included in calculations. ejection process, several theories have emerged to detect separation but remain difficult, if not impossible, to apply in practice, so that the Prandtl criterion is still widely used Denny Kirwan, Henry Chang, Helga S. Huntley because of its simplicity. The Lagrangian theory proposed University of Delaware by Haller in 2004 revisited the concept of unsteady sepa- [email protected], [email protected], [email protected] ration by defining a material instability induced by an un- stable manifold, without requiring to solve Navier-Stokes or boundary layer equations. Based entirely on kinematic MS142 arguments, this theory may in principle be applied to tur- Resynchronization of Circadian Oscillators and the bulent flows, which has never been verified. In this study, East-West Asymmetry of Jet-Lag we propose to explore the nature of separation in 2D un- steady flows. The Lagrangian approach is applied from Cells in the brain’s Suprachiasmatic Nucleus (SCN) are simple vortical flows to numerical data derived from tur- known to regulate circadian rhythms in mammals. We bulence simulations. model synchronization of SCN cells using the forced Ku- ramoto model, which consists of a large population of Se´an Crouzat coupled phase oscillators (modeling individual SCN cells) Polytechnique Montr´eal, Canada with heterogeneous intrinsic frequencies and external pe- [email protected] riodic forcing. Here, the periodic forcing models diurnally varying external inputs such as sunrise, sunset, and alarm J´erˆome V´etel clocks. We reduce the dimensionality of the system using Ecole´ Polytechnique de Montr´eal, Canada the ansatz of Ott and Antonsen and then study the effect [email protected] of a sudden change of clock phase to simulate cross-time- zone travel. We estimate model parameters from previous biological experiments. By examining the phase space dy- MS141 namics of the model, we study the mechanism leading to Lagrangian Coherent Structures and DNS with the difference typically experienced in the severity of jet-lag Discontinuous Galerkin Methods resulting from eastward and westward travel.

High-fidelity numerical tools based on high-order Zhixin Lu Discontinuous-Galerkin (DG) methods and Lagrangian University of Maryland Coherent Structure (LCS) theory for the study of sepa- [email protected] rated, vortex-dominated flows are discussed. A numerical framework is presented that couples a higher-order Kevin Klein-Cardena DG-DNS solver with time-dependent analysis of the DePaul University flow through LCS . At heart of this framework lies an [email protected] algorithm that computes Finite-Time Lyapunov Exponent (FTLE) fields simultaneously with DNS and with spectral Steven Lee accuracy. The algorithm is applied to investigate the role Brooklyn CUNY of LCS control fo unsteady separated flow over a NACA [email protected] 65-(1)412 airfoil at a free-stream Reynolds number of Re=20,000. Thomas Antonsen Jr., Michelle Girvan Gustaaf Jacobs University of Maryland Department of Aerospace Engineering [email protected], [email protected] San Diego State University [email protected] Edward Ott University of Maryland Inst. for Research in Electronics and Applied Physics MS141 [email protected] Ephemeral Transport Boundaries in Geophysical Fluid Flows MS142 Dynamical Systems has been spectacularly successful in Symmetric States Requiring System Asymmetry partitioning mesoscale oceanic and atmospheric transport pathways. Here processes operate on large space and long Spontaneous synchronization has long served as a time scales. However, there is increasing evidence in the paradigm for behavioral uniformity that can emerge from ocean that processes acting in the submesoscale can have interactions in complex systems. When the interacting en- substantial impact on the mesoscale circulation. Specific tities are identical and their coupling patterns are also iden- concerns are the facts that at the submesoscale the fluid is tical, the complete synchronization of the entire network is no longer two-dimensional incompressible and dynamical the state inheriting the system symmetry. As in other sys- processes operate on time scales of hours to days. Thus tems subject to symmetry breaking, such symmetric states the question arises about the efficacy of standard dynam- are not always stable. In this presentation, I will report on ical systems methods for identifying transport boundaries the discovery of the converse of symmetry breaking—the at these scales. To address this issue we developed a dy- scenario in which complete synchronization is not stable for namically balanced model that includes three-dimensional identically coupled identical oscillators but becomes stable DS17 Abstracts 265

when, and only when, the oscillator parameters are judi- dergo repeated oscillations. These devices can illustrate ciously tuned to nonidentical values, thereby breaking the the principles essential to oscillators driven by concentra- system symmetry to preserve the state symmetry. Aside tion gradients (e.g., ion channels). They consist of an from demonstrating that diversity can facilitate and even outer chamber of light fluid underneath an inner chamber be required for uniformity and consensus, this suggests a of heavy fluid, and an orifice between the two chambers mechanism for convergent forms of pattern formation in through which the two fluids of different density can be which initially asymmetric patterns evolve into symmetric exchanged under the influence of gravity and a Rayleigh- ones. Taylor instability. Typically, the two fluids are exchanged via oscillatory jets of laminar flow that can last for many Takashi Nishikawa, Adilson E. Motter hours. For example, when a solution of sodium chloride Northwestern University (NaCl) and deionized water (DI) is placed in the inner [email protected], [email protected] chamber and DI in the outer chamber, downward jets of the heavier fluid (NaCl) and upward jets of DI are produced one after the other at a constant frequency. We show in MS142 this talk how this limit cycle produced by oscillating uni- Two Types of Quasiperiodic Partial Synchrony: directional flows can be perturbed and converted into a fix What Happens When Symmetric States Become point solution where a bidirectional flow is present. We Unstable use the system to show how this bistable region exist as a function of the hole diameter. We then use a modified Population of identical globally coupled oscillators typi- Fitzhugh-Nagumo model to describe the experimental re- cally exhibits either fully synchronous solution or a splay sults and make a relation between the transitions between state. However, quite frequently both these symmetrical limit cycle and fix point as a function of their basins of states are unstable and the system settles at some interme- attractions. diate regime, called self-consistent partial synchrony. This state is characterized by a smooth but non-uniform dis- Flavio H. Fenton tribution of phases, so that the mean field is non-zero. Georgia Institute of Technology Another peculiar feature of the dynamics in this regime fl[email protected] is quasiperiodicity. We distinguish between two different quasiperiodic states: in the first case the frequency of the mean field is generally incommensurate with that of oscil- Hortencia Gonzales, Humberto Arce lators; in the second case average frequencies of the mean Universidad National Autonoma de Mexico field and of the oscillators coincide, but the motion of oscil- [email protected], [email protected] lators is additionally modulated by a generally incommen- surate frequency. We argue that self-consistent partial syn- chrony is a general phenomenon. We illustrate two types MS143 of quasiperiodic dynamics by different models, including Noise, Chaos and Random Numbers phase oscillators with biharmonic coupling function, non- linearly coupled Stuart-Landau systems, and linearly cou- Random number generation is essential for encryption of pled Rayleigh and Hindmarsh-Rose oscillators. information and Monte Carlo simulations. We examine sources and signatures of randomness and determinism in Michael Rosenblum simple optoelectronic nonlinear dynamical systems. Mea- Potsdam University sures of entropy production and dependence on observa- Department of Physics and Astronomy tional precision and time resolution are described. Applica- [email protected] tions of optoelectronic systems to physical random number generation and assessment are explored and state-of-the- art techniques will be discussed. The role of amplification MS142 of intrinsic noise by chaos will be examined and methods Chimera and Chimera-Like States in Populations to quantify it will be outlined. of Nonlocally Coupled Homogeneous and Hetero- geneous Chemical Oscillators Kevin Fei Harvard University We have studied chimera and chimera-like states in popu- [email protected] lations of photochemically coupled Belousov-Zhabotinsky (BZ) oscillators. Simple chimeras and chimera states with multiple and traveling phase clusters, phase-slip behavior, Joseph Hart and chimera-like states with phase waves are described. University of Maryland Simulations with a realistic model of the discrete BZ sys- [email protected] tem of populations of homogeneous and heterogeneous os- cillators are compared with each other and with experi- Thomas E. Murphy mental behavior. University of Maryland, College Park Dept. of Electrical and Computer Engineering Kenneth Showalter [email protected] West Virginia University Department of Chemistry Rajarshi Roy [email protected] University of Maryland Institute for Research in Electronics and Applied Physics [email protected] MS143 Dynamical Transitions Between a Limit Cycle and a Fix Point in a Saline Oscillator MS143 Density oscillators are simple inexpensive devices that un- Coherent Structures in the Motion of Reaction 266 DS17 Abstracts

Fronts and Swimming Organisms in Laminar Flows [email protected]

We present results of table-top experiments that mea- sure burning- and swimming-invariant manifolds (BIMs MS144 and SWIMs) that act as one-way barriers that impede A Path-Integral Approach to Data Assimilation for the motion of reaction fronts and active tracers (swim- Mapping Small Neuronal Networks ming bacteria) in laminar fluid flows. For the reaction front experiments, the flows are 2-D and 3-D chains of vor- We present results of simulated experiments to simultane- tices, generated by magnetohydrodynamic forcing, and the ously estimate properties of individual neurons and synap- propagating front is produced by the excitable Belousov- tic connections in small networks. The calculations are per- Zhabotinsky chemical reaction. The locations of BIMs are formed via a method of data assimilation (D.A.) that can predicted from a model of the velocity field and a set of be derived from a path-integral formulation. The aim in ordinary differential equations that follows the motion of D.A. is to ascertain the predictive power of a model, where an element of the front. For the experiments with active the model may contain parameters to be estimated, there tracers, the flow is a 2-D extensional flow in a microfluidic may exist multiple sets of parameters that fit the experi- channel, and the tracers are a mutated, ”smooth swim- mental measurements, and the available measurements are ming” bacillus subtilis. The locations of predicted SWIMs sparse. The central consideration is the information con- are verified experimentally by following the trajectories of tent of a measurement. We ask: which quantities of (from) individual bacteria in the flow. The results of these sim- a system must be measured, to complete the corresponding plified experiments and the general BIM/SWIM analysis model? We perform D.A. via an optimization procedure, should be applicable to a wide range of systems in which a where the cost function is cast as the physical Action on a front or swimmer with a well-defined speed is moving in a system’s path through its state space. Among degenerate fluid flow. parameter sets, the correct set may be identified via its cor- respondence to the path of least action. We show results Thomas H. Solomon, Minh Doan, Katie Lilienthal, JJ of this procedure on two- and three-neuron networks, and Simons, Payton Johnsom, Joan Gannon the predictive power of the completed models. The long- Department of Physics & Astronomy term aim is to design real laboratory experiments based Bucknell University on this procedure. Such experiments would, in principle, [email protected], [email protected], furnish the data required to seek: 1) a possible quantifi- [email protected], [email protected], able relation between a cell’s electrophysiological proper- [email protected], [email protected] ties and its functional connectivity within a network, and 2) timescales on which properties commonly identified as parameters may in fact vary.

MS143 Eve Armstrong Jamming in Granular Systems of Regular Polygons University of CA, San Diego University of CA, San Diego [email protected] The study of the onset of mechanical stability, known as the jamming transition, of granular systems provides key insights into properties of amorphous materials. A fun- MS144 damental challenge to understanding this transition is to determine the influence of particle properties. Here, we in- Finite Size Effects and Rare Events in Balanced vestigate how nontrivial particle shapes affect the jamming Cortical Networks transition as controlled by the packing fraction. Our exper- iments are performed by compression of two-dimensional Cortical neuron spiking activity is broadly classified as tem- arrangements of photoelastic particles, allowing us to vi- porally irregular and asynchronous. Model networks with sualize force information. To explore the role of parti- a balance between large recurrent excitation and inhibition cle shape, we systematically change the number of sides capture these two features, and are a popular framework of polygonal particles used in the experiments and com- relating circuit structure and network dynamics, though pare the force chain network, contact number and pressure are traditionally restricted to a single attractor. We ana- evolution of compressed systems of polygons to the well- lyze paired whole cell voltage-clamp recordings from spon- studied systems of disks. We also explore the influence of taneously active neurons in mouse auditory cortex slices geometric features, such as face-face contacts and ordering (Graupner & Reyes, 2013) showing a network where cor- within packings, in connection with the jamming transi- related excitation and inhibition effectively cancel, except tion. for intermittent periods when the network shows a macro- scopic synchronous event. These data suggest that while the core mechanics of balanced activity are important, we Cacey Stevens Bester, Yiqiu Zhao require new theories capturing these brief but powerful pe- Department of Physics riods when balance fails. Recent work by Mongillo et.al. Duke University (2012) showed that balanced networks with short-term [email protected], [email protected] synaptic plasticity can depart from strict linear dynam- ics. We extend this model by incorporating finite network Jonathan Bares size, introducing strong nonlinearities in the firing rate dy- Montpellier University namics and allowing finite size induced noise to elicit large [email protected] scale, yet infrequent, synchronous events. We identify core requirements for system size and network plasticity to cap- Robert Behringer ture the transient synchronous activity observed in our ex- Duke University perimental data set. Our model properly mediates between Center for Nonlinear and Complex Systems, Dept of the asynchrony of balanced activity and the tendency for Physics strong recurrence to promote macroscopic population dy- DS17 Abstracts 267

namics. the motor control region of the brain, there has been inter- est in finding electrical stimuli which desynchronize neural Jeffrey Dunworth populations. Recent results on coordinated reset and pe- Mathematics Department riodically forced oscillators suggest that forming distinct University of Pittsburgh clusters of neurons may prove to be more effective than [email protected] achieving complete desynchronization, in particular by pro- moting plasticity effects that might persist after stimula- Bard Ermentrout tion is turned off. We propose a single-input low-power University of Pittsburgh control strategy to achieve such clustering, based on the Department of Mathematics analysis of the reduced phase model for a set of identical [email protected] neurons.

Michael Graupner Jeff Moehlis,TimothyMatchen Universite Paris Descartes Dept. of Mechanical Engineering [email protected] University of California – Santa Barbara [email protected], [email protected] Alex Reyes Center for Neural Science MS145 New York University The Stability and Slow Dynamics of Localized [email protected] Spot Patterns for the 3-D Schnakenberg Reaction- Diffusion Model Brent Doiron Dept. of Mathematics On a bounded three-dimensional domain, we employ a Univ. of Pittsburgh hybrid asymptotic-numerical method to analyze the ex- [email protected] istence, linear stability, and slow dynamics of localized quasi-equilibrium multi-spot patterns of the Schnakenberg reaction-diffusion model with bulk feed rate A.WhenA MS144 is spatially constant, we illustrate an imperfection-sensitive Large-Scale Connectome-Based Brain Networks bifurcation structure of the multi-spot quasi-equilibria, and Models of Seizure Spread demonstrate competition and self-replication instabilities whereby spots may annihilate or split into two. When the Over the past decade we have demonstrated that con- feed rate A is spatially non-uniform, new equilibrium con- straining computational brain network models by struc- figurations can be created. In particular, when A is spa- tural information obtained from human brain imaging tially localized, we show that a spot can become pinned at (anatomical MRI, diffusion tensor imaging (DTI)) allows the source point in finite time. Joint work with S. Xie, T. patient specific predictions, beyond the explanatory power Kolokolnikov, and M.J. Ward. of neuroimaging alone. This fusion of an individuals brain structure with mathematical modelling allows creating one Justin C. Tzou model per patient, systematically assessing the modeled University of British Columbia parameters that relate to individual functional differences. [email protected] The model is expressed as neural fields and their dynam- ics in terms of differential-integral equations, in which the integral kernels represent translationally invariant (local) MS145 and variant (global) connectivity. The functions of the Amoeboid Locomotion in Heterogenous Media brain model allow executing dynamic neuroelectric simu- lation; further modeling features include refined geometry Organisms choose one of various courses of action in re- in 3D physical space; detailed personalized brain connec- sponse to a wide variety of environmental conditions. How tivity (Connectome); large repertoire of mathematical rep- the variety is induced by an external stimulus is an open resentations of brain region models, and a complete set question. We experimentally found that statistical distri- of physical forward solutions mimicking commonly used bution of passage time across the quinine zone switched in non-invasive brain mapping including functional Mag- from unimodal to bimodal when the periodic perturbation netic Resonance Imaging (fMRI), Magnetoencephalogra- of light stimulation is applied homogeneously in space in phy (MEG) Electro-encephalography (EEG) and Stere- addition to the quinine-induced differentiation. This result oElectroEncephalography (SEEG). So far our large-scale implies that differentiation of behavioral types is modified brain modeling approach has been successfully applied to step by step through the compounding of stimulation in- the modeling of the resting state dynamics of individual puts. Based on a mathematical model for cell movement human brains, as well as aging and clinical questions in in Physarum plasmodium, we succeeded to reproduce the stroke and epilepsy. stimulation-induced differentiation observed in the experi- ment. Viktor Jirsa Institut de Neurosciences des Syst`emes Inserm UMR1106 Kei-Ichi Ueda Aix-Marseille Universit´e Faculty of Science, University of Toyama [email protected] [email protected]

Itsuki Kunita MS144 Kumamoto University Clustered Desynchronization of Neural Oscillators [email protected]

Inspired by the hypothesis that some symptoms of Parkin- Shigeru Kuroda son’s disease are related to pathological synchronization in Research Institute for Electronic Science, 268 DS17 Abstracts

Hokkaido university whether oscillators in the network are firing, quiescent, or [email protected] saturated. We use the relationships between external input distribution and oscillator states in the reduced network Dai Akita, Toshiyuki Nakagaki to elucidate properties of oscillators in the original net- Hokkaido University work. Complex behavior in the reduced network includes [email protected], [email protected] the presence of mixed mode oscillations and asymptotically periodic behavior. Extensions of these results could be used to inform system identification in complex networked MS145 oscillatory systems. Wavenumber Selection Via Spatial Parameter Step Elizabeth Davison The Swift-Hohenberg (SH) equation is a prototypical, Princeton University pattern-forming model equation for Turing instability phe- [email protected] nomena. We study the SH equation with a space- dependent parameter that renders the medium stable in Biswadip Dey x<0 and unstable in x>0. We prove the existence of Princeton University a family of steady-state solutions that represent a tran- Department of Mechanical and Aerospace Engineering sition in space from a homogeneous state to a “striped” [email protected] pattern state. Solutions in the family vanish as x →−∞ and are asymptotically periodic as x →∞.Wavenum- Zahra Aminzare, Naomi E. Leonard bers appearing asymptotically in the family are restricted Princeton University to a neighborhood of 1 with width depending linearly on [email protected], [email protected] the size of the parameter jump. Our methods involve us- ing normal form theory and a co-rotating reference frame to represent the instability in SH by the real Ginzburg- MS146 Landau (GL) amplitude equations. The family of solutions Molecular Mechanisms Underlying the Kuramoto is found to leading order as a heteroclinic connection in the Model of Coupled Oscillators GL equations and existence is proved by finding an inter- section of unstable and stable manifolds which persists in The Kuramoto model has been widely used to describe the the SH equation. synchronizations of a large set of coupled oscillators. In particular, the Kuramto model successfully captures the Jasper Weinburd key features of synchronization of 20,000 coupled cellular University of Minnesota - Twin Cities rhythms in the circadian clocks of our brain. However, due [email protected] to the abstractness of the Kuramoto model, specific molec- ular mechanisms underlying the sinusoidal coupling terms Arnd Scheel have not been identified. In this talk, I will discuss that University of Minnesota the combination of intracellular transcriptional repression School of Mathematics mechanisms and intercellular coupling mechanisms in the [email protected] mammalian circadian clocks can lead to such sinusoidal coupling.

MS145 Jae Kyoung Kim Title Not Available At Time Of Publication Department of Mathematical Sciences Korea Advanced Institute of Science and Technology Abstract Not Available At Time Of Publication. (KAIST) [email protected] Shuangquan Xie Dalhousie University Zachary P. Kilpatrick Department of Mathematics and Statistics Department of Applied Mathematics [email protected] 526 UCB, University of Colorado, Boulder, CO, 80309 USA MS146 [email protected] Dynamics of Large Networks of FitzHugh-Nagumo Neuronal Models with Heterogeneous Inputs Matthew Bennett Rice University Large networks of identical nonlinear neuronal oscillators [email protected] exhibit complex dynamics that depend on network struc- ture and distribution of external input. Mathematical in- Kresimir Josic vestigation of the dynamics in oscillator networks can help University of Houston in understanding the behavior of neuronal ensembles in the Department of Mathematics brain. We implement sufficient conditions for synchroniza- [email protected] tion in networks of FitzHugh-Nagumo (FN) neuronal oscil- lators with electrical gap junction coupling to reduce large networks of neurons to corresponding representative net- MS146 works that are more tractable to analysis. The FN model, Approximating Consensus with Graph Limits a two-dimensional simplification of the Hodgkin-Huxley equations for neuronal membrane potential dynamics, ex- The consensus problem in multi-agent systems has long hibits input-dependent regimes of dynamical behavior. In been the subject of intense research within the fields of the reduced network, we explore the contributions of exter- computer science and mathematics. This problem can nal input heterogeneity and coupling strength to determine naturally be expressed as a dynamical system such that DS17 Abstracts 269

the interaction between agents is defined by a network. [email protected] This talk will provide sufficient conditions for when graph limits can be used as a continuum-approximation for the consensus problem on large networks. I will also highlight MS147 other aspects of graph limit theory which have implications Exclusion and Clock Behavior in An Oscillating for the consensus problem. Chemostat

Barton Lee Microbes form a large and central part of the global ecosys- School of Mathematics and Statistics tem. As a consequence of their short reproductive time The University of New South Wales and their proficiency at exchange of genetic material, it [email protected] seems plausible that microbes in communities operate at high efficiency (in terms of free energy and nutrient usage) in many contexts. One obvious issue of interest would be MS146 the description of species within a microbial community Turing Patterns in Reaction-diffusion Systems on and its dependence on the local environment. Descrip- Random Networks tion of niche structure of organisms and how that structure impacts competitiveness has long been a topic of interest Turing instability in reaction-diffusion systems of activa- among ecologists. Here, in the context of Yellowstone Na- tor and inhibitor species leads to formation of non-uniform tional Park microbial mat, we discuss influence of tempo- patterns in continuous media. Turing instability can also ral environment on microbial community species structure. take place in reaction-diffusion systems on general networks The possibilities of competitive exclusion and clocking be- as pointed out by Othmer and Scriven already in 1971. havior are discussed. We consider activator-inhibitor reaction-diffusion systems on large random networks with broad degree distributions Isaac Klapper and show that the Turing instability leads to non-uniform Temple University patterns on networks, in which network nodes differentiate [email protected] into activator-rich and activator-poor groups. We argue that the backbones of the developed Turing patterns can be understood as bifurcation diagrams of individual net- MS147 work nodes through mean-field approximation of the net- work Laplacian. We also discuss some related topics, such Metabolic Processes As Dynamical Systems. A as pattern formation in bistable reaction-diffusion systems Hotbed of Nonlinear Phenomena? on networks, localization of Laplacian eigenvectors in large random networks with broad degree distributions, and ap- Abstract Not Available At Time Of Publication. proximation of the system on a network by a system with nonlocal coupling using recent results on graph limits. Doraiswami Ramkrishna Department of Chemical Engineering Hiroya Nakao Purdue University Graduate School of Information Science and Engineering, [email protected] Tokyo Institute of Technology [email protected] MS147 Separating Putative Pathogens from Background MS147 Contamination with Principal Orthogonal Decom- Dynamics of Cell Signaling Within a Biofilm position: Evidence for Leptospira in the Ugandan Neonatal Septisome Bacterial cell-cell communication, also termed quorum sensing (QS) is a wide-spread process that coordinates mul- Neonatal sepsis (NS) is responsible for over 1 million yearly ticellular behaviors such as virulence, biofilm formation, deaths worldwide. Amplicon DNA sequencing can be used and nutrient acquisition in response to cell density, pop- to identify organisms that are difficult to detect by routine ulation structure and environmental viscosity. There has microbiological methods. However, contaminating bacteria been an explosion in research directed at understanding the are ubiquitous in hospital settings and research reagents. molecular mechanisms of QS, but there is a paucity of in- We sequenced the bacterial 16S rRNA gene obtained from formation on the ecophysiological implications and on the blood and cerebrospinal fluid (CSF) of 80 neonates pre- emergent properties of QS regulatory networks. Short cir- senting with NS to the Mbarara Regional Hospital in cuiting, or self-induction, is a major unanswered question Uganda. Assuming that patterns of background contami- in bacterial QS: How is it that diffusible quorum-signals do nation would be independent of pathogenic microorganism not immediately bind to their cognate receptors in the same DNA, we applied a novel filtering approach using principal cell in which they are produced and activate gene expres- orthogonal decomposition to separate background contam- sion independent of cell density? We will investigate the ination from potential pathogens in sequencing data. We roles of antiactivation and lasR regulation in modulating employed a statistical random matrix bootstrap hypothe- the quorum response and in preventing short-circuiting We ses to estimate significance. These analyses demonstrate will talk about Quorum sensing, Short-Circuiting in Quo- that Leptospira appears present in some infants presenting rum Sensing Systems, and some recent efforts to model within 48 hr of birth, indicative of infection in utero, and some experimental results from the Suel lab on Electri- up to 28 days of age, suggesting environmental exposure. cal Communication via Potassium currents within bacte- This organism cannot be cultured in routine bacteriolog- rial communities. ical settings, and is enzootic in the cattle that the rural peoples of western Uganda often live in close proximity. Jack D. Dockery Our computational strategy can reveal putative pathogens Montana State University in small biomass samples, revealing an important medical Department of Mathematics finding that has the potential to alter therapy and preven- 270 DS17 Abstracts

tion efforts in a critically ill population. that perform near theoretical limits. Recently, informa- tion theory is being pushed in a new direction; rather than Steven J. Schiff defining Shannon limits, this ”applied” approach to infor- Penn State University mation theory seeks to infer causal relationships from esti- Center for Neural Engineering mates of conditional mutual information. In this applica- sschiff@psu.edu tion, it is imperative that the significance of these estimates be computed before any inference is made. Surprisingly, al- most no work has been done to develop significance tests MS148 for information theoretic quantities. In this talk we give an Multivariate Dependence Beyond Shannon Infor- exact test for mutual information, an exact test for Markov mation order, and a typical realizations test for transfer entropy.

Accurately determining dependency structure is critical to Shawn D. Pethel discovering a system’s causal organization. We demon- U.S. Army RDECOM strate that Shannon-like information measures fail at this [email protected] when used to analyze multivariate dependencies. This has broad implications, particularly when employing informa- Daniel Hahs tion to express the organization and mechanisms embed- Torch Technologies ded in complex systems. Here, we do not suggest that any Huntsville, AL aspect of information theory is wrong. Rather, the vast [email protected] majority of its informational measures are simply inade- quate for determining the meaningful dependency struc- ture within joint probability distributions. Therefore, such MS148 information measures are inadequate for discovering intrin- Causation Entropy and Pure Inference of Boolean sic causal relations. We close by demonstrating that such Factors distributions exist across an arbitrary set of variables, and discuss several potential avenues toward rectifying this is- Understanding the dynamics and functioning of complex sue. systems is one of the most challenging tasks faced in modern science. A central problem to tackle is to accu- Ryan G. James rately and reliable infer the underlying cause-and-effect University of California, Davis (i.e., causal) network from observational data, especially [email protected] when the system consists of a large number of interact- ing components and the dynamics is intrinsicaly nonlinear. Utilizing our recently developed theory of causation en- MS148 tropy (J. Sun, D. Taylor, and E. M. Bollt, SIAM Journal Prediction vs Generation in Simple Markov Chains on Applied Dynamical Systems 14, 73–106, 2015), we de- vised an efficient computational approach of optimal cau- We present the minimal generators for all binary Markov sation entropy (oCSE) to infer causal networks from data. chains and calculate their generative complexity. While In particular, we specialize the general approach to infer this is in general a non-convex optimization and thus re- Boolean networks, which are widely used in the modeling lies on numerics, we find an analytic solution for this par- of biological systems. We demonstrate the effectiveness of ticular set of processes. Importantly, processes in this our approach using both synthetic and experimental data., class have a smaller generative complexity than statisti- and discuss open challenges. cal complexity—the minimal memory required by a pre- dictor of the process. We then examine other information Jie Sun theoretic properties of these minimal generators, includ- Mathematics ing their crypticity, oracular information, and gauge infor- Clarkson University mation, and compare the minimal generators to minimal [email protected] predictors. Erik Bollt John R. Mahoney Clarkson University University of California at Davis [email protected] [email protected]

Josh Ruebeck MS149 Carleton College Master Stability Islands for Amplitude Death in Northfield, MN Networks of Delay-Coupled Oscillators [email protected] This talk presents a master stability function (MSF) ap- Ryan G. James proach for analyzing the stability of oscillation death (OD) University of California, Davis in networks of delay-coupled oscillators. Unlike the fa- [email protected] miliar MSFs for instantaneously coupled networks, which typically have a single input encoding for the effects of the eigenvalues of the network Laplacian matrix, for delay- MS148 coupled networks we show that such MSFs generally re- Significance Testing of Information Theoretic quire two additional inputs: the time delay and the cou- Quantities pling strength. To utilize the MSF for predicting the sta- bility of OD of arbitrary networks for a chosen nonlinear Information theory has been tremendously successful in system (node dynamics) and coupling function, we intro- defining the relevant quantities, relationships, and bounds duce the concept of master stability islands (MSIs), which that enable engineers to design communication systems are two-dimensional stability islands of the delay-coupling DS17 Abstracts 271

space together with a third dimension (“altitude”) encod- may achieve synchronization, even when the network (at ing for eigenvalues that result in stable OD. We compute any instant) is severely disconnected. In the second, we the MSFs and show the corresponding MSIs for several considered the implications of communication delay in a common chaotic systems including the R¨ossler, the Lorenz, fixed network of chaotic oscillator and discover that such and Chen’s system, and found that it is generally possible coupling can ”simplify” the chaotic system to phase syn- to achieve OD and that a nonzero time delay is necessary chronized limit cycles. Our focus for this talk is to consider for the stabilization of the OD states. the behavior of a coupled system where we allow the net- work topology to be time-varying, but also specify that Stanley R. Huddy communication pathways provide a time-delay structure State University of New Paltz to the communications. Coupling of chaotic oscillators, we [email protected] observe (and describe) an interesting interplay between the competing effects of chaotic synchronization and the sim- Jie Sun plification of trajectories implied by delay coupling. Mathematics Clarkson University Joseph Skufca [email protected] Clarkson University [email protected]

MS149 MS149 Controlling Nonlinear Biological Systems Based on Network Structures Patterns in Circular Networks of Excitable Systems with Time Delay Modern biology provides many networks describing regula- tions between bio-molecules. It is widely believed that dy- We consider recurrent pulse-coupled networks of excitable namics of molecular activities based on such regulatory net- elements with delayed connections, which are inspired by works are the origin of biological functions. In this study biological neural networks. If the delays are tuned appro- we present a mathematical theory to provide an important priately, the network can either stay in the steady resting aspect of dynamics from information of regulatory linkages state or, alternatively, exhibit a desired spiking pattern. It alone. We also present an application of our theory to a is shown that such a network can be used as a pattern- real biological network, and experimental results to verify recognition system. More specifically, the application of our prediction. In theoretical part, we show that ”feedback the correct pattern as an external input to the network vertex set” (FVS) of a regulatory network is a set of ”de- leads to a self-sustained reverberation of the encoded pat- termining nodes” of dynamics [Mochizuki, A., Fiedler, B. tern. In terms of the coupling structure, the tolerance and et al. (2013) J. Theor. Biol., 335, 130-146]. It assures that the refractory time of the individual systems, we determine i) any long-term dynamical behavior of whole system, such conditions for the uniqueness of the sustained activity, i.e., as steady states, periodic oscillations or quasi-periodic os- for the functionality of the network as an unambiguous cillations, can be identified by measurements of a subset of pattern detector. We point out the relation of the con- molecules in the network, and that ii) the subset is deter- sidered systems with cyclic polychronous groups and show mined from the regulatory linkage alone. The theory also how the assumed delay configurations may arise in a self- claims that iii) dynamical behavior of whole system can be organized manner when a spike-time dependent plasticity switched from one attractor to others just by controlling of the connection delays is assumed. As excitable elements, dynamics of FVS. In experimental application, we analyzed we employ simplistic coincidence detector models as well a gene network including 90 genes, and responsible for cell as Hodgkin-Huxley neuron models. Moreover, the system differentiation in development of ascidian. We identified is implemented experimentally on a Field-Programmable five genes in minimum FVS of the network. The exhaus- Gate Array (FPGA). tive artificial activation/inhibition of five genes showed that Serhiy Yanchuk the system could be controlled totally just by controlling Humboldt University of Berlin activities of five genes in FVS. [email protected] Atsushi Mochizuki Riken Mochizuki Laboratory Leonhard L¨ucken [email protected] Institute of Transportation Systems, German Aerospace Center Berlin, Germany MS149 [email protected] Dynamic Networks and Delay: Communication Delay on Time Varying Networks David P. Rosin, Vasco Worlitzer Technische Universit¨at Berlin Synchronization of coupled systems remains an important Hardenbergstrae 36, 10623 Berlin, Germany focus area for study of networked components. Such sys- [email protected], [email protected] tems admit rich dynamical behaviors, where coupling may berlin.de lead to simplified or complex system as compared uncou- pled node dynamics. For example, simple oscillators may yeild a chaotic coupled system; coupled chaotic oscillators MS150 may synchronize along a chaotic trajectory; coupling of Interfacial Dynamics in Biological Systems chaotic oscillators, under certain heterogeneity conditions, may lead to periodic or steady-state behaviors of the cou- Biological pattern formation has been extensively studied pled system. We previously examined two very distinct using reaction-diffusion models. These models are inher- types of network communication schemes: The first, which ently local, however many biological systems are known we called a Moving Neighborhood Network created a time- to exhibit nonlocality. In this talk we will discuss nonlo- varying network where diffusively couple chaotic oscillators cal pattern forming mechanisms in the context of bacte- 272 DS17 Abstracts

rial colony formation with an emphasis on arrested fronts. rays on the caudal fin. The development of both wild-type This will lead to a nonlocal framework to understand the and mutated patterns on the body and fins will be dis- interfacial motion in biological systems. We will apply our cussed, as well as the non-local continuum limit associated approach to model the evolution of bacterial colonies that with the model. inhibit the growth of genetically identical colonies. Alexandria Volkening, Bjorn Sandstede Scott McCalla Division of Applied Mathematics Montana State University Brown University [email protected] alexandria [email protected], bjorn [email protected] James von Brecht California State University, Long Beach [email protected] MS151 Effective Variables and Interaction Potentials for Coarse Graining of Macromolecular Systems MS150 The Soft Mode Plays A Role In Defect Persistence The complete thermodynamic and kinetic characterization In Pattern-Forming Systems of high dimensional dynamical systems such as large pro- tein complexes still presents significant challenges. Coarse- When the surface of a nominally flat binary material is grained models have long been used as a practical alterna- bombarded with a broad, normally incident ion beam, dis- tive for the characterization of protein mechanisms over ordered hexagonal arrays of nanodots can form. Defects, long time scales. The theoretical justification for these re- such as dislocations in ripple patterns or penta-hepta pairs duced models relies on the separation of time scales present in hexagonal arrays, limit the utility of patterns produced in macromolecular systems: it appears that the dynamics by ion bombardment. We show that a neutrally stable soft of a set of slow variables regulates the behavior of the sys- mode can contribute to the persistence over time of defects tem over long timescales, suggesting that the remaining, that form in the early stages of pattern formation. Topo- much faster, variables could, in principle, be renormal- logical measures of order provide a method for determining ized into the definition of effective interactions between if a defect is removed as the pattern evolves. the slower variables. The definition of a coarse-grained model from a finer-grained one requires two main choices: Patrick Shipman the degrees of freedom to be considered, and the effec- Department of Mathematics tive energy function regulating their dynamics. Rigorous Colorado State University mathematical procedures are not usually applied to ad- [email protected] dress these questions. We will discuss our recent attempts to use emerging mathematical tools to tackle these points systematically. MS150 Spots and Stripes: Vegetation Patterns in Dry- Cecilia Clementi lands Rice University [email protected] Reaction-advection-diffusion systems have been developed to model vegetation patterns in arid regions. These vege- tation patterns have been observed in satellite images of MS151 the surface of the earth, in arid ecosystems all around Transfer Operator Approximation Using Low-Rank the world. Different patterns include gaps, stripes and Tensor Decompositions spots. The formation of patterns has been linked to in- creased aridity (decreased rainfall). The understanding of In this talk, we will present tensor-based reformulations transitions between different vegetation patterns may re- of data-driven methods such as Dynamic Mode Decompo- sult in early warning signals for an upcoming irreversible sition (DMD) and Extended Dynamic Mode Decomposi- desertification. In this talk we focus on the stability of tion (EDMD) for the analysis of complex dynamical sys- striped patterns in two spatial dimensions. On sloped tems. Due to the curse of dimensionality, analyzing high- terrain these patterns naturally form for a general class dimensional systems is often infeasible using conventional of two-component reaction-advection-diffusion systems. Is methods since the amount of memory required to compute the breakup of stiped patterns in spotted patterns a generic and store modes and eigenfunctions grows exponentially next step in the desertification process? We seek to com- with the size of the system. This can be mitigated by bine analytical and numerical model results with satellite exploiting low-rank tensor approximation approaches. We data analysis. will show how tensor-based methods can be used to approx- imate the eigenfunctions of transfer operators. The results Eric Siero are illustrated with the aid of simple fluid dynamics and Leiden University, Mathematical Institute molecular dynamics examples. [email protected] Stefan Klus Freie Universitat Berlin MS150 [email protected] Stripe Formation on Zebrafish Fins

Zebrafish (Danio rerio) is a small fish with distinctive black MS151 and yellow stripes that form on its body and fins due to the The Coarse Graining of Neural Field Models interaction of different pigment cells. Working closely with the biological data, we present an agent-based model for In this talk I will report my recent progress about the these stripes that explores the interplay of cells and bone coarse-graining of neural field models. The fine scale model DS17 Abstracts 273

we study is a class of conductance-based leaky integrate- debugging in neurological disease states where sleep is dis- and-fire (LIF) models. Instead of the common spatial rupted. coarse-graining, we use information-theoretic tools to re- A major challenge in this work beyond getting reliable duce the LIF model to a Markov process about the time brain-stem recordings in freely behaving animals is the sta- evolution of the distributions of (discretized) membrane tistical and mixed nature of the recordings. The dynamics potentials of excitatory and inhibitory neurons. After this represented in the models is that of interacting cell groups. model reduction, we can rigorously study problems like the The measurements are unit and multi-unit recordings from stochastic stability and the spiking patterns. brain locations where those cell groups are. Unstated is the relationship between the activity - the firing times - of a Yao Li singe neuron and the cell group from which it comes. More University of Massachusetts Amherst complex is that a single electrode will typically sense ac- Department of Mathematics and Statistics tivity from multiple neurons from multiple cell groups. I’ll [email protected] address a framework for addressing the linkage from these measures to the model variables. Sedigh-Sarvestani, et al. (2012). PLoS Computational Bi- MS151 ology, 8(11), e1002788. The Cgdna Coarse Grain Model of the Sequence- Dependent Statistical Mechanics of DNA Bruce J. Gluckman Penn State Engineering I will describe the cgDNA model, its predictions of [email protected] sequence-dependent DNA persistence lengths, and how to use the maximum entropy principle to estimate coarse grain model parameters from large scale data sets gener- MS152 ated in fine grain Molecular Dynamics training set simula- Dynamics and Big Biomedical Data Sets tions.

John H. Maddocks Observational Health Data Sciences and Informatics EPFL (OHDSI) is an interdisciplinary, international collaborative john.maddocks@epfl.ch with 160 researchers from 25 countries to improve health by collaboratively generating evidence to answer questions about diagnosis and treatment. Over 600 million patient MS152 records have been encoded in the OHDSI format, and each record is a time series on a non-stationary subject with sig- Physiologic Model Development Using Data As- nificant measurement error and bias. Answering medical similation and Self-Monitoring Data questions depends on dynamics of the physiology, which varies over time (different drugs and medical interventions This talk will address physiologic model development us- available as well as patient variation including aging and ing data assimilation with self-monitoring data and intro- disease worsening) and over space (i.e., geographic loca- duce the larger topic of the mini-symposium data driven tion of the patient, which includes national differences, biomedical dynamics, modeling and data assimilation. In regional differences, racial differences, etc.). Early stud- the context of endocrine (e.g., glucose-insulin) modeling I ies have used traditional techniques to address variation, will introduce self-monitoring data and their constraints. I such as self-controlled case series and cohort studies, and a will then discuss how problem context, e.g., clinical versus map of variation in treatment pathways for chronic disease basic science goals, drive what mathematics and dynam- over time and space has been produced. Current research ics are important. I will show some results within both of attempts to account for bias in the data through lagged these contexts related to model development and dynam- regression methods. ics. I will conclude with a discussion of data assimilation related challenges including identifiability and uncertainty George Hripcsak quantification. Columbia University David J. Albers [email protected] Colombia University Biomedical Informatics [email protected] MS152 Competitive Offline Parameter Estimation for On- line Data Assimilation in Glucose Dynamics MS152 Statistical Framework for Data Assimilation from Data assimilation with computational models of glucose Single & Multi-Unit Recording to Mean Field Net- dynamics offers great promise in personalized diabetes in- work Models terventions and clinical decision making, but carries signif- icant mathematical challenges. We focus here on the chal- WehavebeenworkingtoapplyDataAssimilation(DA) lenges of using offline parameter estimation methods to im- methods to synchronize mechanistic-based computational prove real-time, online forecasts. Specifically, we consider models of neural activity to measurements from real brain. a simple ultradian model of glucose-insulin dynamics, per- In (Sedigh-Sarvestani, 2012) we demonstrated that com- form parameter estimation using Nelder-Mead optimiza- putational models of the putative sleep-wake regulatory tion, Markov Chain Monte Carlo, and unscented Kalman network could be observed through weak, incomplete mea- filtering (UKF) methods, and perform glucose state fore- surements of the embedded variables using an implemen- casting using a UKF and a simple, unfiltered model. We tation of an unscented Kalman filter (UKF). We are cur- evaluate the opportunities and trade-offs when passing pa- rently working to apply these methods for model valida- rameter estimates between offline and online methods using tion, reconstruction and forecasting in real systems - neural both model selection criteria and a diabetes-specific error recordings from live, freely behaving rats; and for system metric, and we discuss how these different approaches can 274 DS17 Abstracts

be translated into real-time decision support systems. fluid-structure interactions. In this talk, we derive a varia- tional integrator for a particular type of fluid-structure in- Matthew Levine teractions, namely, simulating the dynamics of a bendable Columbia University tube conveying ideal fluid that can change its cross-section [email protected] (collapsible tube). We start by deriving a fully three- dimensional, geometrically exact theory for flexible tubes David J. Albers conveying fluid. Our approach is based on the symmetry- Colombia University reduced, exact geometric description for elastic rods, cou- Biomedical Informatics pled with the fluid transport and subject to the volume [email protected] conservation constraint for the fluid. Using these meth- ods, we obtain the fully three dimensional equations of Andrew Stuart motion. We then proceed to the linear stability analysis Computing + Mathematical Sciences and show that our theory introduces important corrections California Institute of Technology to previously derived results, both in the consistency at all [email protected] wavelength and in the effects arising from the dynamical change of the cross-section. Based on this theory, we derive George Hripcsak a variational discretization of the dynamics based on the Department of Bioinfomatics, Columbia University appropriate discretization of the fluids back-to-labels map, [email protected] coupled with a variational discretization of elastic part of the Lagrangian.

MS153 Vakhtang Putkaradze A Variational Lagrangian Formulation for Nonequi- University of Alberta librium Thermodynamics of Continuum Systems [email protected]

We develop a Lagrangian variational formulation for the nonequilibrium thermodynamics of continuum systems. MS153 This variational formulation extends the Hamilton prin- ciple to allow the inclusion of irreversible processes in the Explicit High-Order Symplectic Integration of Ar- dynamics. The irreversibility is encoded into a nonlinear bitrary Hamiltonians nonholonomic constraint given by the expression of entropy production associated to all the irreversible processes in- Symplectic integrators preserve the phase-space volume volved. The variational formulation is naturally expressed and have favorable performances in long time simula- in material representation, while its spatial version is ob- tions. Methods for explicit symplectic integration have tained via a nonholonomic Lagrangian reduction by sym- been extensively studied for separable Hamiltonians (i.e., metry. The theory is illustrated with the examples of a vis- H(q,p)=K(p)+V(q)), and they lead to both accuracy and cous heat conducting fluid (the Navier-Stokes-Fourier sys- efficiency. However, nonseparable Hamiltonians also model tem) and its multicomponent extension including chemical important problems. Unfortunately, implicit methods had reactions and mass transfer. been the only available symplectic approach with long-time accuracy for general nonseparable systems. This talk will Francois Gay-Balmaz construct high-order symplectic integrators for nonsepara- Ecole Normale Superieure ble systems that use explicit updates. These new integra- [email protected] tors are based on a mechanical restraint that binds two copies of phase space together. Using backward error anal- Hiroaki Yoshimura ysis, KAM theory, and some additional multiscale analysis, Waseday University a pleasant error bound is established for integrable sys- [email protected] tems. Numerical evidence of statistical accuracy for non- integrable systems were also observed.

MS153 Molei Tao Interpolation on Symmetric Spaces via the Gener- School of Mathematics, Georgia Institute of Technology alized Polar Decomposition [email protected] Abstract Not Available At Time Of Publication.

Melvin Leok MS154 University of California, San Diego Modulational Stability of Quasiperiodic Solutions Department of Mathematics to Hamiltonian PDEs [email protected] We consider the stability of quasi-periodic solutions to non- MS153 linear dispersive PDEs. We construct a rigorous Whitham Variational Discretization for Fluid-structure In- theory giving stability information about the quasiperiodic teractions solutions in terms of conserved quantities of the evolution evaluated along the manifold of quasiperiodic solutions. Variational integrators for numerical simulations of La- This is joint work with Robert Marangell and Mat John- grangian systems have the advantage of conserving the mo- son. menta up to machine precision, independent of the time step. While the theory of variational integrators for me- Jared Bronski chanical systems is well developed, there are obstacles in University of Illinois Urbana-Champaign direct applications of these integrators to systems involving Department of Mathematics DS17 Abstracts 275

[email protected] the nonlinear Schrodinger equations. In contrast to earlier work, which indicated that there are three eigenvalues near the origin, we prove that an appropriate Evans function has MS154 six roots near the origin. I will elucidate the underlying ge- Application of the Maslov Index to the Stability of ometry and highlight consequences for the stability of the Traveling Waves Nozaki-Bekki holes for the cubic-quintic Ginzburg-Landau equation. This talk concerns the stability of fast traveling waves for the doubly diffusive FitzHugh-Nagumo equation. The Margaret Beck traveling wave system admits a symplectic structure, which Boston University helps to determine the unstable spectrum for the linear op- [email protected] erator L arising from linearizing about the wave. Using a topological invariant called the Maslov index, it is shown Toan Nguyen that there are no eigenvalues for L in the right-half plane, Pennsylvania State University and hence the wave is stable. The calculation of the Maslov [email protected] index (which is often elusive) is facilitated by the presence of multiple timescales in the equations. Bjorn Sandstede Division of Applied Mathematics Paul Cornwell Brown University Department of mathematics bjorn [email protected] University of North Carolina at Chapel Hill [email protected] Kevin Zumbrun Indiana University Chris Jones USA University of North Carolina-Chapel Hill [email protected] [email protected]

MS155 MS154 Unexpected Patterns: Chimera States on Net- Computation and Stability of Patterns in Second works Order Evolution Equations When identical oscillators are coupled together in a net- In this talk we consider nonlinear traveling waves occurring work, dynamical steady states are often assumed to reflect in systems of semilinear damped wave equations. A numer- network symmetries. I’ll show evidence for the existence of ical method which proves to be very useful for the numer- alternative persistent states that break the symmetries of ical approximation of traveling waves and also more gen- the underlying coupling network. These symmetry-broken eral relative equilibria in evolution equations is the freez- coexistent states are analogous to those dubbed ”chimera ing method. We show how the method generalizes to sec- states,” which can occur when identical oscillators are cou- ond order evolution equations. The method introduces pled to one another in identical ways. the waves velocity as a new unknown and transforms the original PDE into a partial differential algebraic equation Daniel M. Abrams (PDAE). Provided the traveling wave of the original PDE Northwestern University has suitable stability properties, we will present a rigor- [email protected] ous theory which shows the main benefit of the freezing method: By solving the Cauchy problem for the PDAE reformulation, not only a solution to the original PDE is MS155 obtained but also a co-moving frame is generated in which Synchronization in Lattices of Interacting Quan- the solution becomes stationary. Moreover, the additional tum Dipoles unknown converges to the velocity of the traveling wave. Numerical examples demonstrate our analytical findings. Arrays of synchronized oscillators are ubiquitous in bio- logical, physical and engineering systems. In this talk I Wolf-Juergen Beyn, Denny Otten will discuss the emergence of synchronization in arrays of Bielefeld University quantum radiating dipoles coupled only via anisotropic and [email protected], [email protected] long-range dipolar interactions. We find that in the pres- ence of an incoherent energy source, dipolar interactions Jens Rottmann-Matthes can lead to a resilient synchronized steady-state. A classi- Department of Mathematics cal mean-field description of the model results in equations Karlsruhe Institute of Technology similar to the classical Kuramoto model for synchroniza- [email protected] tion of phase oscillators. Using the mean-field formulation for the all-to-all coupled case, we study the synchronized state and find that it exists only for a finite range of the MS154 external energy source rates. We compare the results ob- Stability of Nozaki-Bekki Holes Near the Nonlinear tained from the mean-field model with numerical simula- Schrdinger Limit tions of the quantum system and find that synchronization is robust to quantum fluctuations and spatially decaying The Nozaki-Bekki holes are source defects in the cubic coupling. The robustness of additional nonstationary syn- Ginzburg-Landau equation: they are degenerate in the chronization patterns to more accurate descriptions of the sense that they come in a one-parameter family instead quantum system is discussed. of a discrete family. Their stability properties for the un- derlying PDE are not well understood. In this talk, I will Juan G. Restrepo discuss results about their spectral stability in the limit of Department of Applied Mathematics 276 DS17 Abstracts

University of Colorado at Boulder Jie Sun [email protected] Mathematics Clarkson University [email protected] MS155 Control and Optimization of Phase and Chaotic Os- MS156 cillator Networks Analysis of Freeway Traffic Using Koopman Oper- Controlling and optimizing synchronization is vital is ap- ator Methods plications ranging from the power grid to synthetic cell Over the years, there have been various types of mathe- engineering. In this talk we will explore both control and matical models proposed within the field of traffic physics, optimization in the context of phase oscillators and chaotic all with varying degrees of success. The goal of this talk is oscillators, respectively. First, we present a new nonlinear to demonstrate how the Koopman operator theory frame- dynamics-based method for controlling coupled phase os- work can offer a model-free, data-driven method for accu- cillator networks with minimal intervention by identifying rately capturing and predicting traffic dynamics. The effec- a target synchronized state and using its Jacobian to de- tiveness of the proposed framework is demonstrated by an termine which oscillators require control. We also extend application to the Next Generation Simulation (NGSIM) these results to networks of limit-cycle oscillators and ex- data set collected by the US Federal Highway Adminis- plore various control strategies. Second, we consider the tration. The NGSIM data set provides individual vehicle optimization of networks of heterogeneous chaotic oscil- trajectory data for a segment of the US 101 and US 80 lators. In particular, we apply the framework of the syn- Highways. By obtaining a Koopman mode decomposition chrony alignment function - a recently developed technique of the observed data sets, we are able to accurately recon- for optimizing synchronization in networks of phase oscil- struct our observed dynamics, distinguish any growing or lators - to this more complicated case. Using both numeri- decaying modes, and reconstruct a hierarchy of coherent cal simulations and an experimental setup we demonstrate spatiotemporal patterns that are fundamental to the ob- that this framework performs remarkably well. served dynamics. Furthermore, the proposed method can be utilized to predict traffic dynamics by obtaining a de- Per Sebastian Skardal composition of a subset of the data, that is then used to Trinity College predict a future subset of the data. Department of Mathematics [email protected] Allan Avila University of California, Santa Barbara [email protected] MS155 Optimal Synchronization of Complex Networks MS156 Synchronization is central to many complex systems in en- Koopman Operators for Reduced Order Modeling gineering physics (e.g., the power grid, Josephson junction of Nonlinear Dynamical Systems circuits, and electrochemical oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms). Despite these Abstract Not Available At Time Of Publication. widespread applications, for which proper functionality de- pends sensitively on the extent of synchronization, there Karthik Duraisamy remains a lack of understanding for how systems can best University of Michigan Ann Arbor evolve and adapt to enhance or inhibit synchronization. [email protected] We study how network modifications affect the synchro- nization properties of network-coupled dynamical systems MS156 that have heterogeneous node dynamics (e.g., phase os- cillators with nonidentical frequencies), which is often the Koopman Theory, Observables and Sparse Regres- case for real-world systems. Our approach relies on a syn- sion for PDE’s chrony alignment function (SAF) that quantifies the inter- We consider the application of Koopman theory to non- play between heterogeneity of the network and of the os- linear partial differential equations. We demonstrate that cillators and provides an objective measure for a system’s the observables chosen for constructing the Koopman op- ability to synchronize. We conduct a spectral perturbation erator are critical for enabling an accurate approximation analysis of the SAF for structural network modifications to the nonlinear dynamics. If such observables can be including the addition and removal of edges, which sub- found, then the dynamic mode decomposition algorithm sequently ranks the edges according to their importance can be enacted to compute a finite-dimensional approxima- to synchronization. Based on this analysis, we develop tion of the Koopman operator, including its eigenfunctions, gradient-descent algorithms to efficiently solve optimiza- eigenvalues and Koopman modes. Judiciously chosen ob- tion problems that aim to maximize phase synchronization servables lead to physically interpretable spatio-temporal via network modifications. features of the complex system under consideration and provide a connection to manifold learning methods. We Dane Taylor demonstrate the impact of observable selection, including Department of Mathematics kernel methods, and construction of the Koopman opera- University of North Carolina at Chapel Hill tor on two canonical, nonlinear PDEs: Burgers’ equation [email protected] and the nonlinear Schr¨odinger equation. These examples serve to highlight the most pressing and critical challenge Per Sebastian Skardal of Koopman theory: a principled way to select appropriate Trinity College observables. Department of Mathematics [email protected] Nathan Kutz DS17 Abstracts 277

University of Washington firing rate naturally maintains a stable homeostatic equilib- Dept of Applied Mathematics rium with a characteristic mean firing rate; but the condi- [email protected] tions under which multiple slow feedbacks produce a stable homeostatic equilibrium have not yet been explored. We consider a highly general model of homeostatic firing rate MS156 control in which two slow variables provide negative feed- Baroreflex Physiology Using Koopman Mode Anal- back to drive a firing rate towards two different targets. ysis Using dynamical systems techniques, we show that such a control system can stably maintain a neuron’s characteris- We propose new methods for the evaluation of eigenval- tic firing rate mean and variance in the face of perturba- (t,t ) (t,t ) ues λ 0 of the Koopman operator family K 0 , t>t0 tions, and we derive conditions under which this happens. of the non-autonomous dynamic systems. The first step We also derive expressions that clarify the relationship be- in the development is a new data-driven method for very tween the homeostatic firing rate targets and the resulting (t,t0) stable firing rate mean and variance. We show that one accurate evaluation of eigenvalues λ , t>t0 in the hybrid linear non-autonomous case. Then, the approach role that dual homeostasis can serve is to tune a neuronal is extended to continuous linear non-autonomous systems network to act as an integrator. and non-autonomous systems in general. We also propose Jonathan Cannon (t,t0) a relationship between eigenvalues λ , t>t0 and eigen- Brandeis University values computed by Arnoldi-like methods on large sets of [email protected] snapshots. We apply the new approach to baroreflex physi- ology, i.e. to the resonant breathing which is used in PTSD treatment. For the resonant breathing we have both data MS157 and parameterized mathematical model. The model incor- Feedback, Singularities and Singular Perturbations porates a delay and thus is infinite dimensional, hybrid, in Robust Neuromodulation with a stochastic input. Its asymptotic dynamics is close to quasi-periodic. The applied new methods give an im- Neurons are excitable cells whose electrical activity ulti- proved insight to the eigenvalues of the related Koopman mately determine how information is processed through operator family K(t,t0). the brain. In our complex and changing environment, an- imals need to constantly switch between different behav- Senka Macesic ioral states. The most striking and evident changes proba- University of Rijeka, Croatia bly occur at the sleep-wake transitions, and effective neural [email protected] control of these transitions is critical for the fitness and sur- vival of the animal. These behavioral switches are accom- Igor Mezic panied by changes in neural activity patterns in many brain University of California, Santa Barbara areas. A grand challenge of neuroscience is to understand [email protected] how those changes are robustly attained via a plethora of neuromodulators and despite large variability at the net- Maria Fonoberova work, cell, and molecular levels. This talk will introduce a AIMdyn, Inc mathematical framework to tackle this challenge in a con- [email protected] structive way via the joint use of feedback, singularity, and singular perturbation theories. Feedback theory maps neu- ron electrophysiology to multiple timescale feedback mo- Nelida Crnjaric-Zic tifs that capture the qualitative features of neuronal dy- Rijeka University namics. Those motifs are rigorously dissected via singu- [email protected] larity and singular perturbation theories to formulate pre- dictions about the possible activity modes robustly observ- Zlatko Drmac able in neuronal dynamics and the mechanisms underlying University of Zagreb switches between different modes. These theoretical pre- Department of Mathematics dictions are qualitatively mapped back to electrophysiology [email protected] to formulate experimentally relevant predictions about the mechanisms underlying robust neuromodulation. Aleksandr Andrejcuk University of Rijeka, Croatia Alessio Franci [email protected] Universidad Nacional Aut´onoma de M´exico M´exico [email protected] MS157 Dual-Mechanism Firing Rate Homeostasis Stabi- Guillaume Drion lizes Multiple Firing Rate Statistics Neurophysiology Unit and GIGA Neurosciences University of Liege, Liege, Belgium Homeostatic processes that provide negative feedback to [email protected] regulate neuronal firing rates are essential for normal brain function. Indeed, multiple parameters of individual neu- Rodolphe Sepulchre rons, including the scale of afferent synapse strengths and University of Cambridge the densities of specific ion channels, have been observed to Cambridge, UK change on homeostatic time scales in response to chronic [email protected] changes in synaptic input. This raises the question of whether these processes are controlled by a single slow feed- back variable or multiple slow variables. A single home- MS157 ostatic process providing negative feedback to a neuron’s Model Formulation and Experimental Challenges 278 DS17 Abstracts

in Underactuated Neurocontrol ple applications. For N identical particles, even in simplest geometries, such as the sphere in 3D, the computation of Despite recent technological advances, neurostimulation optimal designs that minimize the global energy is a chal- applications typically fall within one of two extremes of lenging problem. When N 1, the situation is further scale: controlling precise spike patterns in small numbers complicated by the rapid growth of local minimum num- of neurons, or driving large ensembles using bulk perturba- bers. The case of nonidentical particles adds further com- tion, such as in deep brain stimulation. However, most the- plexity. As a result, only putative optimal spherical designs ories of neural coding, e.g. in sensory systems, exist at an of N 1 particles are known from global optimization, intermediate scale, where neuron identity carries informa- even for basic potentials. Mathematical proofs of optimal- tion, but overall function arises from large ensembles (e.g. 4 ity of certain arrangements for certain potentials are known ∼ 10 neurons in a cortical column). An essential challenge only for highly symmetric cases. Questions like universal is thus development of neurostimulation strategies at this optimality (existence of a design minimizing a class of en- intermediate scale: independent control of large numbers ergies) generally remain open. In this talk, we present an of ‘individual’ neurons. I will describe our efforts towards algorithm for the computation of N-particle arrangements bridging control theory with in vivo neurophysiology. We on a unit sphere corresponding to local and putative global focus on underactuated control objectives, where the ratio energy minima, for a given pairwise potential. Geometri- of neurons to stimulation channels can be > 100 : 1. We ex- cal properties, such as pairwise distances and coordination ploit neural heterogeneity to design controls for ensembles numbers, are used to characterize, quantify, and distinguish that are naively uncontrollable. We argue that robustness between particle arrangements. Putative globally and lo- may motivate intentional model misspecification. In mouse cally optimal configurations obtained for several different somatosensory cortex recordings, we have begun estimat- pairwise energies, and their properties, are discussed. A ing parameter heterogeneity as applied to control design, companion talk by W. Ridgway focuses on the global-local as distinct from accurate biophysical descriptions. Early optimization numerical algorithm. results suggest a distribution of parameters qualitatively consistent with our model assumptions. Taken together, Alexei F. Cheviakov these results are step towards a unified systems-theoretic Department of Mathematics and Statistics framework for optimal control of dynamics in large neu- University of Saskatchewan ronal networks. [email protected] Jason Ritt Massachusetts Institute of Technology MIT E25-436 MS158 [email protected] Mean First Passage Time with Space-dependent Diffusion Anirban Nandi Washington University, St. Louis We consider the problem of having many traps inside a [email protected] bounded domain where the particle velocity is non-uniform (space-dependent). We ask the following question: how ShiNung Ching should the traps be distributed to optimally trap a ran- Washington University in St. Louis domly moving particle? In the limit of many traps, a mean- [email protected] field description yields some surprising answers.

Theodore Kolokolnikov MS157 Dalhousie University Folded Saddle Nodes: Understanding Transitions [email protected] Between Activity Patterns in a Coupled Neuron Model MS158 The Butera model is a well known model of two synapti- cally coupled neurons in a region of the brain stem known Boundary Homogenization of a Spherical Target as the pre-Botzinger complex, which play a part in gen- and Berg-Purcell Revisited erating breathing rhythms in mammals. In this talk, we will discuss the bifurcations corresponding to the transi- We consider the classic chemoreception problem of capture tions between various bursting and spiking states of the by a structured spherical target with a distribution of small model. In particular, we will investigate a new type of absorbing surface traps. In now seminal work, physicists folded saddle-node bifurcation (FSN III) which arises in Berg and Purcell postulated in 1977 an expression for the the context of slow-fast averaging. capture rate of such a target. From a flux based analysis, they determined that the capture rate should depend on Kerry-Lyn Roberts the combined perimeter of the absorbers. Therefore, for a The University of Sidney fixed absorbing area, a highly fragmented distribution leads [email protected] to a nearly optimal capture flux compared to an all absorb- ing target. In this work, we give a first principles derivation of the classic Berg and Purcell result and also determine its MS158 higher order corrections which give detailed information on Local and Global Optimization of Particle Loca- the spatial configuration of the absorbing locations. The tions on the Sphere: Models, Applications, Math- result is derived in the limit of small absorber radius and ematical Aspects, and Computations gives an explicit discrete intertrap interaction energy. This high order expansion is used to homogenize the boundary Ordered arrangements of particles on the boundary of a of the target and determine an effective Robin boundary domain in two or three dimensions, with potential interac- condition for the target in the limit of large trap numbers. tion forces depending on pairwise distances, arise in multi- Our theoretical results are verified with a spectral bound- DS17 Abstracts 279

ary element method. of blow-up and the faithful reproduction of the main statis- tical properties of the full, high-dimensional model by an Alan E. Lindsay EMR-MSM in two illustrative cases: (i) a conceptual, non- Applied Computational Mathematics and Statistics linear, stochastic climate model of coupled slow and fast University of Notre Dame variables, in which only slow variables are observed; and [email protected] (ii) a partially observed, generalized Lotka-Volterra model of population dynamics in its chaotic regime. Michael Ward Department of Mathematics Michael Ghil University of British Columbia Ecole Normale Superieure, Paris, and [email protected] University of California, Los Angeles [email protected] Andrew J. Bernoff Harvey Mudd College Mickael Chekroun Department of Mathematics University of California, Los Angeles [email protected] Department of Atmospheric and Oceanic Sciences [email protected]

MS158 Dmitri Kondrashov, Andreas Groth Numerical Computation of Locally and Globally Department of Atmospheric and Oceanic Sciences Energy-Minimizing Spherical Designs University of California, Los Angeles [email protected], [email protected] Multiple applications, including ones in material science, solid state physics, and cell biology, involve ordered ar- rangements of particles on the surface of a sphere. Ba- MS159 sic examples include trap arrangements in the narrow es- Representing Time Series with Uncertainties and cape problem for a Brownian particle in biophysics, and Identification of Abrupt Transitions the study of surface defects in spherical crystals. In many situations, the ordered arrangements are local or We put forward a novel representation for observed time global minima of a pairwise potential energy. A system- series: in lieu of considering measurements as a point-like atic search of energy-minimizing configurations, even for estimate with concomitant error, we propose to consider a simple pairwise potential and identical particles on the them as a sequence of time-ordered probability densities. sphere, presents a challenging computational problem. In In our framework, uncertainties of the datasets—arising this talk we present an efficient optimization algorithm for either from spatial or temporal variations, or from mea- the computation of N-particle energy minima from known surement imprecision—are inherently quantified from the (N −1)-particle energy minima. We discuss the implemen- beginning of the analysis and are carried over throughout tation of local and global optimization, exclusion of saddle subsequent analyses of the data as well. We show how a points, convergence properties, and other aspects of the al- matrix of recurrence probabilities can be estimated ana- gorithm. Results are presented for different choices of the lytically from such a sequence of time-ordered probability pairwise potential energy function. This is a joint work densities, and how such a measure of the recurrences of the with Alexei Cheviakov. system can then be used to infer sudden dynamical changes in the dataset. We demonstrate our approach with three Wesley J. Ridgway kinds of real-world examples: paleoclimatic data, historic Department of Mathematics. climatic data, and financial data. Our results capture sev- University of Saskatchewan eral well-known paleoclimatic and historic climatic events, [email protected] and well-known financial crashes, while also revealing sev- eral other events in the data which are understood to a lesser degree. MS159 Nonlinear Stochastic Models for Irregular Time Se- Bedartha Goswami ries: Empirical Model Reduction and Multilayer Potsdam Institute for Climate Impact Research Stochastic Models Potsdam Institute for Climate Impact Research [email protected] We derive stable and efficient data-based models for simu- lation and prediction in the sciences (Kondrashov et al., Aljoscha Rheinwalt Physica D, 2015). The proposed low-order models use Institute for Earth and Environmental Science a multivariate time series of partial observations from a University of Potsdam large-dimensional system; they are compared with the op- [email protected] timal closures provided by the Mori-Zwanzig (MZ) formal- ism of statistical physics. Our multilayer stochastic mod- els (MSMs) generalize existing multilevel, regression-based Niklas Boers approaches to data-based closure, in particular empirical Laboratoire de Meteorologie Dynamique model reduction (EMR). We show that the MSMs’ multi- Ecole Normale Superieure, Paris layer structure can provide a natural Markov approxima- [email protected] tion to the generalized Langevin equation (GLE) of the MZ formalism. A simple criterion for an EMR-MSM model is Norbert Marwan, Jobst Heitzig derived to assess how well it approximates the GLE so- Potsdam Institute for Climate Impact Research lution. Sufficient conditions are given to guarantee the [email protected], [email protected] existence of a global random attractor for a given MSM. This existence ensures that no blow-up can occur for a very Sebastian Breitenbach broad class of MSM applications. We confirm the absence Institute of Geology, Mineralogy & Geophysics 280 DS17 Abstracts

Ruhr-Universitaet Bochum complete codes for the sequence of amplitude orderings. [email protected] Michael Small Juergen Kurths The University of Western Potsdam Institute for Climate Impact Research Australia - WSA [email protected] [email protected]

Michael McCullough, Konstantinos Sakellariou, Thomas MS159 Stemler, David Walker University of Western Australia Determining the Kolmogorov-Sinai Entropy from [email protected], kon- Ordinal Patterns [email protected], [email protected], [email protected] Quantifying the complexity of a time series is an impor- tant subject of research, especially if this helps distinguish- ing deterministic chaotic signals from stochastic ones. The MS160 Kolmogorov-Sinai entropy is, in principle, the right indi- Assessing the Contribution of Distinct Vascular cator to look at, but obtaining reliable estimates is quite Segments in Peripheral Arterial Disease problematic even in the case of relatively low-dimensional chaos. I show that a relevant step forward can be made, by Peripheral arterial disease (PAD) is a major health prob- combining the analysis of ordinal patterns with the mea- lem in which systemic arteries become occluded, leading surement of the width of the corresponding cylinder sets. to a significant reduction in blood supply to tissue. There Many years ago, Bandt and Pompe suggested to encode is an ongoing controversy regarding the potential efficacy finite sequences of data as ordinal patterns, by assigning of PAD therapies that rely on new vessel formation versus at each datum within a given window its relative order the dilation of existing vessels to restore tissue perfusion. (largest, second largest, and so on). The corresponding en- The theoretical model presented here provides a method for tropy, called permutation entropy, is often used as a proxy resolving this controversy by identifying the relative contri- for the Kolmogorov-Sinay entropy. However, even in the butions of short- (acute) and long-term (chronic) vascular simple, one-dimensional, logistic map, strong deviations adaptations to the collateral arteries and distal microvas- are found between the two quantities for the numerically culature following a major arterial occlusion. A computa- accessible window lengths. I propose to look at the disper- tional hemodynamic model analogous to an electrical cir- sion among all trajectories characterised by the same or- cuit is developed in which vessel compartments are defined dinal pattern: a suitably modified ”relative” permutation as resistors connected in both series and parallel accord- entropy, which takes this information into account allows ing to the observed network geometry of the rat hindlimb. for by far much more accurate estimates even in dynamical The model is used to test the sensitivity of blood flow to systems with multiple positive Lyapunov exponents. The changes in vessel number or diameter following an abrupt, method is illustrated by discussing a few different models, total, and sustained femoral arterial occlusion. The model of increasing complexity. shows that an occlusion shifts the site of primary vascu- lar resistance from arterioles to collateral arteries and that Antonio Politi new vessel formation independent of collateral growth has University of Aberdeen, UK little to no effect on restoring tissue perfusion. The model [email protected] is also used to predict the degree of collateral dilation re- quired to achieve experimentally observed flow values in the rat hindlimb following an occlusion under rest and ex- MS159 ercise conditions. From Symbolic Dynamics to Ordinal Partition Net- Julia Arciero works: Constructing Proxies of the Dynamical Sys- Department of Mathematical Sciences tem from Observed Time Series Indiana University-Purdue University Indianapolis [email protected] We consider methods to extract features of a dynamical system Φ : M→Mobserved via a instantaneous scalar measurement function h : M→R. Of course, the gold- MS160 standard of such methods is delay reconstruction via an Modeling, Analysis, and Experiment Design for invocation of Takens’ embedding theorem. However, such Syk-Mediated Signalling Events in B Cells approaches are severely challenged by finite data observa- tional and dynamical noise, and excessively large sampling B cells play an important role in immune system response interval. Instead, we focus on a class of methods that re- and are activated by a complex chain of signalling events, construct a network representation of the underlying dy- either with or without mediation by T cells. Depending on namical system via and ordinal encoding of successive ob- the level and type of binding events associated with a B served time series points. That is, for a chosen window size cell, it may develop in one of several ways; understanding w, we replace the observations (xt,xt+τ ,...,xt+(w−1)τ ) the mechanism behind the associated activation or inac- with a permutation of the integers 1, 2, 3,...,w which re- tivation is an important component of understanding im- flects the order of the original points. In this talk we will mune response more generally. The kinase Syk plays a cen- describe why this is a good thing to do, both theoreti- tral role in early signaling events in B cells and is required cally, and from the pragmatic objective of application. We for proper response when antigens bind to B cell receptors will demonstrate the efficacy of the approach and introduce (BCRs). Experiments using an analog-sensitive version of new extensions of the method to the situation where the Syk (Syk-AQL) have better elucidated its role, but have permutation window size w may be variable. To do this, not completely characterized its behavior. We present a we consider the permutation windows as code words and computational model for BCR signaling, using dynamical apply standard encoding schemes to construct efficient and systems, which incorporates both wild-type Syk and Syk- DS17 Abstracts 281

AQL. Following the use of sensitivity analysis to identify subsequent spread of bacteria leads to active TB disease. significant reaction parameters, we screen for parameter It is difficult to study the activation process in humans vectors that produce graded responses to BCR stimulation because those with latent TB are asymptomatic and are as is observed experimentally. We demonstrate qualita- often undiagnosed. Current animal models all have lim- tive agreement between the model and dose response data itations. Several previous mathematical models describe for both mutant and wild-type kinases. The model pro- the infection or granuloma formation stages of TB. No ap- vides suggestions for how manipulation of Syk-AQL can proach considers the dynamics of matrix metalloproteinase be used to modulate the downstream response associated 1 (MMP-1) regulation and its impact on TB activation. with BCR stimulation. MMP-1 dysregulation has been implicated in TB activa- tion experimentally, but the mechanism is not well under- Reginald Mcgee stood. The overall objective of the study is to predict TB The Ohio State University activation dynamics in response to MMP-1 dysregulation. [email protected] We will discuss a mathematical model to test the hypothe- sis that the dynamics of MMP-1 regulation play a key role Gregery Buzzard in the transition from latent TB to active TB. Purdue University [email protected] Steven Ruggiero, Minu Pilvankar, Ashlee N. Ford Versypt Oklahoma State University [email protected], [email protected], MS160 [email protected] Multi-scale Model of Breast Cancer Predicts Re- sponse to Anti-angiogenic Therapies MS161 Angiogenesis, the formation of new blood capillaries from Subdiffusion with Reactions and Forces pre-existing vessels, is a hallmark of cancer. Thus far, strategies for reducing tumor angiogenesis have focused The mathematics of subdiffusion has been well developed on inhibiting pro-angiogenic factors, while less is known over the past two decades with the governing equations about the therapeutic effects of mimicking the actions of obtained from continuous time random walks, stochastic angiogenesis inhibitors. Thrombospondin-1 (TSP1) is an modelling and fractional calculus. In this talk I will go important endogenous inhibitor of angiogenesis that has through a derivation of the generalized master equation, been investigated as an anti-angiogenic agent. TSP1 im- including the diffusion limits, for an ensemble of particles pedes the growth of new blood vessels in many ways, in- undergoing reactions and subject to external forces, in a cluding crosstalk with potent pro-angiogenic factors such subdiffusive environment. I will also show how the model as vascular endothelial growth factor (VEGF). Given the equations can be solved numerically by revisiting the diffu- complexity of TSP1 signaling, a predictive systems biology sion limit equations through a discrete time random walk model would provide quantitative understanding of the an- formalism. Some open problems and future directions will giogenic balance in tumor tissue. Therefore, we have de- also be discussed. veloped a molecular detailed model of interactions between TSP1 and VEGF. The model predicts the distribution of Christopher N. Angstmann the angiogenic factors in tumor tissue and throughout the UNSW Australia body, providing insight into the angiogenic balance (and [email protected] its disruption) in cancer. We utilize the model to simulate administration of exogenous TSP1 mimetics, alone and in Bruce I. Henry combination with VEGF-targeting agents. The model pre- School of Mathematics and Statistics dicts the ratio of receptor-bound VEGF to receptor-bound University of New South Wales TSP1 under various tumor microenvironmental conditions, [email protected] identifying the ideal tumor profile that responds to these therapeutic strategies. Our model provides a quantitative framework to study the effects of anti-angiogenic treat- MS161 ment. Slow Subdiffusion-Reaction Equation Stacey Finley Johns Hopkins University We will consider the slow-subdiffusion process with reac- Department of Biomedical Engineering tions of type A + B → B in which particles A are assumed sfi[email protected] to be static whereas B are assumed to be mobile. Slow subdiffusion occurs when the Laplace transform of func- tion ω(t) is a slowly varying function. The function ω(t) MS160 is a probability distribution of time which is needed for Dynamic Modeling of Tuberculosis Granuloma Ac- a particle to take its next step. Starting with the differ- tivation ence equation, which describes a random walk of particle in a system with reactions, using the generating function Tuberculosis (TB) is one of the deadliest infectious dis- method and the continuous-time random walk formalism, eases. TB is spread by inhaling Mycobacterium tuberculo- we will derive the slow-subdiffusion equation. We will find sis. The bacteria are attacked by the immune system in the its solution (Green’s function) over a long time limit. Dif- lungs. Aveolar macrophages cluster the bacteria into cellu- ference between the Green’s function which describes ’stan- lar aggregates called granulomas. Granulomas can contain dard’ subdiffusion with reactions and the Green’s function the bacteria in a dormant state for long periods of time, which describes slow subdiffusion with reactions will also even decades, in a condition called latent TB. However, the be discussed. bacteria persist and can be activated when the granulomas are compromised by other immune response events in a Tadeusz Kosztolowicz host, such as cancer, HIV, or aging. The activation and Institute of Physics, Jan Kochanowski University 282 DS17 Abstracts

[email protected] Flows That Includes Internal Waves

There is increasing reliance on autonomous vehicles for MS161 ocean observations. These vehicles are particularly sen- Diffusion and Walks in an Expanding Medium sitive to submesoscale currents and processes. Of special concern are buoyancy effects, vertical motions arising from A classical-like derivation of the Fokker-Planck equation internal waves and horizontal currents with significant ver- for diffusion in an expanding medium is presented. Our tical shear. It is important then to have reliable models starting point is a conveniently generalized Chapman- of these processes so as to study the responses of these Kolmogorov equation. We obtain analytical expressions vehicles in realistic oceanic conditions. General circula- for the Green’s functions, their moments, and first-passage tion models can capture many of these processes, but they properties in expanding hyperspherical geometries. The are computationally expensive and many do not include behavior of these quantities is largely determined by the a prognostic vertical velocity. Here we report on some quantity we name Brownian conformal time τ(t) defined recent simulations with a simple three-dimensional model byτ ˙ =1/a2,wherea(t) is the expansion scale factor of that includes both stratification and rotation. The analysis the medium. We explicitly consider the cases of power-law includes assessments of transport pathways and in partic- expansion and exponential expansion. For the power-law ular how they depend on the interaction of submesoscale case we find interesting crossover effects in the mixing ef- and mesoscale. fectiveness of the diffusion process depending on the value of the power-law exponent. The specific case of an expo- Henry Chang, Helga S. Huntley, Denny Kirwan nential contracting medium is also discussed. University of Delaware [email protected], [email protected], [email protected] Santos B. Yuste Universidad de Extremadura [email protected] MS162 Quantifying Material Transport in Geophysical Enrique Abad Flows Dpt. Fisica Aplicada. Universidad de Extremadura We consider laboratory and numerical experiments on [email protected] time-dependent multi-gyre flows that provide improved un- derstanding of Lagrangian transport. This work presents Carlos Escudero current efforts in understanding the impact of geophysical Departamento de Matematicas fluid dynamics on underwater vehicle control and auton- Universidad Autonoma de Madrid omy. The focus of the talk is on the use of collaborative [email protected] vehicles to perform a variety of sensing tasks.

Felipe Le Vot Eric Forgoston Dpto. Fisica Montclair State University Universidad de Extremadura Department of Mathematical Sciences [email protected] [email protected]

Ani Hsieh MS161 Drexel University Levy Walks: from Lineland to Higher Dimensions [email protected] The L´evy walk model has two main features distinguishing Philip Yecko it from all other random walks: a finite velocity of moving Cooper Union particles, and the length of displacements that is power-law [email protected] distributed. The model proved to be very successful in de- scribing a large variety of superdiffusive phenomena across disciplines [Zaburdaev et al., Rev. Mod. Phys. 2015]. We will discuss the main properties of the model with an em- MS162 phasis on practical applications. We will see how the basic Stochastic Parametrization in Wave Breaking and model can be modified and extended to better match the Weathering Processes challenges posed by complex real-life systems. One of the recent advances in this field is the generalisation of L´evy A stochastic parametrization for breaking dynamics of wa- walks to two and three dimensions, where most of observed ter waves is proposed. The parametrization derives from real-life dispersals occur. In higher dimensions, the L´evy the analysis of Lagrangian particle paths, from computed walk model demonstrates yet another difference to classi- and laboratory data. The Langevin dynamics combines a cal random walks in that the geometry of the underlying drift term largely informed by deterministic dynamics and random walk model is imprinted into the particle density a diffusion process that is based upon Matern processes. profile at arbitrary long times. The data is further analyzed using ellipse ridge analyses, yielding a compact representation of this complex dynam- Vasily Zaburdaev ics as well as the clean calculation of the residual flow from Max-Planck-Institute for the Physics of Complex Systems progressive multichromatic waves. A long term aim of this [email protected] work is to derive sharp estimates of the dissipation due to breaking on waves and currents, a fundamental agent of momentum transfer between oscillatory and mean flows. MS162 Preliminary results of the projection of dissipation in the A Stratified Three Dimensional Model for Ocean Lagrangian frame to the Eulerian frame makes this esti- DS17 Abstracts 283

mate practical in applied oceanography. trast increases. To elucidate the mechanisms underlying these phenomena, we implemented essential findings from Juan M. Restrepo different experiments in the setup of our large-scale neu- Oregon State University ronal network simulation. And we successfully reproduced Department of Mathematics the contrast-dependent phenomena and unveiled its mech- [email protected] anism respectively in simulation and its analysis. Other important populational properties are also consist with ex- Jorge Ramirez perimental findings. Universidad Nacional de Colombia [email protected] Wei Dai Shanghai Jiao Tong University Institute of Natural Science MS162 [email protected] Geophysical Transport Structure, Reduced Order Modeling, and Connections with Field Experi- ments MS163 Individual and Population Models of Insect Olfac- Lagrangian techniques for revealing the most influential tion transport structures in geophysical flows are poised to make a significant impact on the prediction of material When a locust detects an odor, the stimulus triggers a se- transport, with application to the assessment of hazards ries of synchronous oscillations of the neurons in the anten- which spread in the ocean and atmosphere. However, more nal lobe, followed by slow dynamical modulation of the fir- work is necessary to bridge the gap between powerful the- ing rates. I model this behavior using an Integrate-and-Fire oretical concepts and practical field experiments. Here neuronal network with excitatory and inhibitory neurons, we discuss some results relevant for real-time analysis and each with a fast and slow inhibitory conductance response. prediction of geophysical transport structures which have I derived a coarse-grained model for each (excitatory and arisen from field applications. We discuss the incorporation inhibitory) neuronal population, which allows for more de- of Lagrangian structure into reduced order modeling of geo- tailed analysis of the olfaction mechanisms. physical flows and discuss challenges involved in forecast- ing atmospheric Lagrangian coherent structures, including Pamela B. Pyzza the effect of spatial averaging and ensemble averaging. We Ohio Wesleyan University also consider the local approximation of the most influen- Dept. of Mathematics and CS tial material surfaces via tracer release or monitoring of [email protected] meteorological gradients. Gregor Kovacic Shane D. Ross Rensselaer Polytechnic Inst Virginia Tech Dept of Mathematical Sciences Engineering Mechanics program [email protected] [email protected] David Cai MS163 Courant Institute for Mathematical Sciences, NYU Shanghai Jiao-Tong University Oscillations in a Parkinsonian Network [email protected] Muscle rigidity associated with Parkinsons disease (PD) is thought to be correlated with the loss of dopamine and MS163 the emergence of beta oscillations in the basal ganglia. As dopamine-producing neurons die off, synaptic connection High Dimensional Two-sample Test on Neuronal strengths change leading to favoring of specific pathways. Data We first modeling average firing rates of a healthy basal ganglia using a systems-based approach. A PD basal gan- Statistical methods typically require independent samples. glia is then modeled using the system’s stability. Together To apply recently-developed statistical methods to perform these models can show disease progression and which con- high dimensional two-sample test in neuroscience, we pro- nections are the most influential for this change - a ques- pose an efficient method to pretreat neuronal data, so sam- tion that is still under debate in the scientific community. ples become independent. This is achieved by transferring These findings are then applied to particular therapeutic correlations between samples into those between dimen- situations, hoping to find an understanding for how they sions. Our results show that the discriminability of high work. dimensional statistical method under our pretreatment is much better than methods that use only a small number Michael Caiola of neurons in a short recording. Emory University [email protected] zhiqin J. Xu Department of Mathematics and Institute of Natural Sciences MS163 Shanghai Jiao Tong University Dynamics Underlying the Orientation Selectivity [email protected] in Mouse V1

Experiments in mouse V1 have shown that excitatory neu- MS164 rons exhibit sharpening orientation selectivity(OS) while Alternans in Action Potential Amplitude and Not inhibitory neurons show broadening OS as the input con- in Duration As a Mechanism for Conduction Block 284 DS17 Abstracts

and the Initiation of Fibrillation [email protected]

It is widely believed that one of the major life-threatening transition to chaotic fibrillation occurs via spiral-wave MS164 breakup that is preceded by spatiotemporal dispersion of Adaptive Control of Chaotic Dynamics in Excitable refractoriness due to alternations in the duration of the Media cardiac action potential (AP). However, recent clinical and experimental evidence suggests that other characteristics of the AP may contribute to, and perhaps drive, this dan- During cardiac arrhythmias like ventricular fibrillation the gerous dynamical instability. To identify the relative roles spatial-temporal dynamics of the heart determined by spi- of AP characteristics, we performed experiments in rabbit ral or scroll waves is highly chaotic. Observations in exper- hearts under conditions to minimize AP duration dynam- iments and simulations show, that the chaotic behavior is ics which unmasked pronounced AP amplitude alternans not necessarily persistent, but may also be transient. Here just before the onset of fibrillation. We used a simplified we investigate in numerical simulations how the average ionic cell model to derive a return map and a stability con- lifetime of such a chaotic transient can be reduced passively dition that elucidates a novel underlying mechanism for by changing the system properties of 2D and 3D systems AP alternans and spiral breakup. We found that inacti- and discuss the possibilities of how to actively reduce the vation of the sodium current is key to develop amplitude transient time significantly (termination) by recent control alternans and which is directly connected to conduction strategies. block and initiation of arrhythmias. Simulations in 2D in which AP amplitude alternation led to turbulence confirm Thomas Lilienkamp our hypothesis. Our results suggest novel approaches for Max Planck Institute for Dynamics and Self-Organization preventing the dangerous transition to fibrillation. Biomedical Physics Research Group [email protected] Diandian Diana Chen Georgia Institute of Technology Stefan Luther, Ulrich Parlitz School of Physics Max Planck Institute for Dynamics and Self-Organization [email protected] Research Group Biomedical Physics [email protected], [email protected] Flavio H. Fenton Georgia Institute of Technology fl[email protected] MS164 Closed-Loop Feedback Control of Cardiac Alter- nans in the Heart MS164 Previous studies have established a link between alternans, Modeling Framework to Delineate Optimized Drug a beat-to-beat alternation in cardiac action potential dura- Action for Atrial Spiral Wave Termination tion (APD), and ventricular arrhythmias, suggesting that elimination of alternans could lead to the prevention of Atrial fibrillation (AF) is the most common type of arrhythmias in the heart. Over the past years, several at- sustained reentrant cardiac arrhythmia. Antiarrhythmic tempts have been made to control alternans. Some at- drugs for the modulation of the cell nonlinear voltage dy- tempts involved the control of APD or diastolic interval namics have long been used to terminate AF reentrant (DI) in isolated cells based on the feedback from a micro- activity. However, limited drug efficacy and the risk of electrode recordings. In addition, the majority of control important complications lead to decrease usage. Thus, approaches target basic cycle length (BCL) while assum- there is a strong interest in the development of AF-selective ing periodic pacing: APD+DI=BCL. Recently, we demon- antiarrhythmic drugs. Mathematical modeling of cardiac strated, using single cell numerical simulations, that elim- nonlinear models and drug action is central to this opti- inating the dependence of the DI on the preceding APD, mization. Careful identification of pharmacodynamic pa- i.e. maintaining a constant DI pacing, prevents the for- rameters optimized for selective actions on atrial tissue mation of alternans. However, implementation of constant during AF is central to the method. Such an approach DI pacing requires the development of a sophisticated con- also opens the way to multiple-channel blockade therapy trol system that can sense, track and measure APD and (sodium and potassium channel block for example). The DI in real-time. Here, we extended and validate the anti- ultrarapid potassium current (IKur) is an interesting tar- arrhythmic benefits of constant DI pacing in 1D cables of get for atrial-specific block as it is found in atria but not in canine and human cardiac cells. In addition, we devel- ventricles, thus decreasing the ventricular arrhythmia risk. oped a 2D closed-loop feedback control system that en- To date, an important step in the method is to evaluate abled the detection of APDs and providing real time con- the steady-state changes induced to the action potential stant DI pacing from various single pixels based on optical and conduction velocity. Here, we show that this approach mapping recordings from ex-vivo rabbit heart in real-time. can be limited as only small changes in APD are found for Our method incorporates beat-by-beat DI control and pro- IKur block while important transient variations in APD vides the spatiotemporal flexibility to target alternans in are linked in simulated 2D reentry termination. Roughly the whole heart. modeling the pharmacodynamic of the agent by addition of the time-constant in effective block highlight its impor- Alena Talkachova, Kanchan Kulcarni tance on the termination of reentrant activity. University of Minnesota [email protected], [email protected] Philippe Comtois Institute of Biomedical Engineering, Sharon Zlochiver Universite de Montreal Tel-Aviv University DS17 Abstracts 285

[email protected] not correspond to those generated by classical ecological models. A hypothesized explanation is that more than one algal species is present in the chemostat. We assess the sta- MS165 tistical evidence for this claim in terms of three potential Structural and Practical Identifiability of Stochas- causes of lack of fit for standard differential equation mod- tic, Mechanistic Cancer Models els for this system: (i) unmodeled stochastic forcing, (ii) mis-specified functional forms and (iii) mis-specified state Multistage clonal expansion (MSCE) models, a class of variables. Here the proposed explanation of multiple al- mechanistic, continuous-time Markov models, are com- gal species corresponds to hypothesis (iii). Tests between monly used to connect population-level cancer incidence these hypotheses are achieved by representing lack of fit to the putative underlying biology of carcinogenesis by pa- in terms of time-varying parameters and assessing the re- rameter estimation. Identifiability, then, tells us what as- lationship between these time varying parameters and ex- pects of the model we can make inferences about. Few isting state variables with statistical significance assessed tools are available to assess the identifiability of stochas- through bootstrap and permutation methods. While our tic models, but, because age-specific cancer incidence data tests suggest that observed dynamics are well-matched by corresponds to the model hazard, we can leverage the multiple-species models, alternative causes for lack of fit Kolmogorov equations for the probability generating func- cannot be ruled out. We conclude with an examination tions and use the differential algebra approach to assess of the use of control theory to design inputs into chemo- the structural identifiability of this class of models, an ap- stat systems to improve parameter estimation and power proach that can be used for time-to-event stochastic models to detect missing components. more generally. However, this approach assumes available data can identify features along the entire trajectory of Giles Hooker the model hazard, and certain salient features appear only Biostatistics and Computational Biology Department several orders of magnitude beyond the scale of a human Cornell University lifespan. We use a profile-likelihood approach to show that [email protected] this practical limitation restricts our inference to three, biologically-relevant parameter combinations, regardless of the structural identifiability of the MSCE model used. In particular, for each model we can identify only the product MS165 of the initiation rates, the net cell proliferation rate, and Identifiability of Linear Compartmental Models the scaled malignant conversion rate.

Andrew Brouwer Parameter identifiability analysis addresses the question of University of Michigan which unknown parameters of a model can be determined [email protected] from given input-output data. In this talk, we discuss structural identifiability analysis, which addresses whether or not the model parameters can be determined from per- MS165 fect input-output data (noise-free and of any duration re- Identifiability and Parameter Estimation in Mod- quired) and is an important step in the parameter estima- eling Biological Dynamics tion problem. Many linear ODE models used in systems biology are unidentifiable, which means that parameters Connecting dynamic models with data to yield predictive can take on an infinite number of values and yet yield the results often requires a variety of parameter estimation, same input-output data. We study a particular class of identifiability, and uncertainty quantification techniques. unidentifiable models and find conditions to obtain identi- These approaches can help to determine what is possible fiable reparametrizations of these models. In particular, we to estimate from a given model and data set, and help use a graph-theoretic approach to analyze the models and guide new data collection. In this talk, we will discuss show that graphs with certain properties allow a monomial approaches to both structural and practical identifiability scaling reparametrization over identifiable functions of the analysis. Using a range of examples from Hodgkin-Huxley parameters. We also examine conditions to obtain identifi- models, to cholera spread, to cancer, we illustrate some of ability for this class of models, and in particular, show how the potential difficulties in estimating the relative contri- identifiability can be determined by simply looking at the butions of different pathways, and show how alternative graphical structure of these linear compartmental models. data collection may help resolve unidentifiability. We also illustrate how even in the presence of large uncertainties in the data and model parameters, it may still be possible to Nicolette Meshkat successfully forecast the system dynamics. North Carolina State University [email protected] Marisa Eisenberg University of Michigan [email protected] MS166 Central Configurations of the N-Body Problem MS165 with Generalized Potentials Goodness of Fit in Differential Equation Models: Misspecified Rates Or Misspecified States? The behavior of some central configurations in the N-body This talk considers the statistical evidence for evolution problem are investigated when the central potential expo- in laboratory-based ecological experiments. We examine nent is changed. data from a chemostat experiment in which algae are grown on nitrogen-rich medium and rotifers are introduced as a Marshall Hampton predator. The resulting data exhibit dynamics that do University of Minnesota, Duluth 286 DS17 Abstracts

[email protected] Wilfrid Laurier Univerisity [email protected]

MS166 Existence, Stability, and Symmetry of Relative MS167 Equilibria with a Dominant Vortex Absolute Instabilities of Travelling Waves Solutions in a Keller-Segel Model We will consider relative equilibria in the n-vortex problem with one dominant vortex and N much smaller vortices. In this talk we investigate the spectral stability of trav- The dimension of the problem can be reduced by taking elling wave solutions in a Keller-Segel model of bacterial an infinitesimal circulation limit, resulting in the (1 + N)- chemotaxis with a logarithmic chemosensitivity function vortex problem. In this talk we generalize a previously and a constant, sublinear, and linear consumption rate. known reduction to allow for circulations of varying signs We locate the essential and absolute spectrum of the as- and weights, and then focus on the case where N =3. sociated linear operators and find that all travelling wave Here we find conditions for symmetry in the relative equi- solutions have essential spectrum in the right half plane. libria configurations, and consider the stability of the con- However, we show that in the case of constant or sublin- tinuations from the infinitesimal circulation case back to ear consumption there exists a range of parameters such nonzero, small circulation. Moreover, we show that there that the absolute spectrum is contained in the open left are stable asymmetric relative equilibria. Techniques from half plane and the essential spectrum can thus be weighted algebraic geometry are used to count families of relative into the open left half plane. For the constant and sub- equilibria at different circulation values. linear consumption rate models we also determine critical parameter values for which the absolute spectrum crosses Alanna Hoyer-Leitzel into the right half plane, indicating the onset of an absolute Mathematics Department instability of the travelling wave solution. We observe that Bowdoin College this crossing always occurs off of the real axis. This work [email protected] is done in conjunction with Peter van Heijster and Robert Marangell. MS166 Paige Davis Morse Theory and Stability of Relative Equilibria Queensland University of Technology in the Planar N-Vortex Problem [email protected] The planar n-vortex problem is a Hamiltonian system of differential equations that effectively approximates vortic- MS167 ity evolution in fluid dynamics. Since collisions between point vortices must be excluded, the topology of the un- Determining the Critical Spectrum About the Ori- derlying configuration space is nontrivial and Morse theory gin Using Lin’s Method can be applied to yield interesting results. We identify an Pattern solutions often arise as perturbations from a sim- explicit connection between the Morse index of a critical pler limit structure. One can think of a family of periodics point of the Hamiltonian restricted to a level surface of the that accompany a homoclinic pulse, multi-pulse solutions angular impulse and the eigenvalues of the corresponding bifurcating from a primary pulse or patterns arising from a relative equilibrium solution (a rigidly rotating configura- concatenation of slow and fast orbit segments in singularly tion). The Morse inequalities are then used to classify the perturbed problems. Nonlinear stability of these patterns stability (or instability) of all relative equilibria for some is in many cases decided by the spectral properties of the particular cases in the four-vortex problem. An overview of linearization about the pattern. Exploiting the limit struc- topological results and open questions related to this work ture often leads to asymptotic control over the spectrum. will also be provided. However, asymptotic spectral control is insufficient to de- Gareth E. Roberts cide upon stability if there is non-trivial spectrum shrink- Dept. of Mathematics and C.S. ing to the origin in the asymptotic limit. In such cases a College of the Holy Cross higher order analysis is necessary to determine the spec- [email protected] tral geometry about the origin. This can be achieved using Lin’s method.

MS166 Bj¨orn De Rijk Remarks on the Central Configurations of Four University of Stuttgart Bodies Institut f¨ur Analysis, Dynamik und Modellierung [email protected] The study of the central configurations in the n-body prob- lem is one important area of Celestial Mechanics. The finiteness of central configuration was listed by Stephen MS167 Smale as problem 6 on his list of problems for this century. Stability of Fronts in a Diffusive Model for Porous In the four-body case the finiteness was proved by Marshall Media Combustion Hampton and Rick Moeckel in 2006. There are, however, many things we still need to understand in the case of four We consider a model of combustion in hydraulically resis- bodies. One well known conjecture by Alain Albouy states tant porous media. There are several reductions of this that there is only one convex central configuration for each systems that can be used to understand the evolution of cyclic ordering of the masses. In this talk we will revisit the combustion fronts. In this presentation, we discuss some results for the four-body problem by extending them this model with the Lewis number chosen in a specific way. and presenting new proofs. First, we, in addition, consider initial conditions of a spe- cific form. We then show that the stability results for that Manuele Santoprete system extend to the fronts in the full system with the same DS17 Abstracts 287

Lewis number. The fronts are either absolutely unstable Stefano Boccaletti or convectively unstable. CNR-Istituto dei Sistemi Complessi [email protected] Anna Ghazaryan Department of Mathematics Ivan Bonamssa Miami University Bar Ilan University, Israel [email protected] [email protected]

Stephane Lafortune College of Charleston MS168 Department of Mathematics Chimera States in Continuous Media [email protected] The defining property of chimera states is the coexistence Peter McLarnan of coherent and incoherent domains in symmetric coupled Miami University systems. The recent realization that such states might [email protected] be common in oscillator networks raises the question of whether an analogous phenomenon can occur in continu- ous media. In this presentation, I will show that chimera MS167 states can exist in continuous systems even when the cou- pling is strictly local, as in many fluid and pattern forming Coherent Structures in Run-and-Tumble Processes media. Using the complex Ginzburg-Landau equation as a model system, we characterize chimera states consisting of Motivated by the observation of rippling patterns in a coherent domain of a frozen spiral structure and an inco- colonies of Myxobacteria, we study a “simplest’ modes for herent domain of amplitude turbulence. We show that in dynamics of populations, where agents run either left or this case, in contrast with discrete network systems, fluc- right with constant speed, and reverse direction depend- tuations in the local coupling field play a crucial role in ing on local population densities in a nonlinear fashion. limiting the coherent regions. We suggest these findings Linear analysis and simulations reveal how shot noise per- shed light on new possible forms of coexisting of order and turbations can give rise to rippling behavior and predict disorder in fluid systems. wavelengths. The talk will explain the basic mechanism and highlight the relevance of a systematic stability anal- Zachary G. Nicolaou ysis for the coherent structures observed. Northwestern University [email protected] Arnd Scheel University of Minnesota Hermann Riecke School of Mathematics Applied Mathematics [email protected] Northwestern University [email protected] MS168 Adilson E. Motter Synchronization in Networks with Multiple Inter- Northwestern University action Layers [email protected] The structure of many realworld systems is best captured by networks consisting of several interaction layers. Un- MS168 derstanding how a multilayered structure of connections Using Symmetries and Equitable Partitions To- affects the synchronization properties of dynamical sys- gether to Find All Synchronization Clusters and tems evolving on top of it is a highly relevant endeavour in Their Stability mathematics and physics, and has potential applications to several societally relevant topics, such as power grids Many networks of coupled oscillators are observed to pro- engineering and neural dynamics. We propose a general duce patterns of synchronized clusters where all the os- framework to assess stability of the synchronized state in cillators in each cluster have exactly the same dynamical networks with multiple interaction layers, deriving a neces- trajectories in state space, but not the same as oscillators sary condition that generalizes the Master Stability Func- in other clusters. It has been difficult to predict these clus- tion approach. We validate our method applying it to a ters in general. We show the intimate connection between network of Rssler oscillators with a double layer of interac- network symmetry and cluster synchronization. We apply tions, and show that highly rich phenomenology emerges. computational group theory to reveal the clusters and de- This includes cases where the stability of synchronization termine their stability. Other synchronization clusters are can be induced even if both layers would have individually possible in addition to the symmetry clusters (SC). These induced unstable synchrony, an effect genuinely due to the are equitable partitions (EP) of the network. We show that true multilayer structure of the interactions amongst the the EP can be constructed by the merging of appropriate units in the network. SC. We show that this construction also allows the deriva- tion of further simplified stability (variational) equations Jesus Gomez-Gardenes for the EP case thus allowing the SC and EP approaches to Universidad de Zaragoza compliment each other. The connection between symme- [email protected] try and cluster synchronization is experimentally explored using an electro-optic network. Charo I. del Genio School of Life Sciences Louis M. Pecora University of Warwick, U.K. U.S. Naval Research Laboratory [email protected] [email protected] 288 DS17 Abstracts

Francesco Sorrentino Thereafter, we will use this reduced data set for running a University of New Mexico distributed estimation algorithm based on alternating di- Department of Mechanical Engineering rection multiplier method (ADMM). We will illustrate the [email protected] convergence, synchronization, and real-time constraints as- sociated with this entire process with examples drawn from Aaron M. Hagerstrom IEEE power system models. University of Maryland [email protected] Aranya Chakrabortty North Carolina State University [email protected] Rajarshi Roy University of Maryland, College Park, MD, USA [email protected] Yoshihiko Susuki Kyoto University, JST [email protected] Thomas E. Murphy University of Maryland, College Park Dept. of Electrical and Computer Engineering MS169 [email protected] Finding Slow Modes and Accessing Very Long Timescales in Molecular Dynamics MS168 The sampling problem, i.e. the difficulty to sample rare Controlling Synchronous Patterns in Complex Net- events along the slow modes, is a key problem in molecular work of Coupled Chaotic Oscillators dynamics and many other microscopic dynamical systems. In thermal equilibrium, where the dynamics have a sta- Although the set of permutation symmetries of a complex tionary state and are statistically reversible, we can derive network could be very large, few of them give rise to stable a variational approach to approximate the eigenfunctions synchronous patterns. Here we present a general frame- of the Koopman operator / transfer operator. Algorithmi- work and develop techniques for controlling synchroniza- cally, this can be achieved by computing covariance matri- tion patterns in complex network of coupled chaotic oscil- ces from a set of basis functions and obtaining the optimal lators. Specifically, according to the network permutation approximation of eigenfunctions from the eigenvectors of a symmetry, we design a small-size and weighted network, generalized eigenvalue problem. This approach turns out namely the control network, and use it to control the large- to be identical to the more recently introduced Extended size complex network by means of pinning coupling. We dynamic mode decomposition, and it has similarities the argue mathematically that for any of the network symme- dynamic mode decomposition. We will show how the eigen- tries, there always exists a critical pinning strength beyond function approximates can be used in order to approximate which the unstable synchronous pattern associated to this the essential dynamics in complex many-body systems and symmetry can be stabilized. The feasibility of the con- how this information can be exploited to enhance the sam- trol method is verified by numerical simulations of both pling of rare events. artificial and real-world networks and demonstrated ex- perimentally in systems of coupled chaotic circuits. Our • F. Nueske et al: Variational Approach to Molecular studies show the controllability of synchronous patterns in Kinetics. J. Chem. Theory Comput., 10, 1739-1752 complex networks of coupled chaotic oscillators. (2014) • Xingang Wang F. Noe and F. Nueske: A variational approach to mod- Zhejiang University, China eling slow processes in stochastic dynamical systems. [email protected] SIAM Multiscale Model. Simul., 11, 635-655 (2013)

Frank Noe MS169 FU Berlin A Data-Driven Distributed Algorithm for Nonlin- [email protected] ear Mode Estimation in Power Systems

We present a distributed optimization algorithm for esti- MS169 mating nonlinear oscillation modes of power systems us- Assessment of Voltage Collapse Phenomena in ing real-time Synchrophasor data. In our previous works Power Grids Based on Continuous Spectrum of the we have shown that linear mode estimation can be cast Koopman Operator as a linear least squares estimation problem for the coeffi- cients of the characteristic polynomial of the power system Large decrease or destabilization of voltage amplitudes of- model, assuming the order of the polynomial to be known. ten disturbes normal operation of modern power grids. The For nonlinear mode estimation, however, such a polyno- so-called voltage collapse and associated dynamic phenom- mial with a fixed order does not exist any more. Instead ena have been widely studied in power grids engineering an appropriate length of data needs to be selected based on and nonlinear dynamical systems. In this talk, we revisit which the effective order of the estimation is decided. If the the dynamics of voltage collapse phenomena in terms of estimation has to be carried out in a distributed fashion be- spectral properties of the Koopman operator. We show tween multiple phasor data concentrators, then operators that one dynamic signature of decrease of voltage ampli- would prefer this data length to be as small as possible tudes is characterized by continuous spectrum of the Koop- without sacrificing accuracy of the estimation. To address man operator for a underling mathematical model. This this problem, in this talk we will present an algorithm by provides a novel method for detecting the occurrence of which the length of the measured variables can be reduced voltage collapse directly from data without development by appropriate use of singular value decomposition and for- of mathematical models. Effectiveness of the method is getting factors associated with the data Hankel matrices. discussed with its application to data obtained by not only DS17 Abstracts 289

detailed power-grid simulations but also practical measure- New Jersey Institute of Technology ment. &RutgersUniversity [email protected] Yoshihiko Susuki Kyoto University, JST Amitabha Bose [email protected] New Jersey Institute of Technology [email protected] Kyoichi Sako, Fredrik Raak Kyoto University [email protected], MS170 [email protected] A Singulartity Theory Approach to Homeostasis Takashi Hikihara Department of Electrical Engineering Homeostasis occurs when some output variable remains ap- Kyoto University proximately constant as input parameters λ vary over some [email protected] intervals. First, we discuss the effect of coordinate changes on the input-output map associated to homeostasis. Sec- ond, we formulate homeostasis in the context of singular- MS169 ity theory by replacing ‘approximately constant over an Koopman Operator Theory, Memory Effects, and interval’ with ‘zero derivative of the output with respect to Fractional Order Dynamical Systems the inputs at a point’. Unfolding theory then classifies all small perturbations of the input-output function. In par- Koopman operator theory has recently become a practical ticular, in one input systems the ‘chair’ singularity, which tool for data-driven analysis of dynamical systems. The has been shown by Best, Nijhout, & Reed to be especially important in applications, is discussed in detail. Its nor- theory rests on an inherent assumption that the observ- 3 ables follow a Markov rule of memoryless evolution. How- mal form and universal unfolding λ + aλ is derived and ever, memory effects may naturally arise in the dynamics the region of approximate homeostasis is deduced. This of observables as the complexity of the system increases. normal form shows how a one output system can deform For example, the dynamics of a system interacting with from non-homeostasis to homeostasis. In two input sys- an environment is generally non-Markovian, and in spe- tems the hyperbolic umbilic can also organize evolution to cial cases open systems may exhibit long-term memory ef- homeostasis. fects characterized by fractional order derivatives in time. We discuss first steps toward a generalized Koopman op- Martin Golubitsky erator theory that is compatible with memory effects and Ohio State University fractional calculus. We focus on a simple nonlinear sys- Mathematical Biosciences Institute tem that is useful for testing the theoretical ideas. Since [email protected] Koopman operator theory is notably amenable to numeri- cal treatment, we also outline plans for developing general- Ian Stewart ized numerical methods that in principle could be used to University of Warwick detect memory effects in data and subsequently construct Mathematics Institute an appropriate model. [email protected]

Adam Svenkeson, Bryan Glaz Army Research Laboratory MS170 [email protected], [email protected] Chaos, Spike-Adding, and Mixed-Mode Oscilla- tions in a Nonlinear Neuronal Model with Resets MS170 Phase-Locking in a Neuronal Networks with Neuronal models with resets, or hybrid models, can cap- Synaptic Depression ture complicated dynamics in a relatively low-dimensional setting. I will discuss two activity patterns, chaos and We derive a 2-dimensional map to investigate the origins of mixed-mode oscillations (MMOs), in a general class of bistability of phase-locked periodic solutions in a pair of in- these hybrid models. I will explain the mechanisms under- hibitory neurons coupled via depressing synapses. We show lying chaos and MMOs as well as the pathways through that stable solutions that exist in the absence of synaptic which various model parameters can modulate the re- depression can undergo saddle-node bifurcations leading sulting model dynamics. The analysis makes use of a to distinct high- and low-frequency out-of-phase solutions one-dimensional adaptation map. I will discuss how un- becoming concurrently stable. The method of analysis ap- der parameter variations, chaos emerges in association plies to any Type I neuron and involves using phase reset- with period-incrementing transitions, and how, in another ting curves and geometric analysis. regime, rotation theory for circle maps can be used to elu- cidate the MMO dynamics. Zeynep Akcay Queensborough Community College Jonathan E. Rubin [email protected] University of Pittsburgh Department of Mathematics Xinxian Huang [email protected] Department of Mathematical Sciences New Jersey Institute of Technology Justyna Signerska-Rynkowska [email protected] Faculty of Applied Physics and Mathematics Gdansk University of Technology Farzan Nadim [email protected] 290 DS17 Abstracts

Jonathan D. Touboul along the axon is relatively uniform and that vesicular de- The Mathematical Neuroscience Laboratory livery to synapses is reversible. Recent models of vesicular College de France & INRIA Paris delivery to synapses have made explicit the crucial role [email protected] that reversibility in vesicular delivery to synapses plays in achieving uniformity in vesicle distribution, so called Alexandre Vidal synaptic democracy. We will illustrate this principle by University of Evry first discussing a one-dimensional model of motor trans- Laboratoire de Math´ematiques et Mod´elisation d’Evry port and vesicle delivery on a semi-infinite domain and (LaMME) show how reversibility in vesicle delivery facilitates synap- [email protected] tic democracy. We will then discuss more general models that account for exclusion effects between motors traveling in bulk and higher dimensional models on the disk, sphere, MS170 and Cayley tree that illustrate how uniformity in vesicle Temperature Sensitivity of PO/AH Neurons density may be achieved in cells in general. In all cases we find that uniformity in vesicle delivery to target sites is Thermoregulatory responses are partially controlled by the facilitated by allowing for reversibility in vesicle delivery, preoptic area and anterior hypothalamus (PO/AH), which provided that the velocity of cargo-carrying motors is not contains a mixed population of temperature-sensitive and significantly slowed by the load of the cargo. insensitive neurons. In [Wechselberger et al, 2006] based on physiological data, a Hodgkin-Huxley-like conductance Bhargav R. Karamched based model was constructed. This model suggests that University of Utah most PO/AH neurons have the same types of ionic chan- Salt Lake City, UT nels, but different levels of channel expression can explain [email protected] the inherent properties of the various types of temperature- sensitive and insensitive neurons which is encoded in their frequency sensitivity relative to temperature. Here we MS171 present a detailed bifurcation analysis of this model to con- First-Passage Time to Clear the Way for Receptor- firm these observations. We focus on three main physio- Ligand Binding in a Crowded Environment logical bifurcation parameters, the temperature T and the maximum conductances of two specific background potas- I will present theoretical support for a hypothesis about sium leak channels, gtask and gtrek,thatareknowntobe cell-cell contact, which plays a critical role in immune func- expressed in these PO/AH neurons. These three bifurca- tion. A fundamental question for all cell-cell interfaces is tion parameters are sufficient to explain the dynamics of how receptors and ligands come into contact, despite being PO/AH neurons observed in experiments. separated by large molecules, the extracellular fluid, and other structures in the glycocalyx. The cell membrane is a Martin Wechselberger,TimothyRoberts crowded domain filled with large glycoproteins that impair University of Sydney interactions between smaller pairs of molecules, such as the [email protected], [email protected] T cell receptor and its ligand, which is a key step in im- munological information processing and decision-making. A first passage time problem allows us to gauge whether a MS171 reaction zone can be cleared of large molecules through pas- Metastable States for an Aggregation Model with sive diffusion on biologically relevant timescales. I combine Noise numerical and asymptotic approaches to obtain a complete picture of the first passage time, which shows that passive We consider the aggregation equation with kernels that diffusion alone would take far too long to account for exper- generate patterns consisting of two delta-concentrations. imentally observed cell-cell contact formation times. The Without noise, there is a one-parameter family of admissi- result suggests that cell-cell contact formation may involve ble equilibria that consist of two concentrations whose mass previously unknown active mechanical processes. is not necessary equal. When a small amount of noise is added, the heavier concentration leaks its mass towards the Jay Newby lighter concentration over a very long time scale, eventu- University of North Carolina-Chapel Hill ally resulting in the equilibration of the two masses. We [email protected] use exponentially small asymptotics to derive the long-time ODEs that quantify this mass exchange. Our formal com- putations show that adding noise destroys the degeneracy MS171 in the equilibrium solution and leads to a unique symmetric Kinetic Monte Carlo Methods for Computing First steady state. Capture Time Distributions in Models of Diffusive Absorption Joep Evers, Theodore Kolokolnikov Dalhousie University Consider the pistil of a flower waiting to catch a grain of [email protected], [email protected] pollen, a lymphocyte waiting to be stimulated by an anti- gen to produce antibodies, or an anteater randomly forag- ing for an ant nest to plunder. Each of these problems can MS171 be modeled as a diffusive process with a mix of reflecting Effects of Cell Geometry on Reversible Vesicular and absorbing boundary conditions. One can characterize Transport the agent (pollen, antigen, anteater) finding its target (pis- til, lymphocyte, ant nest) as a first passage time (FPT) A major question in biology concerns the mechanics behind problem for the distribution of the time when a particle the motor- driven transport and delivery of vesicles to lo- executing a random walk is absorbed. In this talk we will calized regions of a given cell. In the case of neurons, exper- examine a hierarchy of FPT problems modeling planar or imental evidence suggests that the distribution of vesicles spherical surfaces with a distribution of circular absorbing DS17 Abstracts 291

traps. We will describe a Kinetic Monte Carlo (method that our proposed appproach allows to discriminate differ- that exploits exact solutions to accelerate a particle-based ent dynamic regimes adequately, while using a reduced set simulation of the capture time. A notable advantage of of points from the original RP. these methods is that run time is independent of how far from the traps one begins. We compare our results Elbert E. Macau with asymptotic approximations of the FPT distribution INPE - Brazilian National Institute for Space Research for particles that start far from the traps. Our goal is to LAC - Laboratory for Computing and Applied validate the efficacy of homogenizing the surface boundary Mathematics conditions, replacing the reflecting (Neumann) and absorb- [email protected] ing (Dirichlet) boundary conditions with a mixed (Robin) boundary condition. MS172 Daniel Schmidt Network Inference Meets Nonlinear Time Series: Harvey Mudd College Challenges and Solutions [email protected] Recent years have seen a large increase in the availability Andrew J. Bernoff of data. In fact, increasing amounts of data play a key Harvey Mudd College role in every aspect of our lives, such as biology, e.g. ge- Department of Mathematics nomic data, medicine, e.g. functional magnetic resonance [email protected] imaging or electroencephalography, and data mining in the social sciences or digital economies. Dealing with these Alan E. Lindsay data sets efficiently determines the success of the projects, Applied Computational Mathematics and Statistics treatments, assessments, and analyses. The necessity to University of Notre Dame better understand data has led to an outburst of research [email protected] into advanced methods of data analysis. The inference of networks underlying complex systems is of utmost impor- tance. Especially when dealing with complex data sets the MS172 algorithms for network inference should meet certain re- quirements such as dealing with truly multivariate data, ac- Using Recurrences to Detect Non Stationary Be- counting for various concurrent noise sources, and address- havior of Time Series ing both linear and non-linear systems. A multitude of We explore the utility of a nonlinear recurrence measure, algorithms has been developed to address these extremely the Determinism, to identify and quantify transitions be- challenging requirements. This is partly because a rigor- tween segments with distinct statistical properties in the ous mathematical framework, i.e., a theory of a suitable output signals of nonlinear dynamical systems, and to in- highly versatile class of mathematical models to comprise fer coupling between systems. We demonstrate that this these features, is challenging. In this talk, the challenges measure is sensitive to detect even small variations in the as well as various methods will be introduced, discussed dynamics of a system by studying stationarity breaking in and compared. This results in a comprehensive overview the signal output displayed by another system coupled to of techniques that exists to tackle one of the key challenges the first. To test our approach we apply the technique of data based modelling: The detection of directed inter- to the dynamics of two physiological systems, activation actions from time series in nonlinear systems. in the hippocampus brain area and in the locomotor sys- tem in mice during sleep. We analyze the respective out- Bjoern Schelter put signals of these two systems – evasively obtained hip- Institute for Complex Systems and Mathematical Biology pocampal local field potential and noninvasive locomotor [email protected] accelerometer sensor signal. We demonstrate that small- amplitude short bursts in the accelerometer signal (rep- Marco Thiel resenting micro-arousals during sleep) are associated with University of Aberdeen abrupt transitions in the profile of the recurrence quan- Scotland tifier. Moreover we find that stationarity breaking, as- [email protected] sociated with abrupt transition in the determinism, con- sistently precedes each arousal burst in the Determinism signals by few seconds. MS173 A Computational Model of Hemostasis Sergio R. Lopes Paran´a Federal University, Brazil Hemostasis is the process by which a blood clot forms to [email protected] prevent bleeding. The clot size, structure and time to for- mation all depend on the local hemodynamics and the na- ture of the injury. Our previous computational models were MS172 developed to study intravascular clot formation, a process Recurrency Density Enchanced Approach for Time confined to the interior of a single vessel. Here we present Series Analysis the first computational model of extravascular clot forma- tion (hemostasis) in which blood through a single vessel We present a transformation method, entitled Recurrence initially escapes through a hole in the vessel wall and out a Density Enhancement Approach (RDE), that aims to high- separate injury channel. The model consists of a system of light the main recurrence structures of a given recurrence partial differential equations that describe blood coagula- plot (RP). Our method results in a figure with a reduced tion biochemistry, platelet aggregation, and hemodynam- number of points yet preserving the main and fundamen- ics, solved via the finite element method. In the model, tal properties of the original plot. The existing measures of formation of a blood clot occludes the injury channel and quantification analysis are applied to characterize the un- stops flow from escaping while blood in the main vessel derlying dynamical system. Our evaluation results indicate retains its fluidity. We discuss the different biochemical 292 DS17 Abstracts

and hemodynamic effects on clot formation using distinct to contribute significantly to the functional alternations of geometries representing intra- and extravascular injuries. blood vessels in hypertension. We perform a bifurcation analysis of the model equations to assess the effect of lu- − Nicholas Danes minal pressure, NO and O2 on the behaviors of limit cycle Applied Mathematics & Statistics oscillations. Colorado School of Mines [email protected] Ning Wei Duke University Karin Leiderman [email protected] Colorado School of Mines [email protected] Anita T. Layton Duke University Department of Mathematics MS173 [email protected] Drug Delivery to the Brain: Mathematical Chal- lenges in Breaking the Blood-Brain Barrier MS173 Incidence rates of neurodegenerative diseases and brain Modelling the Interplay Between Cell Signalling cancers continue to rise. Many brain diseases can be and Cell Mechanics treated by drugs and research in biomedicine has re- sulted in powerful compounds to treat Parkinson’s Dis- Protein signalling networks are responsible for regulating ease, Alzheimer’s Disease, and brain tumors. The prob- the shape and polarity of cells. Rac and Rho are mutually lem facing pharmaceutical innovators is to deliver a drug inhibitory signalling proteins that regulate a cell’s ability specifically to a target site, in the required amount, at to form protrusions (Rac) and to contract (Rho). As cells the required rate, and at the appropriate time. In other change shape, migrate individually, or move as group, they words, they must give answers to the primary questions: experience forces from their environment and neighbours. How much? When? How often? Where? Drug delivery to These forces are translated into chemical signals that may the brain is particularly challenging due to the blood-brain increase or decrease the activities of Rac and/or Rho. We barrier: some molecules cannot permeate the barrier at develop and analyze a nonlinear ODE model for Rac and all, and others require active transport to make it through. Rho signalling within a cell and then couple cells mechani- One solution is to package the drugs in nano-particles that cally. To explore the interplay between cell mechanics and release their cargo in response to an external trigger, such cell signalling, we couple Rac and Rho signalling to a simple as temperature or acoustical pressure. In this talk I will (spring-dashpot) mechanical model of a cell. We discover present a mathematical model that describes the delivery that feedback between Rac and Rho signalling and cell me- of a drug to a specific region of the brain with the help of chanics can lead to a variety of dynamics. In particular, liposomes triggered by focused ultrasound. I will discuss in a tissue composed of many cells, we observe waves of the challenges in matching the model to data, and in de- contraction from one end of the tissue to the other. veloping the mathematics needed to describe the transport of the drug from nanoparticle to brain tissue. This is joint Cole Zmurchok work with Peter Hinow (U. Wisconsin, Milwaukee), John University of British Columbia Reynold and Ian Tucker (U. Otago, Dunedin, NZ). [email protected]

Ami Radunskaya Pomona College MS174 Mathematics Department Anomalous Diffusion in Membranes and Cyto- [email protected] plasm of Biological Cells

A surging amount of experimental and simulations stud- MS173 ies reveals persistent anomalous diffusion in both cellular Bifurcation Analysis of Calcium Oscillations in the membranes and the cytoplasm [M.J. Saxton and K. Jacob- Afferent Arteriole Model of Smooth Muscle Cells sen, 1997; F. Hofling and T. Franosch, 2013]. The anoma- lous diffusion is observed for micron-sized objects down to The afferent arteriole (AA) of rat kidney exhibits the myo- labelled single molecules such as green fluorescent proteins genic response, in which the vessel constricts in response [C. Di Rienzo et al., 2014)]. This talk will first present to an elevation in blood pressure and dilates in response results from large scale computer simulations and stochas- to a pressure reduction. Additionally, the AA exhibits tic analysis of the motion of lipids and embedded proteins spontaneous oscillations in vascular tone at physiological in lipid bilayer model membranes [J.-H. Jeon et al., 2012; luminal pressures. These time-periodic oscillations stem J.-H. Jeon et al., 2016)], indicating that increased disorder from the dynamic exchange of Ca2+ between the cytosol leads to longer and longer lasting anomalous diffusion. In and the sarcoplasmic reticulum, coupled to the stimula- particular, the motion of lipids and proteins can become tion of Ca2+-activated potassium and chloride channels, non-Gaussian. In the membranes of living cells anomalous and to the modulation of voltage-gated L-type Ca2+ chan- diffusion of embedded protein channels can last over several nels. The effects of physiological factors, including blood hundreds of seconds [A.V. Weigel et al., 2011]. In particu- pressure and vasoacive substances, on AA vasomotion re- lar, this anomalous diffusion can become non-ergodic and main to be well characterized. In this paper, we analyze a exhibit ageing, two topics explained and discussed in this mathematical model of Ca2+ signaling in an AA smooth talk [R. Metzler et al., 2014)]. The findings of anomalous muscle cell. The model represents detailed transmembrane diffusion in membranes will be complemented by a brief ionic transport, intracellular Ca2+ dynamics as well as ki- summary of anomalous diffusion in the cellular cytoplasm, − netics of nitric oxide (NO) and superoxide (O2 ) formation, referring to both subdiffusion of passive tracers [J.-H. Jeon diffusion and reaction. NO is an important factor in the et al., 2011; I. Golding and E.C. Cox, 2006; I. Bronstein − maintenance of blood pressure and O2 has been shown et al., 2009; M. Weiss et al., 2004] and superdiffusion due DS17 Abstracts 293

to active motion of cells [J.F. Reverey et al., 2015; N. Gal and Development. and D. Weihs, 2010]. Michael Zaks Ralf Metzler Humboldt University of Berlin Inst for Physics & Astronomy [email protected] University of Potsdam [email protected] Alexander Nepomnyashchy Technion Israel Institute of Technology MS174 [email protected] A Fractional Kinetic Process Describing the Inter- mediate Time Behaviour of Cellular Flows MS175 This work studies the intermediate time behavior of a small Active Matter in Time-Periodic Flows: Swimming random perturbation of a periodic cellular flow. Our main with Gradients result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start In this talk, I will discuss the transport dynamics of ac- close enough to cell boundaries is a fractional kinetic pro- tive particles (i.e. swimming bacteria) in two-dimensional cess: A Brownian motion time changed by the local time time-periodic flows using a magneto-hydrodynamic flow of an independent Brownian motion. Our proof uses the cell. This experimental setup allows for the creation of Freidlin-Wentzell framework, and the key step is to estab- flows of different structures and dynamical properties de- lish an analogous averaging principle on shorter time scales. pending on the arrangement of the underlying magnets and As a consequence of our main theorem, we obtain a ho- magnitude of forcing. Flow velocity fields are measure us- mogenization result for the associated advection-diffusion ingparticletrackingmethodsandusedtocalculatethe equation. We show that on intermediate time scales the flows stretching fields and finite-time Lyapunov exponents effective equation is a fractional time PDE that arises in (FTLE). The transport of bacteria in this 2D flow is inves- modelling anomalous diffusion. tigated by injecting dilute suspensions of Vibrio cholerae. Surprisingly, we find that activity (or swimming) can hin- Martin Hairer der overall transport compared to passive particles. While The University of Warwick at short times, bacteria tend to align along regions of high [email protected] stretching, we find that at long times bacteria accumulate near elliptic points of the flow. This accumulation leads to Gautam Iyer a decrease in overall transport. Carnegie Mellon University [email protected] Paulo E. Arratia Mechanical Engineering and Applied Mechanics Leonid Koralov University of Pennsylvania, Philadelphia. Princeton University [email protected] [email protected] MS175 Alexei Novikov Penn State University Lagrangian Stochastic Prediction and Data Assim- Mathematics ilation [email protected] We first present differential equations and high-order nu- Zsolt Pajor-Gyulai merical schemes for predicting the probability of La- Courant Institute of Mathematical Sciences grangian Coherent Structures (LCS) in uncertain flows. New York University We then derive Bayesian Lagrangian data assimilation [email protected] schemes for the joint inference of structures and fields. Re- sults are extended to Lagrangian-Eulerian Bayesian infer- ence, mutual information, and smoothing. Examples are MS174 provided for idealized incompressible fluid and geophysi- cal flows, as well as for realistic data-assimilative multi- Anomalous Dispersion of Particles in Steady Two- resolution ocean simulations. dimensional Flows Pierre F. Lermusiaux It is well known that anomalous dispersion of particles is MIT possible in chaotic or random flows. Surprisingly, it is pos- [email protected] sible also in steady spatially-periodic flows in the absence of both Eulerian and Lagrangian chaos, if the flow pattern includes stagnation points or solid obstacles [M.A. Zaks et Chinmay S. Kulkarni al., Phys. Rev. Lett. 77 (1996) 4338-4341; M.A. Zaks and Massachusetts Institute of Technology A.V. Straube, Phys. Rev. Lett. 89 (2002) 244101]. In [email protected] this talk, we present a quantitative description of this phe- nomenon using a special flow construction: flow over the Florian Feppon, Arkopal Dutt circle mapping with logarithmic and power-law singulari- MIT ties of return time. Analytical estimates, based on compu- [email protected], [email protected] tation of partial sums of series and the CTRW approach, match the numerical data from the direct simulation of the flow. The research was supported by Grant 3140-11899 MS175 from the German-Israeli Foundation for Scientific Research Optimizing Vehicle Autonomy in Geophysical 294 DS17 Abstracts

Flows rons in the Adult Cortex

Autonomous marine vehicles (AMVs) deployed for ocean Synchronization of neurons in the cortex is thought to un- science and environmental monitoring applications must derlie several cognitive processes such as learning and mem- operate with little to no human supervision for long peri- ory formation. Fitting data from experimental papers, we ods of time. Given their limited energy budgets, it makes construct a detailed model with synaptic and electric cou- sense to consider motion plans that leverage the dynamics pling using a modified version of the Hodgkin-Huxley equa- of the surrounding flow to minimize the vehicles’ energy ex- tions. We use this model to show that a small percentage of penditures. We present recent graph search based methods excitatory neurons connected via electrical junctions serves in computing energy and time optimal paths for AMVs in to tighten synchrony in a realistic cortical network. time independent and time-varying flows. Leveraging our graph search methods, we analyze and relate energy and Jennifer Crodelle time optimal paths in deterministic flows with escape tra- Rensselaer Polytechnic Institute jectories in stochastic flows. [email protected]

Ani Hsieh Gregor Kovacic Drexel University Rensselaer Polytechnic Inst [email protected] Dept of Mathematical Sciences [email protected] Shibabrat Naik Engineering Science and Mechanics David Cai Virginia Tech New York University [email protected] Courant institute [email protected] Dhanushka Kularatne Drexel University MS176 [email protected] Emergence of a Balanced Core Through Dynami- cal Computation in Inhomogeneous Neuronal Net- Subhrajit Bhattacharya works Lehigh University [email protected] The balance between excitatory and inhibitory current input is crucial for neuronal computation and has been Eric Forgoston observed in many experiments. Theoretical studies have Montclair State University mainly focused on the analysis of homogeneous networks. Department of Mathematical Sciences However, neuronal networks in the brain are usually inho- [email protected] mogeneous. Here we show that the balanced state can exist even in inhomogeneous neuronal networks and embedded in the original network there is a homogeneous-like core MS175 that underlies origin of the balanced state. Adaptive Learning and Prediction of Motion of Qinglong Gu Controlled Lagrangian Particles Shanghai Jiao Tong University [email protected] Marine robots can be viewed as a new type of Lagrangian particles with their motion under feedback control. When the controlled speed is relatively weak comparing to the MS176 speed of ocean current, the controlled Lagrangian particles Neuronal Network Reconstruction by Transfer En- will demonstrate complex yet interesting trajectories that tropy can be partially modeled by stochastic systems with un- known input and parameters. This talk will explain the We apply low order and pairwise Transfer entropy to re- various factors that affect the trajectories. Furthermore, construct the structural connectivity of neuronal networks adaptive learning and control methods are applied to re- based on only the spike trains. In this work, we use duce the uncertainty in the trajectories. In addition to a current-based, integrate-and-fire network and address theoretical analysis, the results are verified through simu- the issue of why the reconstruction can be successfully lation and experimental effort. achieved. We also obtain a quadratic relationship between the TE and the coupling strength and establish a direct Fumin Zhang link between the causal and structural connectivity in the School of Electrical and Computer Engineering network. Georgia Institute of Technology [email protected] Zhongqi Tian New York University Sungjin Cho [email protected] Georgia Institute of Technology [email protected] MS176 Spike Triggered Regression on Neuronal Network MS176 Reconstruction Mathematical Model of a Network Containing We propose a spike-triggered regression (STR) method Electrotonic Junctions Between Excitatory Neu- to accurately reconstruct the network connectivity of a DS17 Abstracts 295

nonlinear integrate-and-fire (I&F) neuronal network. The [email protected] basic idea of our method is to capture the subthreshold voltage response to a presynaptic spike through linear re- Pavel Buran gression. Through numerical simulations, we demonstrate Physikalisch-Technische Bundesanstalt (PTB) that, by using relatively short-time recordings, the I&F [email protected] neuronal network connectivity can be well reconstructed using our STR method. Sergio Alonso UPC, Barcelona Yaoyu Zhang [email protected] New York University [email protected] Markus Baer Physikalisch-Technische Bundesanstalt (PTB) MS177 [email protected] Modeling and Estimation of Dynamical Variables in a Cardiac Cell Model MS177 Cellular dynamical variables, such as ionic concentrations New Insights into the Termination of Unpinned and gating states, play an important role in the forma- Spiral Waves Using Low-Energy Electric Field tion of cardiac arrhythmias. Since these quantities can Pulses be difficult to measure, a Kalman filtering approach was tested on dynamical model of a myocyte, the Luo-Rudy Successful termination of rapid cardiac arrhythmias using dynamic model, and it was shown that cellular variables low-energy, electric-field based shocks requires a new un- can be reconstructed based on simulated measurements of derstanding of how action potential waves launched by the membrane potential. The observability of the system was applied electric field pulses interact with the rotating waves characterized under a range of conditions. that are responsible for the arrhythmia. In this talk, we present our findings on (1) how waves launched from obsta- Laura Munoz cles inherent in cardiac tissue can merge with the rotating Rochester Institute of Technology waves to create new waves that terminate upon interaction School of Mathematical Sciences with the boundaries, and (2) how plane waves launched [email protected] from boundaries can merge with rotating waves in three dimensions to form waves with C-shaped filaments that Claire Charron, Kalyan Pusarla then shrink and disappear. With respect to the first mech- Rochester Institute of Technology anism, we will also suggest what field strengths and stim- [email protected], [email protected] ulus timings are most likely to terminate a rotating wave, and discuss an ECG-based method might be able to deter- mine the location and phase of the rotating wave, which is MS177 needed to determine the appropriate stimulus timing. For the second mechanism, we will also discuss the importance Wave Control by Cooperative Excitation of Car- of the shape of the boundary as a factor in determining diac Tissue whether the C-shaped filament will shrink and disappear, or revert back to an I-shaped filament that persists indefi- Rotating excitation waves and electrical turbulencein car- nitely. diac tissue have been associated witharrhythmias like the life-threatening ventricular fibrillation. The application of Niels Otani an electrical shock (defibrillation) is an effective therapy, as Rochester Institute of Technology it globally excites the tissue resulting in termination of all [email protected] excitation waves, but also causes severe side effects. Recent experimental studies have shown that a sequence of elec- trical pulses is able terminate fibrillation more gently than Valentin Krinski a single pulse. Only tissue at major conduction hetero- Max Planck Institute for Dynamics and Self-Organization geneities, such as large coronary arteries, may be activated [email protected] by each of these very weak pulses. Therefore, global tis- sue activation and wave termination originates from few Shuyue Han, Kayleigh Wheeler localized activation sites. In order to decipher the inter- Rochester Institute of Technology play of the individual pulses, we performed extensive sim- [email protected], [email protected] ulations of cardiac tissue perforated by blood vessels and tested a variety of cellular models. For models exhibit- Stefan Luther ing a dominant excitation period during fibrillation, the Max Planck Institute for Dynamics and Self-Organization pulses appear to be highly cooperative if the period be- Research Group Biomedical Physics tween these pulses matches the dominant period. These [email protected] findings are elucidated by the analysis of the dynamical variables, such as the fraction of excited tissue and the number of phase defects, both during the state of electrical MS177 turbulence and during cooperative excitation. Moreover, Controlling the Dynamics of Cardiac Tissue During we propose a simple stochastic model which integrates our Ventricular Fibrillation results in an intuitive way. Ventricular fibrillation is a lethal condition of the heart Thomas Niedermayer which is still not well understood and medicine lacks a suit- Research Group Complex Systems in Biophysics and able treatment. The irregular and fast activation patterns Medicine during ventricular fibrillation make it hard to find efficient Physikalisch-Technische Bundesanstalt (PTB) methods to control the dynamics of the myocardium. In 296 DS17 Abstracts

contrast, e.g. monomorphic ventricular tachycardia can contagion) is true, while accounting for over-dispersion in often be terminated by local stimulation with a train of the data. Our analysis finds that some, but not all, of these uniform electric pulses. While such a local stimulation has tragedies do indeed appear to be significantly contagious, a very limited region of influence in ventricular fibrillation, and media attention appears to play a key role in these we demonstrate how fast trains of electric far field pulses dynamics. modify the properties of the dynamics in heterogeneous cardiac tissue and may facilitate efficient termination of Sherry Towers ventricular fibrillation. Mathematical and Computational Modeling Sciences Center, ASU Henrik tom Woerden [email protected] Max Planck Institute for Dynamics and Self-Organization [email protected] MS178 Extra High-Dimensional Linear ODE Model Selec- MS178 tion and Parameter Estimation: A Matrix-Based Modeling Differential Transmission Characteristics Approach of Whitefly-Transmitted Cassava Viruses Ordinary differential equations (ODEs) are widely used to The whitefly, Bemisia tabaci, is one of the most economi- model the dynamic behavior of a complex system. Parame- callyimportantinsectvectorsofplantviruses.Ittransmits ter estimation and variable selection for a Big System with over 200 viruses through either semi-persistent or persis- high-dimensional linear ODEs are very challenging due to tent mechanisms. Cassava mosaic geminiviruses and Cas- the difficulty of nonlinear optimization problems with an sava brown streak viruses have been associated with region- extra-high parameter space. In this article, we develop a wide spread of a dual pandemic of cassava mosaic dis- parameter estimation and variable selection method based ease (CMD) and cassava brown streak disease (CBSD). We on the ideas of similarity transformation and separable formulate a deterministic compartmental model for host- least squares (SLS). Simulation studies demonstrate that vector dynamics. Our simulations show that these models the proposed matrix-based SLS method could be used to not only fit the field data well, with biologically meaningful estimate the coefficient matrix more accurately and per- parameter estimates, but additionally they also capture the form variable selection for a linear ODE system with thou- differences between the two types of viral infections: CMD sands of dimensions and millions of parameters much better and CBSD. compared to the direct least squares (LS) method and ex- isting vector-based two-stage method. We apply our new Ariel Cintron-Arias method to two application data sets: a yeast cell cycle gene Department of Mathematics and Statistics expression data set with 30 dimensions and 930 unknown East Tennessee State University parameters; and the Standard & Poor 1500 index stock [email protected] price data with 1250 dimensions and 1,563,750 unknown parameters. MS178 Hulin Wu Comparative Estimation of Parameters for Dengue Department of Biostatistics and Chikungunya in Costa Rica from Weekly Re- University of Texas Health Science Center at Houston ported Data [email protected]

Dengue virus has been a cause of major public health con- cern in Costa Rica. In 2014 the Chikungunya virus was MS179 introduced. We look at the 2015-2016 dengue and chikun- Topological Existence of Periodic Orbits in a Two- gunya outbreaks in order to establish a comparison using Center Symmetric Pair Problem weekly longitudinal data. A nonlinear differential equation single-outbreak model is validated against the aggregate The Two-Center Symmetric Pair Problem, derived from data. We estimate epidemic model parameters, including the Rhomboidal Symmetric-Mass problem by fixing one the basic reproductive number, R0, and explore the impact symmetric pair, limits, as the mass of the non-fixed sym- of undiagnosed cases. metric pair goes to zero, to the integrable Euler Two- Center Problem (with the centers having equal mass). Fabio Sanchez Standard KAM Theory indicates that many of the quasi- University of Costa Rica periodic orbits found in the integrable limit case persist [email protected] for small values of the mass of the non-fixed symmetric pair. Through topological methods, we investigate the ex- istence of several periodic orbits in the Two-Center Sym- MS178 metric Pair Problem with and without collisions when the Using a Mathematical Model and Likelihood Meth- mass of the symmetric pair is not necessarily close to zero. ods to Assess Contagion in Mass Killings and School Shootings Lennard F. Bakker, Mitchell Sailsbery In recent years there have been several high-profile public Brigham Young University mass killings in the US, increasing societal focus on the [email protected], [email protected] persistent problem of firearm violence in American society. Here, I present the results of our recent analysis of data related to US mass shootings (three or more people shot, MS179 not necessarily killed), school shootings, and mass killings High Energy Scattering: Orbit Structure (four or more people killed). We examine the data using a mathematical model of contagion. Using likelihood meth- We report on the motivation and results of “Lagrangian ods, we assess the probability that the null hypothesis (no Relations and Linear Point Billiards” (arXiv:1606.01420) DS17 Abstracts 297

joint with Knauf and Fejoz. Motivated by the high- their relationship to dynamical systems. A review of the energy limit of the N-body problem we construct non- literature indicates that there are many open questions deterministic billiard process. The billiard table is the com- concerning the relationships among the myriad definitions plement of a finite collection of linear subspaces within a of entropy. Our approach focuses on the Hamiltonian dy- Euclidean vector space. A trajectory is a constant speed namical system foundation of the statistical thermodynam- polygonal curve with vertices on the subspaces and change ics definition of entropy. Entropy has played a role in many of direction upon hitting a subspace governed by ‘conser- areas of physics from the 1850’s when Clausius coined the vation of momentum’ (mirror reflection). The itinerary of term to indicate a dissipation of energy, and has contin- a trajectory is the list of subspaces it hits, in order. Two ued on in the later part of that century with the work of basic questions are: (A) Are itineraries finite? (B) What Boltzmann, Gibbs and Planck. Information theory devel- is the structure of the space of all trajectories having a oped by Shannon, used the Boltzmann definition of en- fixed itinerary? In a beautiful series of papers Burago- tropy. More recently a topological entropy is used as a Ferleger-Kononenko [BFK] answered (A) affirmatively by measure of sensitivity in dynamical systems. We plan to using non-smooth metric geometry ideas and the notion of establish rigorous connections among several of the most a Hadamard space. We answer (B) by proving that this important definitions of entropy. These definitions include space of trajectories is diffeomorphic to a Lagrangian re- the classical thermodynamics, statistical entropy to topo- lation on the space of lines in the Euclidean space. Our logical entropy, with a common dynamical systems thread. methods combine those of BFK with the notion of a gen- The results of our approach will then be applied to some erating family for a Lagrangian relation. physical models.

Richard Montgomery Raymond Addabbo University of California at Santa Cruz Vaughn College of Aeronautica and Technology [email protected] [email protected]

Denis Blackmore MS179 New Jersey Institute of Technology Towards a Restricted N-Body Stability Result [email protected] The rhomboidal four-body orbit and Broucke’s isosceles triangle orbit are two examples of periodic Newtonian n- PP1 body orbits featuring regularizable binary collisions. These Neuronal Motifs - Multistability Using Hybrid periodic orbits can still be found for each as a certain mass Computational Approaches parameter is varied between them. In my talk, I will discuss how the limiting cases for both orbits is the same (in an Large scale analysis of the dynamical behavior of Central appropriate sense), and present stability results for both Pattern Generators (CPGs) formed by neuronal networks orbits as this parameter varies, suggesting the stability of of even small sizes is computationally intensive and grows their common limit. exponentially with network size. We have developed a suite of tools to exhaustively study the behavior of such net- Skyler Simmons works on modern GPGPU accelerators using the directive Southern Utah University based approach of OpenAcc. We also achieve paralleliza- [email protected] tion across clusters of such machines using OpenMPI. Di- rective based approaches simplify the task of porting se- MS179 rial code onto GPUs, without the necessity for expertise in lower level approaches to GPU programming, such as Remarks on the N-Body Dynamics on Surfaces of CUDA and OpenCL. 3-cell neuronal CPGs have been ex- Revolution plored previously using various GPGPU tools. As motifs We explore the dynamics of N mass points constrained to form the building blocks of larger networks, we have em- move on a surface of revolution and with some binary po- ployed our framework to study 4-cell CPGS and two con- tential interaction. We discuss symmetries and determine nected 3-cell motifs. We discuss the performance improve- certain invariant manifolds. We also show that the equiv- ments achieved using this framework and present some of alent of Saari’s conjecture for such motions fails. Further, our results. we define homographic motions to be those for which the Sunitha Basodi, Krishna Pusuluri configuration formed by the bodies is planar, orthogonal to Georgia State University the axis of revolution and remains self-similar in the am- [email protected], [email protected] bient space. For equal masses, using discrete reduction, we show that such motions form a two-degree of freedom mechanical system with symmetry for which one may pro- Andrey Shilnikov vide a complete topological description of its phase space. Neuroscience Institute and Department of Mathematics We also comment on the role of Gaussian curvature on the Georgia State University stability of regular N-gon relative equilibria. [email protected]

Cristina Stoica Department of Mathematics PP1 Wilfrid Laurier University Computing the Optimal Path in Stochastic Dynam- [email protected] ical Systems In stochastic systems, one is often interested in finding PP1 the optimal path that maximizes the probability of escape Entropy and Dynamical Systems from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to This paper will discuss different definitions of entropy and find an analytic form of the optimal path, and in high- 298 DS17 Abstracts

dimensional systems, this is almost always the case. We propose that extra care must be taken when applying in- have formulated a constructive methodology that is used tervention methods such as quarantine, hospitalization, or to compute the optimal path numerically. The method uti- vaccination to basic models in order to avoid inaccurate lizes finite-time Lyapunov exponents, statistical selection predictions. Delay differential equations, which use fixed criteria, and a Newton-based iterative minimizing scheme, exposed and infectious periods, can provide better realism and can be used for high-dimensional systems where other but are more difficult to use and analyze. We introduce a methods are known to fail. We demonstrate the method multi-infected compartment model to interpolate between with a variety of examples. the ODE model and the DDE model in order to investi- gate the effect a time delay will have on the dynamics of Martha Bauver the system when an intervention method is also included. Montclair State University Using steady state stability and bifurcation analysis, this [email protected] project will provide guidelines for when simpler infectious disease models can be used or more realistic time periods Lora Billings, Eric Forgoston must be incorporated. Montclair State University Department of Mathematical Sciences [email protected], Adrienna Bingham [email protected] The College of William and Mary [email protected]

PP1 Leah Shaw Gabaergic Synaptic Mechanisms in Information College of William and Mary Transmission [email protected]

In this work we analyze a mathematical model (a map parameterized from experimental data) to explore infor- mation transmission at GABAergic PV basket cell- pyra- PP1 midal cell synapses in mouse hippocampus. Specifically, we quantify the information contained in the amplitude of Chaotic Advection in the Alboran Sea: Lagrangian the postsynaptic response induced by Poisson spike trains Analysis of Transport Processes in and Out of the in deterministic and stochastic models of the synapse. In Western Alboran Gyre the stochastic model, the additional variability predictably reduces the overall information transmission compared to the deterministic model. However, we found that there is The Alboran Sea, east of the Strait of Gibraltar, is where a peak in the information transmission around 2 Hz, in Atlantic water enters the Mediterranean as the Atlantic the theta frequency range. With an information efficacy Jet, AJ. The AJ interacts with coastal recirculations, no- measure we confirmed that the action of the map is not tably the Western Alboran Gyre (WAG) a persistent an- uniform over all frequencies, it has the greatest effect in ticyclonic mesoscale eddy. Past studies of local Finite-Size reducing information transmission at very low and high Lyapunov Exponents (e.g. Sayol et. al, Sea surface trans- frequencies. We also show that calcium concentration in port in the Western Mediterranean Sea: A Lagrangian per- the model follows a gamma distribution with shape and spective”, J. Geophys. Res.: Oceans 118.12 2013) highlight scale parameters that depend on mean frequency rate and the periphery of the WAG, implying that it has chaotic flow calcium decay time constant. In the special case when the motion characterized by exponential stretching and folding calcium decay time is notably smaller than the interspike of fluid parcels. Our work examines the near-surface ex- interval, the postsynaptic response follows beta distribu- change between the AJ and WAG in a high-resolution re- tion with parameters that depend on mean frequency rate gional run of the MIT general circulation model (Marshall and minimum recovery rate of the synapse. et al., Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling”, J. Geophys. Res., 102(C3) 1997). This Elham Bayat-Mokhtari model solves the Boussinesq Navier-Stokes equations for an University of Montana - Missoula incompressible fluid with finite-volume spatial discretiza- [email protected] tion on a curvilinear grid. We use Lagrangian methods from dynamical systems, manifolds and lobe analysis, to Emily F. Stone define the moving WAG boundary and quantify advective Dept. of Mathematical Sciences transport across it. We identify the stirring region, where The University of Montana exchanges with the AJ occur, and the WAG core, where [email protected] they do not. Qualifying transport across the boundary of this modeled mesoscale eddy may contribute to the design of observational campaigns. PP1 Contrasting Epidemic Control in Ordinary and De- Genevieve Brett lay Differential Equations Massachusetts Institute of Technology Modeling the spread of epidemics has been a useful tool Woods Hole Oceanographic Institution in predicting the outcome of an infectious disease. Our [email protected] focus is how to model the distributions of exposed and infectious time periods and how applying a disease con- Larry Pratt trol strategy affects the models accuracy. While ordinary Woods Hole Oceanographic Inst. differential equations are widely used for their simplicity, [email protected] they incorporate an exponential distribution for time spent exposed or infectious. This allows a high probability of un- Irina Rypina realistically short exposed and infectious time periods. We Woods Hole Oceanographic Institution DS17 Abstracts 299

[email protected] Department of Mathematics University of California, Davis [email protected] PP1 A Novel Speech-Based Diagnostic Test for Parkin- Mark Goldman son’s Disease Integrating Machine Learning with UC Davis Application Development for Cloud Deployment msgoldman (at) ucdavis (dot) edu Parkinson’s disease remains one of the most poorly diag- nosed neurological conditions despite the critical need of PP1 early diagnosis for effective management and treatment. Hybrid Statistical and Mechanistic Model Guides This work presents a new method of diagnosing Parkin- Mobile Health Intervention for Chronic Pain sons disease and accompanying scalable web and mobile applications towards the goal of employing this diagnostic Patients with sickle cell disease (SCD) often suffer from test on the cloud. This method provides a more simple, daily chronic pain, traditionally managed at home using inexpensive, and time-effective approach than traditional physician-guided recommendations. Recently, physicians diagnosis strategies by requiring the patient to only speak have streamlined their recommendations by utilizing mo- into a microphone attached to their computer or mobile bile health applications (mHealth). These applications col- device before providing a highly accurate diagnosis within lect objective patient data, such as heart rate and temper- seconds. This work employs speech processing algorithms, ature, and subjective information, such as perceived pain an artificial neural network for machine learning, and an and nausea, with the goal of giving immediate personal- application framework with a user-friendly interface that ized pain interventions. In order to balance pain with the packages the speech test and diagnosis results for easy ac- number of opioid and non-opioid drugs taken, we use a hy- cess by patients and physicians. This test was specifically brid statistical-dynamical systems model of perceived pain designed for rapid in-home Parkinsons testing, a capabil- that predicts a time series of pain levels for a variety of ity that is currently not offered by any existing tests avail- interventions (including drugs). This will assist physicians able on the market. The diagnosis test developed here in offering optimal pain interventions remotely. wastestedwithactualpatientdataandwasshowntobe 96.55% accurate. The results of this research thus take a Sara Clifton significant step towards building the first globally available Northwestern University, USA speech test for the diagnosis of Parkinson’s disease. [email protected]

Pooja Chandrashekar Daniel Abrams Harvard University Northwestern University [email protected] [email protected]

PP1 Chaeryon Kang University of Pittsburgh Synchronization of Electrically Coupled Hybrid [email protected] Neuron Models

Electrical coupling between neurons is typically assumed Jessica Li to synchronize network activity. In the inferior olive and University of Californina Los Angeles many other brain regions, the coupled cells have intrinsic [email protected] resonant properties that may complicate this simple pic- ture. We explore this issue by studying synchronization Qi Long in small networks of electrically coupled resonate-and-fire University of Pennsylvania neurons, using the theory of weakly coupled oscillators. [email protected] With our choice of a solvable minimal model, the weak coupling approach permits an analytical model reduction Nirmish Shah to single-variable phase oscillators, from which properties Duke University such as the existence and robustness of synchronized solu- [email protected] tions are easily inferred. Our study has two main results. First, we develop a broadly applicable extension of the phase reduction approach to hybrid models with discon- PP1 tinuous dynamics. Second, we apply our phase reduction Bifurcation Theory and Phase-Lag Variance in 3- method to the resonate-and-fire model, uncovering subtle Node Neural Networks effects on synchronization. Specifically, we show that an even component arises in the phase interaction function Central pattern generators (CPGs) are functional units due to ”reset-induced shear,” a geometric feature of thresh- commonly tied to neurons coupled in half-center oscil- old crossing in two or more dimensions. Although often lators (HCO). We examine these in 3-cell motifs, often overlooked because it has no effect on mean-field coupled forming centers for larger networks, and explore connec- oscillator networks or coupled pairs, we show that this even tivity giving rise to network bursting. We show that in- component in general has complex effects that can either hibitory Fitzhugh-Nagumo-type (FN) networks can pro- support or oppose network synchrony. duce a range of phase-locked states: pacemakers (Pace), traveling waves (Wave), or peristaltic rhythms with recur- Thomas Chartrand rent phase-varying lags (River). We replicate and build UC Davis on results using Hodgkin-Huxley-type models, promoting [email protected] simplified modeling of biologically plausible CPG circuits. Our reduction maintains all generic behaviors and permits Timothy Lewis a broader exploration of the network state space, aiding in 300 DS17 Abstracts

the search of biologically relevant parameters resulting in Madrid the robustness and stability observed in nature. We do this [email protected] with a particular eye for multi-stable rhythms, in which a single biologically relevant parameter induces rhythm Eva Barrena changes in otherwise robustly oscillatory networks. Universidad de Granada [email protected] Jarod Collens Georgia State University Juan A. Mesa Neuroscience Institute and Mathematics Department Department of Applied Mathematics II [email protected] Universidad de Sevilla [email protected] Deniz Alacam Georgia State University Mathematics Department PP1 [email protected] A Robust Torus Used to Control Chaos in Wave- Particle Interactions Aaron Kelly Georgia State University We analyze the dynamics of relativistic particles moving Neuroscience Institute in a uniform magnetic field and interacting with a station- [email protected] ary electrostatic wave. When particles and wave are in resonance, the wave transfers a great amount of energy Drake Knapper to the particles and they are accelerated. By increasing Georgia State University the amplitude of the wave, we increase the amount of en- [email protected] ergy transferred to the particles. However, there is a limit to this process. For large values of the wave amplitude, Andrey Shilnikov the system becomes chaotic and it is no longer possible Neuroscience Institute and Department of Mathematics to regularly accelerate the particles. In this presentation, Georgia State University we create a robust torus in the phase space of the system. [email protected] When the robust torus is properly adjusted, we show that it may be used to control chaos in the system and to re- store the process of regular particle acceleration from low PP1 initial energies. On a Mathematical Model for the Multiplex Line Meirielen C. De Sousa Graph Instituto de F´ısica, Universidade de S˜ao Paulo, Brazil [email protected] Multilayer and multiplex networks explicitly incorporate multiple channels of connectivity in a system and consti- tute the natural mathematical setting for representing sys- Ibere L. Caldas tems whose elements are interconnected through different Institute of Physics kinds of connections (or levels); thus each level (channel, University of Sao Paulo relationship, activity, category) is represented by a layer [email protected] containing all the elements that have connections at that particular level. Notice that two elements may belong to PP1 two different layers if they are connected at more than one Lyapunov-Type Inequalities for α-Th Order Frac- level (family level, or work level or friendship and vicin- ≤ ity levels etc.). The concept of line graph naturally arises tional Differential Equations with 2 <α 3 and when edges in a specific network (a graph with a huge num- Fractional Boundary Conditions ber of nodes and edges) are given more importance than We study linear fractional boundary value problems con- nodes. Recall that the line graph L(G) of an undirected sisting of an α-th order Riemann-Liouville fractional dif- graph G is another graph whose nodes are the edges of G, ferential equation with 2 <α≤ 3 and certain frac- and two nodes of L(G) are adjacent if the corresponding tional boundary conditions. We derive several Lyapunov- edges in G share a common node. Surprisingly line graphs type Inequalities and apply them to establish nonexistence, have only been considered in a reduced number of stud- uniqueness, and existence-uniqueness for solutions of re- ies an applications that involve networks. In the case of lated homogeneous and nonhomogeneous linear fractional multiplex networks even the concept of line graph has not boundary value problems. As a special case, our work cov- been properly addressed. In fact the richer structure of ers and improves some existing results for the third-order a multiplex network makes it difficult to come up with a linear boundary value problems. sound definition. Thus in our poster we will analyze some different approaches to the definition of the line graph as- Sougata Dhar, Qingkai Kong sociated to a multiplex network as well as some potential Northern Illinois University applications. [email protected], [email protected] Regino Criado, Julio Flores, Alejandro Garc´ıa del Amo Universidad Rey Juan Carlos PP1 [email protected], julio.fl[email protected], A Mathematical Model of Parallel Quorum Sensing [email protected] Quorum Sensing (QS) is a cell-cell communication process Miguel Romance that uses signal-receptor binding to regulate genes based Centre for Biomedical Technology (CTB) on cell density, resulting in collective behaviors such as Technical University of Madrid, Pozuelo de Alarc´on, biofilm formation, bioluminescence and stress response. In DS17 Abstracts 301

certain bacterial species such as V. harvey, several paral- and study the existence and stability of standing wave so- lel QS signaling pathways drive a single phosphorylation- lutions of the form dephosphorylation cycle (PdPC), which in turn regulates n iwt i mkθk QS target genes. We propose a coupled cell model for par- e e k=1 φw(r1,r2, ··· ,rn), allel QS in V. Harveyi, and show how different combina- 2 where (rk,θk) are polar coordinates in R and mk ∈ N∪{0}, tions of parallel signals can be used by the bacteria to gain ··· information on their social and physical environments. k =1, 2 ,n. We show the existence of such solutions as minimizers of a constrained functional and conclude from Gaoyang Fan there that such standing waves are stable if 1 0 equipped with a nonlinear local feedback mechanism, we 302 DS17 Abstracts

explore the capability of the system to generate a variety value/eigenfuntion problems, we solve using finite element of global dynamic patterns. We show that by appropriate methods. One feature of the method is that we recover the choice of the feedback rules it is possible to mimic com- dynamics on the manifold in addition to its embedding. plex dynamic patterning seen in cephalopods, such as the We implement the method for some example problems on passing cloud display. various two dimensional domains.

Aaron Fishman, Jonathan Rossiter Jorge L. Gonzalez University of Bristol Florida Atlantic University aaron.fi[email protected], [email protected] [email protected] Jason Mireles-James Martin Homer Departement of Mathematics University of Bristol Florida Atlantic University Department of Engineering Mathematics [email protected] [email protected] Necibe Tuncer Florida Atlantic University PP1 [email protected] Cascades of Saddle Periodic Orbits and Their Man- ifolds Close to a Homoclinic Flip Bifurcation. PP1 Studying the qualitative dynamics of a system of differ- Diffusion and Drift in Volume-Preserving Maps ential equations is an important tool for comprehending the behaviour of natural phenomena. Of particular impor- Nearly integrable volume-preserving maps and incompress- tance are the creation and disappearance of saddle periodic ible flows have many invariant tori on which the dynam- orbits and the interaction of their associated stable and ics is quasiperiodic in the angle variables. There are also unstable manifolds with other invariant objects in phase chaotic orbits in regions between tori. Although invariant space; such that they can bound and destroy basins of at- tori are barriers to chaotic transport for systems with a traction. We study a phenomenon in three-dimensional single action, they are not for systems with multiple ac- vector fields, called a homoclinic flip bifurcation, that is tions. When the nearly integrable dynamics are also sym- characterised by a change in the associated stable (or un- plectic or Hamiltonian, the famous theorem of Nekhoro- stable) manifolds of a real saddle equilibrium from being shev guarantees that this action instability can only occur orientable to non-orientable. The homoclinic flip bifurca- on exponentially long time scales. We study the lack of tion is a codimension-two phenomenon and, under certain Nekhoroshev stability in general volume-preserving maps. conditions, it leads to cascades of period-doubling and ho- We find a reduced model for action drift in low-rank reso- moclinic bifurcations curves in parameter plane; this par- nances and provide supporting numerical results for some ticular homoclinic flip bifurcation is known as case C.We two degree-of-freedom symplectic maps and their closely are interested in understanding how the global invariant related volume-preserving counterparts. manifolds associated with a homoclinic flip bifurcation of case C interact and what this means for the overall dy- Nathan Guillery namics. The cascades of period-doubling and homoclinic University of Colorado - Boulder bifurcations create saddle periodic orbits with manifolds Applied Math that can also be orientable or non-orientable, and their [email protected] interaction may lead to regions in parameters plane with horseshoe dynamics and chaotic attractors. In our study James D. Meiss the invariant manifolds are computed via the continuation University of Colorado of solutions of two-point boundary value problems. Dept of Applied Mathematics [email protected] Andrus A. Giraldo Phd student at the University of Auckland [email protected] PP1 Streamwise Localization of Traveling Wave Solu- Bernd Krauskopf, Hinke M. Osinga tions in Channel Flow University of Auckland Department of Mathematics Channel flow of an incompressible fluid at Reynolds num- [email protected], [email protected] bers above 2400 possesses a number of different spatially localized solutions which approach the laminar flow far up- stream and downstream. We use one such relative time- PP1 periodic solution that corresponds to a spatially localized Parameterization Method for Parabolic Pdes version of a Tollmien-Schlichting wave to illustrate how the upstream and downstream asymptotics can be computed We consider the problem of computing unstable mani- analytically. In particular, we show that for these spanwise folds for equilibrium solutions of parabolic PDEs posed uniform states the asymptotics predict exponential local- on irregular spatial domains. Our approach is based on ization that has been observed for numerically computed the parameterization method, a general functional ana- solutions of several canonical shear flows but never prop- lytic framework for studying invariant manifolds of dynam- erly understood theoretically. ical systems. The method leads to an infinitesimal invari- ance equation describing the unstable manifold, which we Daniel Gurevich, Joshua Barnett solve recursively via power matching scheme. The recursive School of Physics scheme leads to linear homological equations for the jets Georgia Institute of Technology of the manifold which, along with equilibrium and eigen- [email protected], [email protected] DS17 Abstracts 303

Roman Grigoriev tion Method of De la Llave et al. for center manifolds Georgia Institute of Technology that originate from an equilibrium of an ordinary differ- Center for Nonlinear Science & School of Physics ential equation. This allows us to describe the dynami- [email protected] cal behaviour of the ordinary differential equation near the equilibrium, and to prove (e.g. with the help of validated numerics) whether the equilibrium undergoes a bifurcation. PP1 The method will be implemented numerically, and we aim Dynamics of Delay Logistic Difference Equation in to derive explicit error bounds, so as to optimize the neigh- the Complex Plane bourhood on which the center manifold can be proven to exist. This is joint work with Jan Bouwe van den Berg and The dynamics of the delay logistic equation with complex Bob Rink. parameters and arbitrary complex initial conditions is in- vestigated. The analysis of the local stability of this dif- Wouter A. Hetebrij ference equation has been carried out. We further exhibit Vrije Universiteit Amsterdam several interesting characteristics of the solutions of this [email protected] equation, using computations, which does not arise when we consider the same equation with positive real parame- ters and initial conditions. Some of the interesting obser- PP1 vations led us to pose some open problems and conjectures Time Series Analysis of Tropically Linear Systems regarding chaotic and higher order periodic solutions and global asymptotic convergence of the delay logistic equa- Tropical mathematics is the study of semirings with min or tion. It is our hope that these observations of this complex max as their ‘addition’ operation. For example the max- R R ∪{−∞} ⊕ ⊗ difference equation would certainly be an interesting ad- plus semiring max =[ , , ], where dition to the present art of research in rational difference a ⊕ b =max{a, b},a⊗ b = a + b, for all a, b ∈ Rmax. equations in understanding the behaviour in the complex domain. Interestingly there are certain types of queuing system that admit a max-plus linear dynamical systems description. Sk Sarif Hassan Note that these systems are linear with respect to the op- Dept. of Mathematics, UPES, Dehradun, India erations max and plus but non-linear with respect to plus ?Department of Mathematics and times. Previous research into max-plus linear dynam- [email protected] ical systems has focused on the forwards problem of deter- mining a solution from a presupposed model. My poster PP1 will report my recent work on the inverse problem of fitting a max-plus linear dynamical model to a time series record- Beyond Ensemble Averaging in Turbulent Combus- ing. As an example I will focus on a parallel computing tion system. The state of the system recorded by the time series Transient combustion processes constitute important flow includes the times at which different processors complete problems for many practical applications (for example local tasks. I use tropical algebraic regression techniques high-altitude relight, ignition in internal combustion en- to fit a model to this data. This research has possible ap- gines, unstart in scramjets). While classical turbulent com- plications in performance analysis and optimization. bustion models are reasonably accurate for predicting “av- James Hook erage” behavior of systems, their usefulness in capturing University of Bath low frequency but consequently events is questionable. For University of Bath such problems, it is recognized that multiple individual tra- [email protected] jectories should be analyzed. In engineering problems, the trajectory variability stems from lack of knowledge of ini- tial conditions, model approximations, but also external PP1 forcing that can be used to drive the solution in certain Graph Automorphisms and Dynamical Patterns in portions of the attractor. A dynamical system approach is Complex Networks used along with the Lyapunov theory for chaotic systems to obtain more precise insights about the attractor struc- Past theoretical work has indicated that the automor- ture of a canonical turbulent combustion system. Using phisms of the graphs representing complex networks ought the tangent solution space directions given by the so-called to constrain their realized patterns of dynamics. Experi- covariant Lyapunov vectors, an exploration technique of mental and simulation work observing the evolution of dy- the solution attractor is proposed and tested. namics on microfabricated square lattices of inhibitory cou- pled Belousov-Zhabotinsky (BZ) reactions was performed Malik Hassanaly in order to test these theories for highly nonlinear com- Department of Aerospace Engineering plex networks. The nodes of these experimental networks, University of Michigan unlike commonly theoretically considered Kuramoto cou- [email protected] pling, couple repulsively and asymmetrically. These net- works exhibit clustering of like-dynamical behavior which Venkat Raman vary predictably upon the automorphisms of the network. University of Michigan Beyond illustrating these results, work in controlling these [email protected] dynamical clusters through external light perturbations to the light-sensitive BZ reaction will be discussed.

PP1 Ian M. Hunter The Parametrization Method for Center Manifolds Brandeis Martin A. Fisher School of Physics Fraden Lab In this poster we will present a version of the Parametriza- [email protected] 304 DS17 Abstracts

Seth Fraden, Michael M. Norton tic reaction networks converge to piecewise determinis- Brandeis University tic Markov processes in the large system-size limit. The Department of Physics statistics of the piecewise deterministic process can be ob- [email protected], [email protected] tained much more efficiently than those of the exact pro- cess. By coupling sample paths of the exact model to the REMI Boros piecewise deterministic process we are able to reduce the Brandeis Martin A. Fisher School of Physics variance, and hence the computational complexity of the Fraden Lab Monte Carlo estimator. [email protected] Ethan Levien University of Utah Thomas Litschel [email protected],edu Brandeis University Department of Biochemistry [email protected] Paul C. Bressloff University of Utah Department of Mathematics PP1 bressloff@math.utah.edu Synchronization and Clustering of Stochastically- Driven Mixed-Mode Oscillators PP1 As complex systems with many interacting components, 28 Models Later: Best Practices for Modeling the neural systems exhibit rhythms, reflecting emergent coher- Zombie Apocalypse with Real Data ence across the activity of the constituent neurons. In the case of the olfactory system in mammals, where the pri- Zombies! A popular apocalypse scenario in movies, books, mary neural population features noisy mixed-mode oscil- and games, but do these portrayals have a biological ba- lations in the membrane voltage, two prominent rhythms sis? Would a zombie outbreak really end humanity, or appear. We study the role of the two time scales present would such an outbreak burn itself out? In movies, zom- in each neuron, corresponding to their large- and small- bie outbreaks are caused by a rapidly spreading infection. amplitude oscillations, and of the delayed pulse coupling Diseases tend to be self-limiting but, unlike most diseases, between the neurons in the rhythmogenesis. We explore zombie infection does not spread passively. Since the dis- the system using both a reduced phase oscillator model, ease is spread by zombies, who themselves are hunting hu- valid for weak noise and weak coupling, and direct nu- mans, we view the spread of infection as a predator-prey merical simulation. We assess the range of validity of the interaction and the question of human extinction becomes reduced model and, going beyond it, explore the effects of a question of how predator and prey interact. To deter- stronger noise and coupling on the emergent system-level mine the nature of this interaction we ask questions like: behavior. In particular, we find that the noise exhibits Are humans able to protect themselves better in groups? a resonance with the small-amplitude oscillations, causing Are all zombies equally effective at hunting? Do zombies the variability in the occurrence of the large-amplitude os- compete with each other for human prey or cooperate and cillations to change non-monotonically with noise strength. share prey? We design models with these mechanisms Further, we explore a regime where increasing the noise, and fit them to data gathered from “Humans vs. Zom- while uncorrelated across the oscillators, increases the co- bies’ (HvZ), a popular role-playing variant of tag common herence among them. We elucidate the role of the mixed- on college campuses. These models compete using infor- mode characteristic of the oscillators and delayed cou- mation theoretic criteria to determine which models best pling in these phenomena and their effect on the emergent balance predictivity and simplicity. From this we deter- rhythms. Supported by NSF-CMMI 1435358. mine which mechanisms best explain the observed dynam- ics, giving insight into how a zombie outbreak might play Avinash J. Karamchandani out. Our models indicate that, as might be expected of Department of Engineering Sciences and Applied zombies, they do not cooperate or share resources. Ulti- Mathematics mately, results indicate 30% human survival and zombie Northwestern University extinction after a large scale outbreak. [email protected] Ian McGahan Utah State University James Graham [email protected] Northwestern University [email protected] PP1 Hermann Riecke Fractional Order Compartment Models Applied Mathematics Northwestern University Compartment models have been used to model a variety [email protected] of applications including epidemiology, in-vitro disease and pharmacokinetics. Compartment models have been gener- alised to include fractional derivatives in order to capture PP1 a history effect. This generalisation has often been ad-hoc. Coupling Sample Paths to the Partial Thermody- Regularly leading to a violation of flux-balance and dimen- namic Limit in Stochastic Chemical Reaction Net- sion disagreement. This poster addresses these problems works and considers the derivation of a fractional order compart- ment model as a stochastic process, solving the outlined We present a technique for reducing the variance in Monte problems and considering some of the applications. Carlo estimators of stochastic chemical reaction networks. Our method makes use of the fact that many stochas- Anna V. Mcgann DS17 Abstracts 305

UNSW Hardin-Simmons University [email protected] Department of Mathematics [email protected] Christopher N. Angstmann UNSW Australia [email protected] PP1 Spatially Localized Comb-like Turing Patterns Em- Bruce I. Henry bedded in Hopf Oscillations School of Mathematics and Statistics University of New South Wales A distinct mechanism for the emergence of spatially lo- [email protected] calized states embedded in an oscillatory background is presented in the context of 2:1 frequency locking. The John Murray localization is of Turing type and appears in two space School of Mathematics dimensions as comb-like states in either π phase shifted University of NSW Hopf oscillations or inside a spiral core. Specifically, the [email protected] localized states appear outside the 2:1 resonance region and in absence of the well known flip-flop dynamics (as- James Nichols, Austen Erickson sociated with collapsed homoclinic snaking) that is known UNSW to arise in the vicinity of Hopf-Turing bifurcation in one [email protected], [email protected] space dimension. Derivation and analysis of three Hopf- Turing amplitude equations in two space dimensions re- veals a distinct pinning mechanism for Hopf fronts which PP1 in turn allows the emergence of perpendicular (to the Hopf Interjump Statistics of State-Dependent Jump- front) Turing states. The results are shown to agree well Diffusion Processes with the comb-like core size that forms inside spiral waves. Implications to chlorite-iodide-malonic-acid and periodi- We propose a formal PDE approach to study the jumps of cally driven Belousov-Zhabotinsky chemical reactions, and a jump-diffusion process Xt with state-dependent jump in- shaken granular media are also addressed. tensity λ(Xt). The statistics of the jump component alone are of interest in many applications but are not always Paulino Monroy feasible to extract through Monte Carlo or full PDE simu- Instituto de Ciencias F´ısicas, Universidad Nacional lations. We formulate an iterative map on the sequence of Aut´onoma densities of jump locations, resulting in a governing equa- [email protected] tion that can be studied directly. From this density, other statistics, including those of the interjump times can be Arik Yochelis computed. This resulting relationship also illustrates a Jacob Blaustein Institutes for Desert Research, natural and useful connection between the jump statistics Ben-Gurion University and the stationary density of the full process. Applica- [email protected] tions of this framework include studying the thresholding of integrate-and-fire and processivity of molecular motors. PP1 Christopher E. Miles Information Processing Based on Mutually Delay- University of Utah Coupled Optoelectronic Oscillators Department of Mathematics [email protected] A novel information processing method based on delayed dynamical systems has been proposed in [Appeltant et al. James P. Keener Nat. Commun. 2011], which is called reservoir computing University of Utah (RC). RC is based on information processing in the human [email protected] brain and applicable to tasks which it is hard to compu- tationally solve, such as speech recognition and time-series prediction. In this method, an input information is in- PP1 jected into a delayed dynamical system, which is called Amplitude and Frequency for a Nonlinear Oscilla- ”reservoir,” the reservoir nonlinearly transforms the signal tor by Homotopy Analysis Method into a high-dimensional state space in which the signal is represented. The nonlinear transform enables to linearly The Homotopy Analysis Method (HAM) has been shown separate signals in the high-dimensional state space. If to approximate solutions to nonlinear problems of various the reservoir can transform input signals into higher di- types. The method is useful because of the flexibility in mensional state space, it is expected that the performance defining the auxiliary parameter h and that no assumption of RC is improved. In this study, we propose RC based concerning relative size of parameters is required. We use on mutually delay-coupled optoelectronic oscillators, where HAM to approximate frequency and amplitude of a con- the reservoir is the mutually delay-coupled optoelectronic servative nonlinear oscillator which has many applications oscillators. The mutually coupled oscillators have multiple including lasers, epidemics, and microparasites. The os- delay in delayed feedback and coupling. It is expected that cillator is given by the second-order ordinary differential the mutually coupled oscillators with different delay times equation u + u + uu = 0. We compare our results to in feedback and coupling can provide complex nonlinear those gained from traditional perturbation techniques and mapping, which can improve the performance of RC. We briefly discuss conditions for convergence. applied a chaotic time-series prediction task to RC based on mutually coupled oscillators. It is found that our RC Jonathan Mitchell system can produce higher prediction accuracy than RC 306 DS17 Abstracts

based on a single optoelectronic oscillator in the task. Lora Billings, Eric Forgoston Montclair State University Keisuke Nagatoshi, Kazutaka Kanno, Masatoshi Bunsen Department of Mathematical Sciences Department of Electronics Engineering and Computer [email protected], Science [email protected] Fukuoka University [email protected], [email protected], [email protected] PP1 Finite-Time Attractors and Clustering of Inertial PP1 Particles in Fluid Flows An Investigation of a Structured Fisher’s Equation with Applications in Biochemistry Dynamical systems arising from real-life problems often pose the challenge that they are temporally aperiodic, and Abstract Recent biological research has sought to under- that observations are only available over finite time inter- stand how biochemical signaling pathways, such as the vals. In such settings, classical concepts from dynamical mitogen-activated protein kinase (MAPK) signaling path- systems theory typically become ill-defined and standard way, influence the migration of a population of cells during numerical methods do not apply. wound healing. Fisher’s Equation has been used exten- Here, we extend the classical notion of an attractor to sively to model experimental wound healing assays due finite-time flows with arbitrary time dependence (finite- to its simple nature and known traveling wave solutions. time attractors). We provide algorithms for numerically This partial differential equation with independent vari- constructing finite-time attractors in two and three dimen- ables of time and space cannot account for the effects of the sions. As an application of our approach, we investigate MAPK signaling cascade on wound healing, however. To the dynamics of finite size or inertial particles in various this end, we couple a traveling wave analysis with concepts two- and three-dimensional fluid flows. from structured population models to derive a structured Fisher Equation with independent variables of time, space, David Oettinger and biochemical pathway activity and prove the existence ETH Zurich of a self-similar traveling wave solution. We also derive a [email protected] nonautonomous version of the structured Fisher’s Equation in time and space to investigate how different patterns of biochemical activity can influence cell migration in a more George Haller complicated model. ETH, Zurich, Switzerland [email protected] John Nardini Department of Mathematics University of Colorado Boulder PP1 [email protected] Chaos and Global Bifurcations in the Rock- D. M. Bortz Scissors-Paper Bimatrix Game Department of Applied Mathematics University of Colorado, Boulder We consider a Rock-Scissors-Paper game assuming the per- [email protected] fect memory of playing agents X, Y. The interaction ma- trices depend on two parameters X , Y ∈ (−1, 1) and the dynamics are described by the coupled replicator equa- PP1 tions. We provide the description of naturally appear- Investigation of Disease Invasion Using a Stochastic ing heteroclinic network and investigate asymptotic and SIRκ Model chaotic behavior in its neighbourhood. It turns out that certain types of behaviours are never possible or appear Disease invasion is the seemingly spontaneous introduc- in the system only for some parameter values, e.g. fi- tion of a disease into a population from an external source. nite switching. In particular the infinite switching hap- The study of disease invasion is salient to understanding pening near the network cannot be described by the full- zoonotic spillover events, and can give insight into disease shift on two symbols and its form strongly depends on the spread throughout human networks. The random nature parameter values. In the system we observe different bi- of disease invasion requires an investigation that includes furcation scenarios: e.g. transition from order to chaos the use of stochastic disease models. Here we consider both (through Hamiltonian case where invariant tori and chaos a stochastic and deterministic variation of the Susceptible- might be observed), loss of one dimension of the local sta- Infectious-Recovered (SIR) model that explicitly accounts ble manifold of the subcycle or disappearance/appearance for infection from an external source; for the sake of dis- of the local stable/unstable manifolds of the different sub- tinction we call the new model SIRκ. The SIRκ model can cycles at the same time. As well we investigate numerically be seen as a relatively simple analog to the more complex the existence of the heteroclinic connection between differ- SEIDHR Ebola model and will be used to analytically in- ent heteroclinic subcycles (i.e. a superheteroclinic orbit) vestigate behavior that was observed numerically in the full and its bifurcation to the different heteroclinic connections Ebola model. Of particular interest is the outbreak vulner- (forward and backward) from the hyperbolic fixed point ability which describes the susceptibility of a population to (i.e. Nash equilibrium) to the subcycles. [C.Olszowiec disease invasion. ”Complex behavior in cyclic competition bimatrix games”, https://arxiv.org/pdf/1605.00431v4.pdf] Garrett Nieddu Montclair State University Cezary Olszowiec [email protected] Imperial College London DS17 Abstracts 307

[email protected] [email protected]

PP1 PP1 Quantifying the Role of Folding in Nonautonomous Adaptive Neuronal Networks in Olfaction: Mech- Flows: the Unsteady Double Gyre anism of Network Evolution and Information Pro- cessing The combination of stretching and folding will be a reason to establish the chaos in the chaotic dynamics. As a result While learning in the nervous system is usually associated of this, it is required to analyze the stretching and folding with changes in synaptic weights, in the olfactory system to quantify the chaos. Although there are many techniques it is characterized by structural plasticity, i.e., the persis- which can be used to analyze stretching, there are only tent addition and removal of neurons (neurogenesis) and few studies which are useful to quantify folding. In this the extensive formation and removal of spines that provide poster presentation, we will show how the folding can be connections between neurons. We present a computational characterized in terms of curvature along stable manifold model to investigate the network connectivities emerging and unstable manifold. Also we will provide more detailed from both of these structural plasticity mechanisms and the analytical computation and numerical study of the canon- associated processing of inputs by these networks. We find ical example of the nonautonomous double gyre. While that Hebbian spine stability both explains the experimen- the Smale-Birkhoff Theorem is well-known to imply chaos tally observed enhancement of spine stability with spine for either transversal homoclinic intersections or hetero- age and leads to pattern separation. A key element of the clinic intersections associated with a heteroclinic network, network evolution and the resulting information processing we show why folding-driven chaos is also implied in the is the presence of both bottom-up input from the sensory double gyre, in which the heteroclinic intersection is not neurons and top-down input from the cortex. The neu- part of a transversally intersecting heteroclinic network. ronal turnover via adult neurogenesis renders a network Our method of using curvature to identify folding applies structure that allows cortical inhibition of a very specific to general nonautonomous flows in two dimensions, defined set of neurons, which endows the network with the ability for either finite or infinite times. to suppress familiar, distracting stimuli. Kanaththa G. Priyankara SIAM Jaesuk Park, Hongyu Meng [email protected] Applied Mathematics Northwestern University Erik Bollt [email protected], Clarkson University [email protected] [email protected]

Martin Wiechert Sanjeeva Balasuriya Department of Physiology University of Adelaide U. Bern, Switzerland [email protected] [email protected]

Hermann Riecke PP1 Applied Mathematics Prediction of Dynamical Systems by Symbolic Re- Northwestern University gression [email protected] We study the modeling and prediction of dynamical sys- tems based on conventional models derived from mea- PP1 surements. Such algorithms are highly desirable in situ- ations where the underlying dynamics are hard to model Activity Patterns of Neuronal Network with from physical principles or simplified models need to be Voltage-Sensitive Piecewise Smooth Coupling found.We focus on symbolic regression methods as a part of machine learning. These algorithms are capable of learning We present an analysis of activity patterns in a neuronal an analytically tractable model from data, a highly valu- network of three mutually inhibitory cells with voltage- able property. Symbolic regression methods can be consid- sensitive piecewise smooth coupling. While standard fast- ered as generalized regression methods. We investigate two slow analysis fails to describe the dynamics during fast particular algorithms, the so-called fast function extraction jumps due to the voltage-sensitive nature of coupling, which is a generalized linear regression algorithm, and ge- piecewise smoothness of coupling enables us to consider netic programming which is a very general method. Both a sequence of fast subsystems in piecewise way. Our anal- are able to combine functions in a certain way such that ysis shows that slow dynamics as well as fast dynamics a good model for the prediction of the temporal evolution incorporate to determine where fast jumps actually go. of a dynamical system can be identified. We illustrate the algorithms by finding a prediction for the evolution of a harmonic oscillator based on measurements, by detecting Choongseok Park an arriving front in an excitable system, and as a real-world Department of Mathematics application, the prediction of solar power production based North Carolina A&T State University on energy production observations at a given site together [email protected] with the weather forecast.

Jonathan E. Rubin Markus Quade University of Pittsburgh University of Potsdam Department of Mathematics Ambrosys GmbH 308 DS17 Abstracts

[email protected] timal value for producing robust estimates and predictions, using for example the L-curve criterion with Tikhonov reg- Markus W Abel ularization. We extend this result to nonlinear dynamical Universit¨at Potsdam inverse problems, and using a variational approximation to [email protected] a path-integral formulation of data assimilation, show that the regularization strength has an optimality condition for Kamran Shafi prediction quality and robustness of state and parameter UNSW, ADFA, Canberra BC ACT 2610, AUSTRALIA estimates. This result has important implications for varia- k.shafi@adfa.edu.au tional methods such as 4DVar, in which case overenforcing the model constraints may severely limit ones ability to produce high-quality predictions of an observed system. Robert K. Niven he University of New South Wales, Canberra, Australia. Paul Rozdeba [email protected] University of California, San Diego [email protected] Bernd Noack LIMSI-CNRS Technical University Braunschweig PP1 [email protected] Center Manifolds Via Lyapunov-Perron

PP1 The Lyapunov-Perron operator is a theoretical method Ensemble-Based Topological Entropy Computation used to show the existence of the invariant manifold. In 2005, M.S. Jolly and R. Rosa presented an algorithm for We introduce the Ensemble-Based Topological Entropy solving systems with center manifolds based on discretiz- Computation (E-Tec) algorithm. It computes how an ing the Lyapunov-Perron (L-P) operator. However, this initial codimension-one simplex evolves under an ensem- discretization can be difficult and expensive to implement. ble of trajectories. In order to generalize this topologi- First, we provide detailed proofs of the construction of the cal data analysis to higher dimensions, we evolve an ini- center manifold by the L-P operator under the Jolly-Rosa tial codimension-zero triangulation, with vertices of the framework. Second, we present an algorithm based on a simplex and triangulation attached to trajectories from boundary value formulation of the operator. Importantly, the ensemble. When other trajectories intersect the sim- the algorithm is simple and can be adopted by any generic plex, rather than penetrate it, the simplex is stretched like scheme, such as the Runge-Kutta methods. We implement a piece of elastic. In this manner, the simplex will be the algorithm, test it with several examples, and discuss stretched and folded producing an exponential prolifera- applications. tion of codimension-one simplices. The exponential growth rate is the topological entropy, a quantitative measure of Emily Schaal, Yu-Min Chung the complexity and diversity of orbits in a dynamical sys- College of William and Mary tem. [email protected], [email protected] Eric Roberts University of California, Merced [email protected] PP1 Using the Atomic Force Microscope to Measure Kevin A. Mitchell Stiffness at the Nano-Scale University of California, Merced Physics Originally confined to mapping topography, new methods [email protected] have advanced the operating potential of an atomic force microscope (AFM) to include measurements of the stiff- Suzanne Sindi ness, magnetism and thermal conductivity of a sample. University of California, Merced The ability to measure stiffness is of particular importance [email protected] to the nuclear power industry, specifically for models pre- dicting the life of steel components, as few techniques work at the micro and nanometre scale that do not cause sub- PP1 stantial damage to the sample; potentially destroying it or Optimal Regularization for Prediction in Nonlinear causing unknown effects on results. One method for mea- Dynamical Inverse Problems suring stiffness at the nanoscale, contact resonance AFM, uses the nonlinear properties of an AFM probe and consid- In dynamical inverse problems, the goal is to infer the val- ers its shift in resonant frequency as it comes into contact ues of a model systems state variables and parameters from with the sample. This shift is a result of the nonlinear an observed system, and then use this model system to tip-sample interaction, influenced by the geometry, mate- make predictions of future observations. Data assimila- rial properties, and the surrounding fluid. It yields stiff- tion solves this problem by transferring information from ness variation for both hard and soft samples with up to the observed system to the model. Due to measurement nanometre resolution, but has so far been limited to achiev- and model errors, the data assimilation is statistical. One ing relative measurements. In this talk, we describe new should thus seek expected values of functions on the models modelling results that show how to obtain absolute stiffness states and parameters using a probability distribution con- values. We achieve this by developing a dynamical system ditioned on the observed data, and with a prior distribution model of the micro-mechanical cantilever, combined with containing the dynamical model constraints as a regulariza- the fluid dynamics of the surrounding medium, and new tion term. It is a well-known result in linear inverse prob- cantilever calibration methods. We demonstrate the im- lems that the strength of this regularization term has an op- proved technique in action, via experiments on a range of DS17 Abstracts 309

materials. ger tapping to human-computer interfaces have been in- vestigated experimentally. In the 1980s, Haken, Kelso and Namid Shatil Bunz formulated a coupled nonlinear two-oscillator model, University of Bristol whichhasbeenshowntodescribemanyobservedaspects [email protected] of coordination tasks. We present here a bifurcation study of this model, where we consider a delay in the coupling. Martin Homer The delay is associated with delays in processing sensory University of Bristol information and human reaction times. We use numerical Department of Engineering Mathematics continuation to show that delay has a significant effect on [email protected] the observed dynamics. In particular, we find a much larger degree of bistablility between in-phase and anti-phase os- Loren Picco cillations in the presence of a frequency detuning. Interface Analysis Centre, University of Bristol Piotr Slowinski [email protected] University of Exeter, UK [email protected] Oliver Payton Department of Engineering Mathematics, Krasimira Tsaneva-Atanasova University of Bristol University of Exeter [email protected] [email protected]

Bernd Krauskopf PP1 University of Auckland Cone-Dynamical Discriminants: Decoding Neural Department of Mathematics Velocity [email protected]

In neural computation, researchers frequently study com- plex brain networks in order to understand how these PP1 systems represent information. These approaches, which Predicting Financial Stock Crashes Using Ghost increasingly involve machine learning have proved fruit- Singularities ful for gross brain states such as attention or sleep, but with rather poor accuracy for higher cognitive processes. The analysis of a financial dynamical model which exhibits One reason for such limitation may be that the current bubbles and crashes is presented. In addition to bifurcation approaches based upon mean amplitude or frequency are analysis of equilibria and periodic orbits, scaling laws gov- poorly suited to studying information encoded in chaotic erning the period and amplitude of bubbles are provided dynamics implicated in higher cognition. For these applica- analytically. Moreover, a notion of ’ghosts of finite-time tions, we propose a new method by which to decode infor- singularities’ is introduced. This concept will be used to mation from high dimensional, potentially chaotic dynam- predict time of a crash by estimating the periodicity of ics motivated by recent extensions of monotone dynamic bubbles with periodicity of solutions for truncated normal theory. In brief, these theoretical advances show how the form close to a bifurcation point. Finally, the idea is tested behavior of a system may be constrained by the invariance on the theoretical stochastic system as well as on real fi- of its derivatives in certain generalized cones. We apply nancial data. conic invariance to differentiate dynamic processes by de- coding the underlying vector field of an observed time se- Damian T. Smug, Peter Ashwin ries. These classifications are performed in real time to University of Exeter, UK individual data points thus enabling second-to-second dis- [email protected], [email protected] crimination of the system dynamics and, potentially, la- tent state. We provide an analytic solution to differentiate Didier Sornette states using balancing properties of derivatives. We present Department of Management, Technology and Economics results demonstrating the power of the approach over cur- ETH Zurich rent methods in diverse settings including brain-computer [email protected] interfaces, clinical diagnosis/monitoring, and decoding hu- man cognition. PP1 Matthew Singh Stability of Entrainment in Coupled Oscillators Washington University in St. Louis [email protected] Complex systems are ubiquitous and often difficult to con- trol, hence models for the control of such systems are of ShiNung Ching growing interest. As an example, we examine the design Washington University in St. Louis of the entrainment process for a system of coupled phase Dept. of Electrical and Systems Engineering oscillators that are all subject to the same periodic driving [email protected] signal. It was recently shown that in the absence of cou- pling, an appropriately designed input can result in each oscillator attaining the frequency of the driving signal, with PP1 a phase offset determined by the oscillator’s natural fre- Effects of Time-Delay in a Model of Motor Coor- quency. We consider a special case in which the coupling dination tends to destabilize the phase configuration to which the driving signal would send the oscillators in the absence of Motor coordination is an important feature of intra- and coupling. In this setting we use a mean-field approach to inter-personal interactions, and several scenarios from fin- derive stability results that capture the trade-off between 310 DS17 Abstracts

driving and coupling, and compare these results to the un- Here, we study the effect of different parameters in stochas- forced version (i.e. the standard Kuramoto model). tic consensus networks on the emergence of L´evy flights, and we make use of Local Active Information Storage, a Jordan Snyder framework based on transfer entropy, to develop tools for University of California, Davis predicting the onset of a jump based on past data from the [email protected] temporal process. Such tools may be specifically applica- ble to predict similar opinion jumps in social systems and Anatoly Zlotnik, Aric Hagberg generally useful for developing new, multi-scale models for Los Alamos National Laboratory the dynamics of information across networks. [email protected], [email protected] Xin Su Arizona State University PP1 [email protected] Time-Variant Estimation of Connectivity Theodore Pavlic Interactions of network structures are of particular interest Arizona State University in many fields of research since they promise to disclose the School of Computing/Informatics/Decision Systems underlying mechanisms. Additional information on the di- Engineering rection of interactions is important to identify causes and [email protected] their effects as well as loops in a network. This knowledge may then be used to determine the best target for inter- ference, for example, stimulation of a certain brain region, PP1 with the network. Renormalized partial directed coherence Information Theoretical Noninvasive Damage De- has been introduced as a means to reconstruct the network tection in Bridge Structures structure by investigating Granger causality in multivariate systems. A major challenge in estimating respective coher- Damage detection of mechanical structures such as bridges ences is a reliable parameter estimation of vector autore- is an important research problem in civil engineering. Us- gressive processes. We discuss two shortcomings typical ing spatially distributed sensor time series data collected in relevant applications, i.e. non-stationarity of the pro- from a recent experiment on a local bridge in upstate cesses generating the time series and contamination with New York, we study noninvasive damage detection using observational noise. To overcome both, we present an ap- information-theoretical methods. Several findings are in proach combining renormalized partial directed coherence order. First, the time series data, which represent ac- with state space modelling. We present the application of celerations measured at the sensors, more closely follow this approach to different neural signals. Laplace distribution than normal distribution, allowing us to develop parametric estimates for various information- Linda Sommerlade theoretic measures such as entropy and mutual informa- Institute for Complex Systems and Mathematical Biology tion. Secondly, as damage is experimentally introduced University of Aberdeen by the removal of bolts of the first diaphragm connection, [email protected] the interaction between spatially nearby sensors as mea- sured by mutual information become weaker, suggesting Claude Wischik that the bridge has been “loosened”. Finally, using a pro- TauRx Therapeutics Ltd posed oMII procedure to prune away indirect interactions, [email protected] we found that the primary direction of interaction or in- fluence aligns with the traffic direction on the bridge even Bjoern Schelter after damaging the bridge. Institute for Complex Systems and Mathematical Biology [email protected] Amila N. Sudu Ambegedara Clarkson University [email protected] PP1 Dynamical Characterizations of Complex Behavior Jie Sun in Consensus Networks with Stochastic Link Fail- Mathematics ures Clarkson University [email protected] Consensus dynamics of multi-agent systems on networks has long been studied in both engineering and social- Kerop Janoyan, Erik Bollt science contexts with both stochastic and deterministic Clarkson University models. Recently, it has been discovered that L´evy-like [email protected], [email protected] hyper-jump diffusion emerges in consensus networks when stochastic link failures are incorporated into their network models; that is, the size of these jumps is power-law dis- PP1 tributed, as in L´evy flight behavior. In the engineering 3D Super-Lattice Solutions in Reaction-Diffusion literature where this phenomenon has been studied, the fo- Systems cus has been on developing update rules that ensure con- vergence in scenarios with stochastic link failures. How- Earlier research on reaction-diffusion systems has focused ever, given that L´evy flights emerge in natural systems as on two-dimensional domains and found a wealth of solu- efficient search strategies, it could also be potentially use- tions defined on simple and super-lattices. Even in three ful to induce them in stochastic search algorithms to im- dimensions the solutions on the simple, face centered and prove upon metaheuristics like simulated annealing. Fur- body centered cubic lattices have been studied. We ex- thermore, detailed characterization of these behaviors may plore three-dimensional solutions on the super-lattices cor- help to better understand sudden shifts in public opinion. responding to 24- and 48-dimensional representations of DS17 Abstracts 311

the octahedral group. James P. Keener University of Utah Timothy K. Callahan, Rebecca Tobin [email protected] Embry-Riddle Aeronautical University Department of Physics [email protected], [email protected] PP1 Study of Some Nonlocal Transport Equations: Ap- plication to Stochastic Neural Dynamics PP1 This work deals with the analysis of the limit equation Topological Complexity of the Greenberg-Hastings (mean field) of a stochastic model of spiking neurons in- Cellular Automata teracting with spikes and described as a piecewise deter- ministic Markov process. The limit equation, recently de- The modeling and analysis of excitable media is predomi- rived by N.Fournier etal., describes the distribution of the nantly done by PDEs, in particular of FitzHugh-Nagumo- membrane potential as a nonlinear and non-local transport type. A phenomenological alternative are models with dis- equation with boundary condition. We first focus on the crete time, space and states, i.e. cellular automata (CA). case where there is no electrical synapses stochastic model A special one-dimensional CA for excitable media was in- reduces to a pure jump process. In this case, the mean field troduced by Greenberg and Hastings in the late 1970’s. admits a 1D parametrised family of stationary solutions. We revisit a topological complexity result for the three- The linearised equation around a stationary solution in- state model (rest state, refractory state, excited state) by 0 volves a positive C -semigroup which enables to precisely R. Durrett and J. Steif in the 90’s, and give an explicit de- pinpoint the spectrum of the infinitesimal generator. The scription of the non-wandering set. Next, we generalize this result by changing the alphabet by an arbitrary number of main difficulty is that the nonlinear semigroup is non dif- refractory states. It turns out that the model’s complexity ferentiable. We overcome this issue with a time rescaling results from the annihilation events of pulses upon collision and a cut-off. We then show the exponential stability of at certain space-time locations. Moreover, we give an esti- the stationary solutions. This very interesting result can mation of the topological entropy in case of more than one be recast as the existence of an attracting center manifold excited state. Finally, we formulate some open questions for a transport equation. Due to their non regularising and outline connections to the PDE perspective. properties, the existence of such manifold is notoriously difficult. In the case with electrical synapses, numerical Dennis Ulbrich, Jens Rademacher, Marc Keeb¨ohmer simulations show the occurence of a Hopf bifurcation. In University of Bremen particular, this (numerically) breaks a recent conjecture [email protected], [email protected], about the stability of the stationary distribution. [email protected] Romain Veltz INRIA Sophia-Antipolis [email protected] PP1 A Mathematical Model for Post-Replicative DNA Methylation Dynamics PP1 Resource-Transport Dynamics Induces Criticality DNA methylation is an important epigenetic mechanism in Networks of Excitable Nodes used by cells to repress gene expression. Interestingly, DNA replication, a function necessary for cell division, disrupts Brain function needs to be constantly regulated to avoid the methylation pattern. Since perturbed methylation pat- too much or too little activity. More precisely, it has been terns are associated with aberrant gene expression and can- hypothesized that synapse strengths are tuned so that the cer, DNA methyltransferases (Dnmts) must restore the cor- brain operates at the critical point of a phase transition, rect pattern following DNA replication. However, the exact characterized by experimentally observed power-law dis- mechanisms of this restoration remain under investigation. tributed patterns of activity known as neuronal avalanches. We propose a mean-field model to study the dynamics of A fundamental question is what mechanisms regulate the post-replicative restoration of methylation patterns. Dn- synapse strengths so that the network operates at this crit- mts perform methylation by adding a methyl group to cy- ical point. We propose a model where synapse strengths tosines at CpG sites, in which cytosine and guanine are are regulated by metabolic resources distributed by a sec- consecutive in the DNA. These CpG sites are found in re- ondary network of glial cells. We find that this mul- gions of high density, termed CpG islands (CGIs), and re- tilayer network robustly produces power-law distributed gions of low density in the genome. Nearly every CpG avalanches over a wide range of parameters, and that the site in a CGI has the same state, either methylated or un- glial cell network protects the system against the destabi- methylated. Meanwhile, nearly all CpG sites in regions lizing effect of local variations in parameters. For homo- of low density are methylated. We developed a stochastic geneous networks, we derive a reduced 3-dimensional map model of the restoration of the methylation pattern follow- which reproduces the behavior of the full system and gives ing replication by considering the methylation activity of insights into the robustness of the critical state to param- Dnmts. Our model predicts that the methylation of CpG eter changes. sites exhibits bistable behavior, with methylation in CGIs maintained primarily by the processivity of Dnmts, while Yogesh Virkar the methylation of CpG sites outside of CGIs is maintained Department of Computer Science by localization of Dnmts to the replication fork. University of Colorado, Boulder [email protected] Kiersten Utsey University of Utah Woodrow Shew Department of Mathematics University of Arkansas [email protected] [email protected] 312 DS17 Abstracts

Juan G. Restrepo ity. In the parametric zone of bistability, weak-noise am- Department of Applied Mathematics plitudes may strongly inhibit the neurons spiking activity. University of Colorado at Boulder Surprisingly, increasing the noise strength leads to a mini- [email protected] mum in the spiking activity, after which the activity starts to increase monotonically with increase in noise strength. Edward Ott We investigate this phenomenon in details and show that University of Maryland it always occurs when the initial conditions lie in the basin Inst. for Research in Electronics and Applied Physics of attraction of the stable limit cycle. For initial condi- [email protected] tions in the basin of attraction of the stable fixed point, the phenomenon however disappears, unless the time-scale separation parameter of the model is bounded within some PP1 interval. Using the tools from stochastic sensitivity anal- Hamiltonian Structure and Stability Results for ysis of attractors, we provide a theoretical explanation to Idealised Flows on a Three Dimensional Periodic this phenomenon. Domain Emar Marius Yamakou,JuergenJost The Euler Equations for inviscid incompressible flows pro- Max Planck Institute for Mathematics in the Sciences vide a simplified model for studying fluid dynamics. We [email protected], [email protected] consider an anisotropic three dimensional periodic domain. We first show that this problem has a new Poisson structure in Fourier space for the vorticity formulation , opening the PP1 possibility of Energy-Casimir type stability results. These Demographic Noise Slows Down Cycles of Domi- equations also admit a family of shear flows with vortic- nance in Ecological Models ∗ ity Ω (x)=cos(κ1p1x1 + κ2p2x2 + κ3p3x3)forpi integers. We first show that the linearised system around these sta- We study the phenomenon of cyclic dominance in the tionary solutions almost always reduces to the equivalent paradigmatic Rock–Paper–Scissors model, as occuring in linearised system in the two-dimensional problem. This both stochastic individual-based models of finite popula- demonstrates there are linearly stable solutions of the form tions and in the deterministic replicator equations. The cos(κ1p1x1) for sufficiently large values of κ2,κ3.Thismean-field replicator equations are valid in the limit of is an extension to the shear flow stability result due to large populations and, in the presence of mutation and Arnold. We then show that for most other values of pi unbalanced payoffs, they exhibit an attracting limit cy- and κi, there is linear instability. These results are an cle. The period of this cycle depends on the rate of mu- extension of results from [Dullin & Marangell & Worthing- tation; specifically, the period grows logarithmically as the ton, Instability of Equilibria for the 2D Euler Equations on mutation rate tends to zero. However, this behaviour is the torus, SIAM Journal on Applied Mathematics, 76(4), not reproduced in stochastic simulations with a fixed fi- (2016) 1446-1470] and [Dullin & Worthington, Stability Re- nite population size. Instead, demographic noise present sults for Idealised Shear Flows on a Rectangular Periodic in the individual-based model dramatically slows down the Domain, arXiv:1608.06109 [math.DS], 2016]. progress of the limit cycle, with the typical period growing as the reciprocal of the mutation rate. Here we develop a Joachim Worthington theory that explains these scaling regimes and delineates University of Sydney, Australia them in terms of population size and mutation rate. We [email protected] identify a further intermediate regime in which we con- struct a stochastic differential equation model describing the transition between stochastically-dominated and mean- PP1 field behaviour. Bifurcations in the Piecewise-Linear Standard Nontwist Map Qian Yang University of Bath A piecewise-linear version of the area-preserving standard [email protected] nontwist map [D.del Castillo, J.M. Greene and P.J. Morri- son, Physica D 91, 1 (1996)] is considered. Using symme- try lines and involution maps I compute periodic orbits to PP1 analyze various bifurcations, including periodic orbit col- A Stable Numerical Algorithm for Calculating the lisions and separatrix reconnections, that can suppress or Rate of Exponential Forgetting in HMM enable global transport. We consider a finite state hidden Markov model (HMM) Alexander Wurm with multidimensional observations. Under some mild as- Department of Physical & Biological Sciences sumptions, the prediction filter forget almost surely the Western New England University initial condition exponentially fast. However, it is very [email protected] difficult to calculate this asymptotic rate of exponential loss of memory analytically. We restate this problem in the setting of random dynamical system and use the Lya- PP1 punov exponents of the induced random dynamical sys- n−1 Weak-Noise-Induced Transitions with Inhibition tem defined in the projective space R to approximate and Modulation of Neural Spiking Dynamics the convergence rate. Finally, we propose a stable numer- ical algorithm to calculate the rate of exponential forget- We analyze the effect of weak-noise-induced transitions on ting semi-analytically. The numerical simulation result and the dynamics of a neuron model in a bistable state con- comparison with current upper bound in literature will be sisting of a stable fixed point and a stable unforced limit shown in the presentation. cycle. Bifurcation and slow-fast analysis give conditions on the parameter space for the establishment of this bistabil- Xiaofeng Ye,YianMa DS17 Abstracts 313

University of Washington rium approach to this situation. Distinct transitions are [email protected], [email protected] observed which depend on this cost function, and ramifi- cations are discussed. Hong Qian Department of Applied Mathematics Andrew Belmonte University of Washington Department of Mathematics, Pennsylvania State [email protected] University [email protected]

PP1 Matthew Young Interactions of Solitary Pulses of E. Coli in a One- Department of Mathematics Dimensional Nutrient Gradient Pennsylvania State University [email protected] We study an anomalous behavior observed in interacting E. coli populations. When two populations of E. coli are PP1 placed on opposite ends of a long channel with a supply of nutrient between them, they will travel as pulses toward Lateral Inhibition Networks in Rat Olfactory Bulb one another up the nutrient gradient. We present exper- The first structure involved in mammalian olfactory pro- imental evidence that, counterintuitively, the two pulses cessing, the olfactory bulb, is largely comprised of func- will in some cases change direction and begin moving away tional units, glomeruli, that receive direct inputs from ol- from each other and the nutrient back toward the end of factory receptors. Glomeruli are thought to communicate the channel from which they originated. Simulations of the with each other through a network of inhibitory short axon Keller-Segel chemotaxis model reproduce the experimental cells. The extent and specificity of the inter-glomerular in- results. To gain better insight to the phenomenon, we in- hibitory network is actively debated. However, it is agreed troduce a heuristic approximation to the spatial profile of that the odor-evoked patterns of glomeruli activity result- each population in the Keller-Segel model to derive a sys- ing from the network probably play an important role in tem of ordinary differential equations approximating the odor coding. In this study we make a network model of temporal dynamics of its center of mass and width. This the inter-glomerular connectivity in the olfactory bulb by approximate model simplifies analysis of the global dynam- using data to constrain the number of inhibitory cells in ics of the bacterial system and allows us to efficiently ex- each glomerulus, the distribution of the number of out- plore the qualitative behavior changes across variations of going connections per cell, and the number and location parameters, and thereby provides experimentally testable of targets each cell contacts. We then derive analytically hypotheses about the mechanisms behind the turnaround the in-degree distribution of this network and compute its behavior. other graph-theoretic properties as we allow the specificity Glenn S. Young of connections to vary. We find that the network properties Pennsylvania State University can be substantially different from those of the well-studied Department of Mathematics classes of scale-free, small-world and random networks. Fi- [email protected] nally, we add firing-rate dynamics to the network. Using realistic receptor activation profiles as inputs to our net- work, we study the effect of the connection specificity on Mahmut Demir the resulting patterns of glomerular activity. Department of Molecular, Cellular and Developmental Biology Daniel R. Zavitz Yale University University of Utah [email protected] Department of Mathematics [email protected] Hanna Salman University of Pittsburgh Alla Borisyuk Department of Physics and Astronomy University of Utah [email protected] Dept of Mathematics [email protected] Bard Ermentrout, Jonathan E. Rubin University of Pittsburgh Matt Wachowiak, Isaac Youngstrom Department of Mathematics University of Utah School of Medicine [email protected], [email protected] Department of Neurobiology & Anatomy [email protected], [email protected] PP1 Non-Cooperative Games with Cost of Information PP2 In classical game theory, competitive games involving two Social Experience Reconfigures Decision-Making players typically deal with full information situations, in Circuits in Zebrafish which each players know all of the rules of the game, as well as all strategy choices and associated payoffs for both Understanding how social factors influence nervous sys- players; the only uncertainty is what the opposing player tem function is of great importance. Using zebrafish as will choose - what strategy will be played. We study two- a model system, we demonstrate how social experience af- player games in which one or both players may obtain par- fects decision-making to enable animals to produce socially tial information about the other players choice, but with appropriate behavior. Based on experimental evidence and some cost. We propose a formalism in which partial in- computational modeling, we show that behavioral deci- formation can be purchased, and adapt the Nash equilib- sions reflect the interplay between competing neural cir- 314 DS17 Abstracts

cuits whose activation thresholds shift in accordance with on manifolds are flows on a sphere with geophysical ap- social status. We demonstrate this through analysis of the plications. For such flows, besides considering the generic behavior and neural circuit responses that drive escape and Riemannian manifold approach, we present an alternative swim behaviors in fish. Our computational model of the es- method by using Euclidean representation of the equations cape and swim circuits replicates our findings and suggests with a projection that yields to a set of coupled differential that social status-related shift in circuit dynamics could algebraic equations. In order to validate this numerical so- be mediated by changes in the relative excitability of the lution of differential algebraic equations, we have derived escape and swim networks. an exact solution for the eigenvalues and eigenvectors of the deformation tensor for vortex flows on the sphere and Sungwoo Ahn demonstrate that the numerical results are in agreement Arizona State University with the exact solutions up to the numerical integration [email protected] tolerance.

Thomas Miller, Katie Clements Siavash Ameli East Carolina University University of California, Berkeley [email protected], [email protected] [email protected] PP2 Choongseok Park Department of Mathematics Symbolic Dynamics Applied to a Numerical Simu- University of Pittsburgh lation of a Perturbed Hill’s Spherical Vortex. [email protected] 3D homotopic lobe dynamics (HLD) is a new symbolic method of describing topological dynamics for fully 3D Eoon Hye Ji systems. Here we apply this new method to a numeri- UCLA cally computed perturbed Hill’s spherical vortex flow. We [email protected] consider the scattering of passive tracers that are drawn into and then ejected from the vortex. We focus on the Fadi Issa numerical computation of fractal scattering functionsthe East Carolina University time advected particles are trapped within the vortex as a [email protected] function of two impact parameters. We compare the frac- tal self-similarity of these scattering functions to those pre- dicted by 3D HLD. Our new method also produces a lower PP2 bound on the topological entropy of our system which ap- Network Bursting in Melibe Swim CPG proaches the true topological entropy as we add more nu- merical data. Central pattern generators(CPGs) are neural networks, which can produce rhythmic activity and are responsible Joshua Arenson for generating various oscillatory behaviors like swim lo- University of California Merced comotion. We develop a Hodgkin-Huxley type highly de- [email protected] tailed and biologically plausible model using the extensive data intracellularly recorded from swim CPG of the sea Spencer A. Smith slug Melibe Leonida. We study the rhythmogenesis of oscil- Mount Holyoke College latory patterns emerging in such network circuits consisting [email protected] of half-center oscillators, which are the building blocks of many neural networks including CPGs. The four interneu- Kevin A. Mitchell rons in the modeled CPG are coupled by multiple chemical University of California, Merced inhibitory and excitatory synapses, which are represented Physics by alpha and dynamical models. [email protected] Deniz Alacam Georgia State University PP2 Mathematics Department Channel Capacity of Cardiac Action Potential [email protected] Transmission and Spiral Wave Initiation

Andrey Shilnikov Alternans of action potential duration (APD) is an oscilla- Neuroscience Institute and Department of Mathematics tory dynamics of cardiac cells that could lead to conduction Georgia State University block and spiral wave initiation. However, it is unclear as to [email protected] how this intrinsic property of the cardiac tissue impacts in- formation transmission at the tissue level. To quantify the channel capacity of information transmission in in the pres- PP2 ence of APD alternans and conduction block, we performed Theory and Computation of Nonlinear Deforma- numerical simulation of cardiac tissue using the Fenton- tion Spectra of Flows Karma model that is known to exhibit these properties. We stimulated one cell in a one-dimensional cable and a two- We present the mathematical theory and efficient computa- dimensional lattice with a fixed basic cycle length (BCL) tional framework for spectral information of the nonlinear that ranged from 200 msec and 300 msec. For each cell, the (large) deformation tensor for flows on general Riemannian time series of cardiac action potential was coarse-grained to manifolds characterized by a generic metric tensor. The 1 when excited (during the APD at 90% repolarization, or eigenvalues of the nonlinear deformation tensor have ap- APD90) or 0 when resting. We considered the cell of stim- plications in fluid mechanics. A common example of flows ulus as an information source/transmitter with an input X, DS17 Abstracts 315

any other cell in the system as a receiver/destination with [email protected] an output Y, and the intervening cells between them as a binary asymmetric channel. By calculating the probability of transmission errors, we obtained the channel capacity as PP2 a function of BCL and distance away from the information Three-Dimensional Chattering of Rigid Bodies source. Our approach is clinically applicable to quantify the propensity of the cardiac tissue to develop lethal ar- Ideally rigid objects establish sustained contact with one rhythmia in human patients. another via an infinite sequence of collisions accumulating in finite time. Similar phenomena in hybrid dynamical sys- Ameneh Asgari-Targhi tems are often referred to as Zeno behaviour or complete University of Glasgow chatter (CC). We examine the chattering of a rigid body School of Mathematics and Statistics, hitting a stationary rigid plane. It is assumed that several [email protected] points of the object reach the plane almost simultaneously inducing a rapid sequence of impacts, during which the Hiroshi Ashikaga effect of external forces other than the contact forces is Johns Hopkins University School of Medicine negligible. The analysis of a chattering rod by Goyal and [email protected] others revealed that they undergo CC or they leave the sur- face after a finite sequence of impacts (incomplete chatter). The exact conditions of these two types of behaviour were PP2 determined analytically. We seek analogous conditions for Computing the Optimal Path for Stochastic Popu- objects with more than 2 potential contact points. The lations Coupled Through Migration difficulty of the general problem lies in that these systems - unlike the rod - may undergo irregular impact sequences. We consider a two-population stochastic model where both We use and extend the recently developed theory of com- populations are connected through migration. Each pop- mon invariant cones of linear operators to develop sufficient ulation of species undergoes birth and annihilation events. conditions of CC. Numerical simulations indicate that our Using a master equation approach and a WKB (Wentzell- sufficient condition is sharp in many situations. Potential Kramers-Brillouin) approximation, we find the theoretical engineering applications including spacecraft landing un- mean time to extinction. Numerical Monte Carlo simula- der microgravity and earthquake protection are outlined. tions agree well with the analytical results for a single pop- ulation, and also allow for an exploration of how changes Tam´as Baranyai in migration rates affect the behavior of extinction events PhD student, Dept. of Mechanics, Materials and in coupled populations. Using an iterative method, we nu- Structures merically compute the optimal path to extinction and the Budapest University of Technology and Economics mean time to extinction for coupled populations. Further- (Hungary) more, we explore how changes in migration rates in a cou- [email protected] pled system affect the curvature of the optimal path. Peter L. Varkonyi Alexa Aucoin Budapest University of Technology and Economics Montclair State University [email protected] [email protected]

Lora Billings, Eric Forgoston PP2 Montclair State University Closed-Form Solutions and New Algorithm for Department of Mathematical Sciences Two-Dimensional Frictional Problems [email protected], [email protected] We’ve presented closed-form solutions for two-dimensional frictional problem loaded by constant external loads. These solutions define the response during the slip period includ- PP2 ing the exact time and position of the transition from slip to Conservation Laws in Nonlocal Equations stick. By assuming that the external loads can be treated as constant if the time during is sufficiently small, we pro- In this poster we consider the dynamics of the shift map posed a new numerical algorithm for two-dimensional fric- on the set of bounded solutions to the nonlocal equation tional problems excited by time-varying external loads. We

 also developed an analytical patch for the recommencement Au + K ∗ u + f(u)=0, of slip after stick, since zero initial velocity at the bound- ary of friction circle results in difficulties for both analyti- where K is a symmetric convolution kernel. Such equations cal and numerical solutions. In order to identify the states arise naturally when studying coherent structures in e.g. of stick and slip, we introduced a two-state boolean vari- spatially discretized parabolic equations, neural field equa- able B with B=0,1 indicating stick and slip respectively. tions, or nonlocal NLS. We will show how to construct con- We utilized this variable step-size algorithm to obtain the served quantities (or, when A>0, a Lyapunov function) response of the system during dynamic slip periods includ- for the shift dynamics. We then discuss some dynamical ing an accurate determination of the time and position of consequences of these structures. This is joint work with slip/stick transitions. We’ve presented numerical exam- Arnd Scheel (University of Minnesota), Jan Bouwe van den ples of two-dimensional problems involving multi-stops to Berg (VU Amsterdam), and Rob Vandervorst (VU Ams- evaluate the performance of the numerical algorithm. The terdam). results show much more realistic behaviour in the neigh- bourhood of slip/stick transitions, and the new algorithm Berry Bakker is also significantly more efficient computationally than the VU University Amsterdam conventional method when time steps are chosen so as to 316 DS17 Abstracts

ensure comparable levels of precision. University of Bristol [email protected] Jim Barber University of Michigan [email protected] PP2 Particle Diffusion and Competitive Receptor Bind- ing PP2 Glycinergic Neurons in the Pre-Bo¨tzinger Complex We consider a setup in which N particles are initially re- Are Crucial for Eupnea: An Optogenetic Study leased into an environment and diffuse freely. Much of the domain boundary is absorbing, where the particles can es- The core structures of the respiratory central pattern gen- cape the domain. The rest of the boundary consists of erator (CPG) are the pre-B¨otzinger (pBC) and B¨otzinger patches that we call “receptors” and that can switch be- (BC) complexes. Synaptic inhibition in these CPG cir- tween being reflecting and absorbing. An absorbing recep- cuits has been proposed to be essential for the eupneic res- tor turns reflecting (“unavailable”) once a particle collides piratory pattern. We employed viral vectors to express with it; a reflecting receptor turns absorbing (“recharges”) channelrhodopsin (ChR2) or archaerhodopsin (Arch) un- spontaneously after spending a random exponentially dis- der control of the glycine transporter 2 promoter in the tributed amount of time in the reflecting state. We are pBC of Wistar rats. Optoexcitation of pBC glycinergic interested in the distribution of the number of particles neurons significantly increased respiratory frequency due that are absorbed by the receptors. We find that the num- to the loss of E2. Optoinhibition also increased the respi- ber of the receptor-absorbed particles has an upper-bound ratory frequency, but unlike optoexcitation, activation of of the order of (log N). By using a Markov chain approx- Arch suppressed the post-I phase. We used an established imation in a special case, we also estimate the complete computer model of the respiratory CPG to mechanistically distribution of the number of the receptor-absorbed par- explain these results. In the model, we found that simi- ticles, and find that it matches simulation results in one- lar perturbations could be induced by altering the activity and two- dimensional environments. Finally, we use our of the inhibitory population of pBC inspiratory neurons. approach to model synaptic cleft, and find that variability Simulated excitation of this population suppressed the BC in the number of postsynaptic bindings is strongly affected E2 neurons and augmented oscillation frequency. Inhibit- by the recharging time of the receptors, and only weakly ing this population disinhibited post-I and E2 expiratory by the number of molecules released. We also find that lo- populations causing tonic discharge, thus eliminating pha- calization of receptors in clusters and directly across from sic post-I activity and creating conditions for endogenous the release site allows for faster and more reliable onset of bursting of excitatory inspiratory neurons at a higher respi- the postsynaptic response, especially for slowly recharging ratory frequency. We demonstrated the functional impor- receptors. tance of glycinergic interneurons in the pBC for generation of the eupneic pattern and provided mechanistic insights Alla Borisyuk for these responses using mathematical modeling. University of Utah Dept of Mathematics William H. Barnett [email protected] Neuroscience Institute Georgia State University Gregory A. Handy [email protected] University of Utah Mathematics Department [email protected] Yaroslav Molkov Department of Mathematics & Statistics Sean Lawley Georgia State University Duke University [email protected] [email protected]

Ana Abdala, Beihui Liu School of Physiology PP2 University of Bristol Improving Network Inference of Oscillatory Sys- [email protected], [email protected] tems: The Impact of False Positive and False Neg- ative Conclusions About the Presence Or Absence Davi Moraes of Links Department of Physiology University of Sao Paulo Many research groups have focused on the inference of net- davimoraes@rfi.fmrp.usp.br works from data such as brain networks from observed EEG. Brain networks but also others are typically clas- sified into main prototypic networks, e.g., Erdos-Renyi, Sergey Kasparov, Lucas Koolen Small World and Scale Free Networks. Two challenges School of Physiology are particularly important: to reliably obtain links in a University of Bristol network and to use network characteristics to uniquely de- sergey.kasparov@ bristol.ac.uk, [email protected] termine the type of network. The correct reconstruction of networks is hampered by mis-estimation of links due to Jeffrey Smith statistical uncertainties, unobserved processes, noise con- Cellular and Systems Neurobiology Section tamination to name just a few. Mis-estimated links in a NIH network typically alter the key characteristics. We inves- [email protected] tigate to which extent classical statistical approaches to estimate links in a network are reliable; furthermore, we Julian Paton investigate if common rules of type I and II errors should School of Physiology be modified to achieve superior inference. Our approach is DS17 Abstracts 317

based on the topological analysis of the detected networks tors is a crucial component for long distance transfer of taking into account the role of two important parameters: cargo inside neurons. We mathematically model the move- false positive and false negative decisions about the pres- ment of a motor-cargo complex along a parallel arrange- ence of links. We present simulation results showing that ment of microtubules. The first passage time for a diffus- typical alpha values for statistical inference are suboptimal ing motor to reattach to a microtubule is computed, and when, e.g., the degree distribution is to be estimated. We key quantities of interest are derived with a macroscopic suggest a novel procedure to optimally balance false pos- viewpoint of effective transport process over multiple mo- itive and negative conclusions. This enables us to obtain, tor attachment and detachment events. e.g., better estimates for the average shortest path length or the degree distribution. Abhishek Choudhary Rensselaer Polytechnic Institute Gloria Cecchini [email protected] University of Aberdeen [email protected] PP2 Bjoern Schelter Noise-Induced Stabilization of Collective Behavior Institute for Complex Systems and Mathematical Biology in Oscillatory Ensembles [email protected] We study the role of noise in the emergence of non-trivial collective dynamics of coupled oscillators. Contrary to the Marco Thiel standard expectation that noise contributes to destabilize University of Aberdeen or to smooth out coherent patterns, we discuss some ex- Scotland amples where a weak noise can stabilize a collective state: [email protected] the so-called self-consistent partial synchrony. This regime is characterized by a smooth non-uniform distribution of Linda Sommerlade the phases of the single oscillators, where the frequency Institute for Complex Systems and Mathematical Biology of the mean field differs from that of the single units. University of Aberdeen In a simple, noiseless, biharmonic Kuramoto-Daido setup, [email protected] self-consistent partial synchrony can be found in a finite range of parameter values, but it is not everywhere sta- ble. Numerical simulations of large ensembles show that PP2 a small amount of noise acting on each oscillator can re- Partial Synchronization in Pulse-Coupled Oscilla- instate partial synchrony. We study the phenomenon at a tor Networks macroscopic level by transforming the noiseless nonlinear continuity equation into a suitable Fokker-Plank equation We consider N leaky-integrate-and-fire oscillators coupled and solving it by means of a perturbative approach. The by a sum of α-function pulses (whose shape depends on a corresponding linear stability analysis reveals the overall delay time τ) weighted by a coupling parameter K.Signif- bifurcation scenario, separating the region where partial icant insight can be gained when the oscillators are identi- synchrony is stable from that where the asynchronous dy- cal, and identically coupled in an all-to-all network. Pre- namics prevails. The results are in close agreement with vious studies have revealed how the stability of the fully microscopic simulations. A similar situation is observed in synchronous and the fully asynchronous splay states de- an ensemble of weakly coupled Rayleigh oscillators, sug- pend on the value of τ,andthesignofK (inhibition vs gesting that the above instance about noise is not isolated. excitation). We focus on understanding the stability of new classes of attractors describing other stable oscillator con- Pau Clusella Cobero figurations that are typically partially synchronized. This University of Aberdeen includes (N − 1, 1) states (where N − 1 oscillators fire in Universita di Firenze sync and the remaining outlier fires at at different phase), [email protected] clustered splay states (composed of equal size subsets of synchronized oscillators with the relative phases between Antonio Politi clusters organized in a splay configuration) as well as limit University of Aberdeen, UK cycle states. We analyze the bifurcations between all fixed [email protected] points and limit cycles for small N. Of particular interest is how long this stability persists when the idealization of identical oscillators and the identical coupling network is PP2 relaxed. Developing a Hybrid Model for Chemotactically Driven Neuritogenesis Bolun Chen, Jan Engelbrecht Department of Physics, Boston College Here we introduce a model for chemotactically-driven neu- [email protected], [email protected] ritogenesis. Neuritogenesis, or neurite outgrowth, is a pro- cess in which neurons send out finger-like subcellular struc- Renato Mirollo tures through cytoskeletal rearrangements. It is an impor- Department of Mathematics, Boston College tant step in many neural processes, including nerve forma- [email protected] tion and development, as well as nerve regeneration and injury response. It is known that neurite outgrowth can be influenced by chemotactic gradients, but few models for PP2 these processes have been developed. We present a 2D Axonal Transport in Cells: Modeling of Motor- spatial hybrid model, using a PDE diffusion model for the Cargo Dynamics neuropeptides that stimulate neuritogenesis, and a simple, phenomenological agent-based model for the neurite out- Intracellular transport along the axons by molecular mo- growths. The neurite submodel is simulated using the gra- 318 DS17 Abstracts

dient provided by the diffusion model as an input, and in servations, but that the observation impact is short-lived. turn secretes and removes neuropeptide in the PDE sub- The forecast errors quickly increase to the level of a free- model. We provide an overview of the model development running forecast with no assimilation. We show that the and several simulation results using a range of physiologi- time taken for the forecast to lose the added skill from the cally based neuropeptide and neurite configurations. assimilation of observations depends on the length of the domain of interest. This is because the assimilation cor- Jeremy P. D’Silva rects water depths in all parts of the domain (even those University of Michigan School of Public Health which are unobserved) and error growth in the forecast step [email protected] propagates downstream. We demonstrate that this pattern of error growth can be due to incorrect friction parameter Marisa Eisenberg specification, rather than errors in upstream inflow. Joint University of Michigan state-parameter estimation leads to an improvement in the [email protected] forecast skill, and is less dependent on domain length.

Sarah Dance PP2 University of Reading Numerical Solutions of Three Dimensional Tele- [email protected] graphic Equations Using Differential Quadrature Method Elizabeth Cooper University of Reading, UK This paper employs a novel differential quadrature scheme [email protected] with B-spline basis functions, that can be used for solving linear and nonlinear partial differential equations in higher Javier Garcia-Pintado dimensions. B-spline functions are employed for space vari- MARUM able and their derivatives. The partial differential equa- University of Bremen, Germany tion results into a system of first-order ordinary differential [email protected] equations (ODEs). The obtained system of ODEs has been solved by a fourth stage Runge-Kutta method. Efficiency Nancy K. Nichols and reliability of the method has been established with five University of Reading linear problems and one nonlinear test problem. Obtained Department of Mathematics numerical solutions are found to be better as compared [email protected] to those available in the literature. Simple implementa- tion, less complexity and computational inexpensiveness are some of the main advantages of the scheme. Further, Polly Smith the scheme gives approximations not only at the knots but University of Reading, UK also at all the interior points in the domain under consid- [email protected] eration. The scheme is found to be providing convergent solutions and handles different cases. The stability of the scheme is dependent on the time step condition used by the PP2 four stage strongly stable Runge-Kutta method. However, Synaptic Plasticity in Excitatory and Inhibitory when we apply stability analysis for the scheme derived, Hippocampal Neuron Circuit Dynamics it is found to be unconditionally stable. The scheme can therefore be used effectively to handle higher dimensional Microcircuits of inhibitory and excitatory neurons in the PDEs. hippocampus have been studied over the past 30 years be- cause of their importance in creating neural spike rhythms Sumita Dahiya that have been implicated in processes such as the con- Indian Institute of Technology Roorkee, India solidation of episodic memories. Models for these circuits [email protected] range from extremely simple (coupled phase oscillators) to extremely complex (complete, spatially extended, biophys- Ramesh Mittal ically correct representations of the cells in Neuron). Vari- Indian Institute of Technology, Roorkee ous complex phenomena have been observed and explained [email protected] through these models, such as synchronization, phase pre- cession and bursting. Here we consider how time depen- dent connectivity of these cells effects these phenomena by PP2 incorporating a model for short-term synaptic plasticity Observation Impact in Data Assimilation for Flood that we developed and parameterized (from whole cell ex- Inundation Forecasting periments) for two specific interneuron-pyramidal cell con- nections. To make analysis possible, we use the maps to Accurate flood inundation forecasting provides vital infor- describe both the plasticity and spike timing in the circuit mation, enabling local residents and emergency services to (inspired by the work of Ermentrout and Kopell, 1998). make necessary preparations. Data assimilation is a pow- erful technique for combining forecasts from mathemati- Claire Seibold cal models with observations to give an improved forecast. University of Montana In this study, numerical inundation simulations are car- [email protected] ried out in a river channel topography using Clawpack. Data assimilation is performed using an Ensemble Trans- Dominika Dec form Kalman Filter (ETKF) and synthetic observations of Dept. of Mathematical Sciences water depth in identical twin experiments. In agreement [email protected] with other studies, we find that using data assimilation to combine observations of water depth with forecasts from a Emily F. Stone hydrodynamic model works very well at the time of the ob- Dept. of Mathematical Sciences DS17 Abstracts 319

The University of Montana bjorn [email protected] [email protected]

PP2 PP2 Large Time Behavior of a Conserved Phase-Field Determining the Mean Field Dynamics in Complex System Networks Via the Kramers-Moyal Expansion We investigate the large time behavior of a conserved Populations of phase-coupled oscillators can exhibit a va- phase-field system riety of qualitatively different dynamics. The settings in − − − which these dynamics can be determined analytically are τφt Δ(δφt Δφ + g(φ) u)=0, rather limited to special symmetries and comparably sim- epsilonut + φt − Δu =0, ple frequency distributions. In the current work we recover the dynamics of such populations from the time series of which describes phase separation in a material with viscos- d some common variables (e.g., mean fields). We employ the ity effects, and occupying a bounded domain Ω ⊂ R ,d= Kramers-Moyal expansion that capitalizes on the condi- 1, 2, 3. τ>0 is a relaxation time, δ ≥ 0istheviscosity tional cumulants of these variables [R. Friedrich, J. Peinke, parameter, ≥ 0 is the heat capacity, φ is the order param-  PRL 1997; J. Gradisek, S. Siegert, R. Friedrich, I.I. Grabec, eter, u is the absolute temperature and g = G ,whereG is PRE 2000]. After testing for Markovianity and the appli- a double-well potential. We prove a well-posedness result. cability of the Pawula theorem (to guarantee the diffusive Also, on the levels of global attractor, exponential attrac- characters of the dynamics) we focus on the so-called drift tors and inertial manifolds, we establish the convergence coefficient that represents the deterministic part of the dy- of the dynamics of this system to those of the well known namics. We illustrate the accuracy of the method exploring viscous Cahn-Hilliard equation, as the heat capacity goes a two-dimensional case including the dynamics of the mean to zero, i.e., → 0. This work extends and improves in an phase and the phase divergence. These variables are semi- interesting way, some recent results in this direction. nal for studying a system of phase-coupled oscillators with bimodal frequency distributions or its equivalent of two in- Cyril D. Enyi terconnected network system [B. Pietras, N. Deschle, A. King Fahd University of Petroleum and Minerals. Daffertshofer, PRE 2016]. After this illustration we apply [email protected] the method to cases were analytical results are not present. For this we break symmetry by imposing a different net- work topology in one of the two networks and show that PP2 the Kramers-Moyal expansion can yield is a meaningful, Exploring the Risk of Desynchronization in Com- analytical description of the mean field dynamics. plex Networks

Nicol´as Deschle In the field of Complex Network Synchronization an un- MOVE & iBB, Vrije Universiteit Amsterdam explored notion is that of the risk of desynchronization. ICSMB, Univesity of Aberdeen The risk of desynchronization is a quantity related to the [email protected] size of the basin of synchronization and a quantity which is easily understood within the context of coupled oscillators. Bj¨orn Schelter The risk of desynchronization is a measure of how likely a Institute for Complex Systems and Mathematical Biology particular node, or a set of nodes, within a network is to University of Aberdeen, UK cause desynchronization upon perturbation away from the [email protected] synchronous state. The concept of risk of desynchroniza- tion leads to several questions; how is the risk related to the number of nodes perturbed within the network? What Andreas Daffertshofer is the relationship between the risk and the degree of a MOVE Research Institute Amsterdam particular node in the network? Is the risk of desynchro- VU University Amsterdam nization related to the size of the perturbation? We will a.daff[email protected] now examine these questions, beginning with some theory.

PP2 Jeremie A. Fish Spectral Properties of Spiral Waves in the Barkley Clarkson University Model fi[email protected]

Spectral properties of spiral waves formed in the Barkley Jie Sun model are investigated. Absolute spectra, essential spectra, Mathematics and pseudospectra of spirals are calculated using asymp- Clarkson University totic wave trains. These results are compared with the [email protected] point spectra of spirals on large disks. The spirals are computed by numerical continuation, and point spectra by iterative methods. PP2 Measurement Sequences and Chernoff Information Stephanie Dodson Brown University First, I illustrate how a Bayesian analysis can guide mea- Stephanie [email protected] surement selection for a short sequence of measurements for a contrived example. Then I extend the small sample anal- Bjorn Sandstede ysis that requires priors and costs to an asymptotic analysis Division of Applied Mathematics that provides a comparative valuation independent of pri- Brown University ors and costs. While the motivating examples and analysis 320 DS17 Abstracts

are novel, the conclusions match work of Chernoff in 1952. [email protected], [email protected]

Andrew M. Fraser PP2 Los Alamos National Laboratory [email protected] Clustered Ventilation Defects in Asthmatics Asthma is a chronic lung disease of reversible airway con- striction, and during an asthma attack we see heteroge- PP2 neous ventilation, or specifically, clustered ventilation de- fects. These effects are seen in imaging studies of lungs in Parameter Estimation for Dynamical Systems Us- asthmatics during asthma attacks in the form of the images ing Bifurcation Analysis showing clustered regions of very low ventilation, which is expected from a person struggling to draw breath, but in- terestingly, the images also show hyper-ventilated regions. Dynamical systems are widely used to model real world These clusters vary from event to event, even in the same systems in biology, physics, finance, and many engineer- subject, and thus, it is believed that the causes are dy- ing applications. This work builds on existing methods namic rather than structural. We want to understand the of parameter estimation and uncertainty quantification by dynamic mechanisms behind the formation of these ventila- extending them to make use of information from bifurca- tion clusters, both in terms of the underlying mathematics tion analysis. After performing structural and practical and the physiological implications. Mechanisms of spatial identifiability analysis of various systems, we compare the clustering have been suggested, and we use these concepts efficiency of parameter estimates before and after leverag- to help us formulate and analyse models representative of ing the bifurcation structure. the formation of clustered ventilation defects. In particular we decompose the problem two kinds of airway coupling. Jace E. Gilbert This poster looks at these two different coupling styles and University of Nevada, Reno the results we get from each one, being particularly inter- Mathematics and Statistics ested in how each model can lead to clustered ventilation. Jace [email protected]

Paul J. Hurtado Austin J. Ibarra Mathematical Biosciences Institute University of Auckland The Ohio State University Department of Mathematics [email protected] [email protected]

PP2 PP2 Mathematical Tales of Fairy Circles

Saddle Slow Manifolds and Canard Orbits in the Dryland ecosystems exhibit intriguing self-organising Hodgkin-Huxley Model mechanisms in order to account for the strong plant compe- tition for limited resources, especially water. These mech- Many physiological phenomena have the property that anisms typically manifest themselves in the form of veg- some variables evolve much faster than others. For exam- etation patterns, ranging from bare-soil gaps in uniform ple, neuron models typically involve observable differences vegetation to vegetation stripes and vegetation spots. The in time scales. The Hodgkin-Huxley model is well known underlying mathematical model that describes these pat- for explaining the ionic mechanism that generates the ac- terns is of singularly perturbed reaction-diffusion type. A tion potential in the giant squid axon. Rubin and Wech- fascinating example of gap patterns is the so-called fairy selberger nondimensionalized this model and obtained a circle phenomenon, found in Namibia and recently in Aus- singularly perturbed system with two fast, two slow vari- tralia.Fairy circles are circular barren gaps in grasslands of ables, and an explicit time-scale ratio parameter. The dy- significant size, a few meters in diameter, that form nearly- namics of this system are very complex and feature pe- periodic patterns on large, landscape scales. Mechanisms riodic orbits with a series of action potentials separated for the appearance of this phenomenon in Namibia and by small-amplitude oscillations; also referred to as mixed- in Australia, and corresponding reaction-diffusion models, mode oscillations. The system features two-dimensional lo- have recently been proposed by Zelnik et al. [PNAS(2015)] cally invariant manifolds called slow manifolds, which can and by Getzin et al. [PNAS(2016)]. Unlike the more be either attracting or of saddle type. We introduce a new widely studied homoclinic pulse-type vegetation patterns, method for computing two-dimensional saddle slow mani- fairy circles have an underlying multi-front structure. This folds. This allows us to show how the interaction of slow poster presents a mathematical analysis of the model pro- manifolds creates a new mechanism for robust maximal posed for Namibian fairy circles. The mathematical ap- canard orbits, which play an important role in organizing proach is along the lines of geometric singular perturbation mixed-mode oscillations and determining the firing rates of theory and is accompanied by numerical simulations of the action potentials. full model. Olfa Jaibi Cris Hasan Univeristeit Leiden University of Auckland [email protected] [email protected] Martina Chirilus-Bruckner Hinke M. Osinga, Bernd Krauskopf University of Leiden University of Auckland Mathematical Institute Department of Mathematics [email protected] DS17 Abstracts 321

Arjen Doelman geophysics to plasmas to neuroscience. For linear systems, Mathematisch Instituut this estimation is solved exactly via the Kalman filter, but [email protected] nonlinear systems demand either linearizing approxima- tions or statistical representations. Recent improvements Ehud Meron in nonconvex optimization have led to adopting methods Ben-Gurion University that instead treat the estimation in a variational approx- [email protected] imation, thereby casting the sequential filtering problem as a global minimization. In continuous time, the vari- ational approximation arises as the stationary phase ap- PP2 proximation of a path integral, analogous to transition am- High-Order Finite-Difference Time-Domain Simu- plitudes in mechanics. Carrying this analogy further, the lation of Electromagnetic Waves at Complex Inter- formulation admits an action functional A[x(t)] and asso- faces Between Dispersive Media ciated Lagrangian L(x, x,˙ t), which can be used to derive Euler-Lagrange equations with natural boundary condi- We propose a new numerical scheme for the simulation of tions. Alternatively, the optimal estimate can be calculated electromagnetic waves in linear dispersive media, materi- by transforming to a Hamiltonian space, the idea being als where the permittivity depends on the frequency of the that the constraints provided by symplectic symmetry will wave. We formulate Maxwell’s equations as a second order improve estimation accuracy. Since symplectic structure wave equation with an additional time-history term which also arises quite naturally in discrete Lagangian systems, is associated with the macroscopic electronic response of the implications of this symmetry are compared in both the material to the external electric field. This formula- Lagrangian and Hamiltonian formulations for a sparsely tion allows us to address material interfaces with complex observed chaotic model. It is found that working in Hamil- curvilinear geometries, thereby avoiding the drawbacks of tonian coordinates can in some cases considerably improve the well-known staggered Yee numerical discretization. We the accuracy of the optimal estimate. use the physical interface conditions and boundary condi- tions for Maxwell’s equations to develop numerical compat- Nirag Kadakia, Daniel Rey, Jingxin Ye ibility conditions which preserve the accuracy and stability University of California San Diego of the scheme at the interfaces and boundaries. High ac- [email protected], [email protected], curacy simulation of transient waves at dispersive material [email protected] interfaces is of particular interest in the fields of plasmonic metamaterials and nano-photonics, with various applica- Henry D. Abarbanel tions in imaging and sensing. This is joint work with Jef- Physics Depratment frey W. Banks and William D. Henshaw. Univ of California, San Diego [email protected] Michael Jenkinson Rensselaer Polytechnic Institute Department of Mathematical Sciences PP2 [email protected] Consensus in One and Two Population Models of Opinion Dynamics

PP2 We consider a continuous version of the Krause model of Torus Bifurcation in Purkinje Cell opinion dynamics. As per the model, agents update their opinions, which are one-dimensional, based on a compactly Purkinje cell is an important kind of neuron in the cere- supported interaction function. Interaction between agents bellum. Recently, the cell was found to have amplitude- either leads to a state of consensus, where agents starting modulation spiking characters in the experiments. This out with differing initial opinions converge to a single opin- phenomena can be explained by torus bifurcation from dy- ion as time evolves, or to a fragmented state with multiple namic system view, which will be shown in this poster. opinions. We linearize the system about a uniform den- sity solution and predict consensus or fragmentation based Huiwen Ju upon the most unstable mode of the dispersion relation, Neuroscience Institute which is then compared against numerical simulations for Georgia State University varying domain sizes and interaction functions. We then [email protected] extend our model to two populations, each having a dis- tinct interaction function, and test our hypothesis devel- Andrey Shilnikov oped for the single population model. Neuroscience Institute and Department of Mathematics Georgia State University Ratna Khatri [email protected] Department of Mathematical Sciences George Mason University Alexander Neiman [email protected] Ohio University Department of Physics and Astronomy [email protected] PP2 Topological Principles of Control in Dynamical Networks PP2 A Hamiltonian Formulation for the Estimation of Networked biological systems, such as the brain, feature Partially Observed Chaotic Systems complex patterns of interactions. To predict and correct the dynamic behavior of such systems, it is imperative to The problem of tracking hidden states in nonlinear dynam- understand how the underlying topological structure af- ical systems is confronted in a variety of contexts, from fects and limits the function of the system. Here, we use 322 DS17 Abstracts

network control theory to extract topological features that form one-sided barriers to reaction fronts in advection- favor or prevent network controllability, and to understand reaction-diffusion (ARD) systems. To model more realistic the network-wide effect of external stimuli on large-scale time-aperiodic or unsteady flows, recent theoretical work brain systems. Specifically, we treat each brain region as a has suggested that similar one-sided barriers, termed burn- dynamic entity with real-valued state, and model the time ing Lagrangian coherent structures (bLCSs), exist for fluid evolution of all interconnected regions using linear, time- velocity data prescribed over a finite time interval. In this invariant dynamics. We propose a simplified feed-forward presentation, we use a stochastic wind to generate time scheme where the effect of upstream regions (drivers) on dependence in a double-vortex channel flow and demon- the connected downstream regions (non-drivers) is charac- strate the (locally) most attracting or repelling curves are terized in closed-form. Leveraging this characterization of the bLCSs. the simplified model, we derive topological features that predict the controllability properties of non-simplified net- Rory A. Locke works. We show analytically and numerically that these UC-Merced predictors are accurate across a large range of parameters. [email protected] Among other contributions, our analysis shows that het- erogeneity in the network weights facilitate controllabil- Kevin A. Mitchell ity, and allows us to implement targeted interventions that University of California, Merced profoundly improve controllability. By assuming an un- Physics derlying dynamical mechanism, we are able to understand [email protected] the complex topology of networked biological systems in a functionally meaningful way. PP2 Jason Kim Inference and Comparison of Dynamical Systems University of Pennsylvania As Models for Glacial Climate Variability [email protected] The most pronounced variability in the climate during the Fabio Pasqualetti last glacial period are the so-called Dansgaard-Oeschger Department of Mechanical Engineering (DO) events. These abrupt climate changes are elusive University of California, Riverside, CA in simulations of state-of-the-art coupled climate mod- [email protected] els. Furthermore, the underlying dynamical mechanism remains unknown. We investigate whether the climate sys- Danielle S. Bassett tem is exhibiting self-sustained oscillations of vastly vary- School of Engineering and Applied Science ing periods or rather noise-induced jumps in between two University of Pennsylvania quasi-stable regimes. To this end, we employ statistical [email protected] model comparison to compare different classes of stochas- tic dynamical systems, representing different dynamical paradigms, to the NGRIP ice core record from Greenland. PP2 We avoid calibrating the models by time series fitting, but The Acceleration and Deceleration of Mixing Rates rather focus on the most important qualitative features of by Discontinuities in Chaotic Flows the data, as captured by summary statistics. These are used to perform inference and comparison of the models Studies on the rate of mixing in highly viscous fluids have via Approximate Bayesian Computation. Based on our primarily been concerned with time dependent smooth de- choice of summary statistics, we find evidence that sim- formations. A well-known phenomena in non-uniformly hy- ple stochastic motion in a double well potential is better perbolic systems with diffusion is the emergence of strange supported by the data than noisy relaxation oscillations or eigenmodes with exponential decay rate. Yet mixing de- excitable systems. With our model comparison approach vices or fluid properties may allow for the introduction of we furthermore investigate to which extent the dynami- discontinuities, the dynamics of which are more subtle and cal process underlying the observed climate record can be less understood. We present a model combining the mech- regarded as stationary. anisms of stretching, cutting, and shuffling, in which the presence of diffusion reveals time periodic eigenfunctions Johannes Lohmann, Peter Ditlevsen whose decay rates show unexpected sensitivity to the gov- Centre for Ice and Climate, Niels Bohr Institute erning parameters. University of Copenhagen, Denmark [email protected], [email protected] Hannah E. Kreczak, Rob Sturman, Mark Wilson University of Leeds [email protected], [email protected], PP2 [email protected] A Variant of the Aubry-Andr´eModelforGranular Materials

PP2 We study linear and nonlinear modes in granular crystals Reaction Front Barriers in Unsteady Fluid Flows with an incommensurate Aubry-Andr´e (AA) potential. We introduce an on-site external potential in each particle in Many chemical and biological systems can be character- the form of a harmonic oscillator with elastic constant kn, ized by the propagation of a front within an underlying and we set kn = γfn with fn =cos(2πξn + φ). We show fluid flow. One approach to formalizing a general theory that, an AA localization transition can be induced in gran- is to apply frameworks developed in nonlinear dynamics. ular crystals, as it is in Schr¨odinger chains, by the action of It has been shown that invariant manifolds form barriers the external potential. We also explore the different fami- to passive transport in time-dependent or time-periodic lies of nonlinear modes that emerge and their stability. fluid flows. More recently, analogous manifolds termed burning-invariant-manifolds (BIMs), have been shown to Alejandro J. Martinez DS17 Abstracts 323

Mathematical Institute tive Sobolev seminorm, which simulates the smoothing of University of Oxford high order frequencies by diffusion. We use experimental [email protected] results and frequency and stability analysis to construct stirring schemes to improve mixing. Mason A. Porter University of Oxford Rebecca A. Mitchell Mathematical Institute University of Colorado Boulder [email protected] [email protected]

Panayotis Kevrekidis James D. Meiss Department of Mathematics and Statistics University of Colorado University of Massachusetts Dept of Applied Mathematics [email protected] [email protected]

PP2 PP2 A Heterodimensional Cycle in the Flow of a 4D Global Invariant Manifolds and Slow Manifolds Differential Equation Near a Singular Hopf Bifurcation

We investigate a four-dimensional ordinary differential Invariant manifolds of equilibria and periodic orbits are equation model for intracellular calcium dynamics. This key objects that organize the behavior of a dynamical sys- model exhibits a heterodimensional cycle, which is a het- tem both locally and globally. If multiple time scales are eroclinic connection between two saddle-periodic orbits present, there also exist so-called slow manifolds, along whose corresponding stable manifolds are of different di- which the flow is very slow compared with the rest of the mensions. Heterodimensional cycles are associated with dynamics. Slow manifolds are locally invariant objects and new types of chaotic dynamics in diffeomorphisms of three they may interact with invariant manifolds, which are glob- or more dimensions, and hence in differential equation sys- ally invariant objects. Little is known about such interac- tems of four or more dimensions. In particular, it has been tions and we show that they may produce new types of shown by Bonatti and Diaz that, given a C1 diffeomor- complicated dynamics. To this end, we consider a slow- phism containing a codimension-one heterodimensional cy- fast system near a singular Hopf bifurcation and study the cle, there is an arbitrarily C1-close set of diffeomorphisms transition through a quadratic tangency between the two- that have robust codimension-one heterodimensional cyles. dimensional unstable manifold of the associated equilib- We wish to examine how the heterodimensional cycle in rium and a two-dimensional repelling slow manifold. We the flow of this differential equation affects the nearby dy- compute the respective manifolds as families of orbits seg- namics. To observe the behaviour of this practical exam- ments with a two-point boundary value problem setup, and ple, we require computational tools and visualisation tech- track their intersections as a parameter is varied. We de- niques for invariant manifolds of periodic orbits in four di- scribe the local and global properties of the manifolds as mensions. We present a three-dimensional Poincar section, they interact, as well as the role of their interactions in which provides an overview of the model’s behaviour. Pro- organizing large-amplitude oscillations in the dynamics. jections of the four-dimensional flow into three dimensions Jose Mujica, Bernd Krauskopf, Hinke M. Osinga are then used to explain the behaviour of objects in the University of Auckland Poincar´e section in more depth. Department of Mathematics Gemma Mason [email protected], [email protected], Applied and Computational Math [email protected] Caltech [email protected] PP2 Global Regularity in a Surface Growth Model Us- Andy Hammerlindl ing a-Posteriori Methods Monash University [email protected] We study the time evolution of a surface growth model of 2 2 the type ∂th = −Δ h − Δ|∇h| , where standard methods Bernd Krauskopf, Hinke M. Osinga fail to verify global uniqueness and smoothness of solu- University of Auckland tions. Based on an arbitrary numerical approximation, we Department of Mathematics apply a-posteriori error-analysis in order to prevent a blow [email protected], [email protected] up analytically, which is a method that in a similar way also applies to 3d Navier-Stokes. Here we derive energy- estimates for the error between solution and approximation PP2 that yields a scalar ODE controlling the norm of the error Designing a Finite-Time Mixer: Optimizing Stir- and whose coefficients solely depend on the numerical data. ring for Two-Dimensional Maps Also, the solution of the ODE can be bounded using only numerical data. A key technical tool is a rigorous eigen- Mixing results from a two part process in which large gra- value bound for the nonlinear operator linearized around dients are created by advection and then smoothed by the numerical approximation. The presented method suc- diffusion. We investigate methods of designing efficient ceds to show global uniqueness for relatively large initial finite-time mixers to optimize stirring of a passive scalar conditions, which was studied in numerical simulations. in a two-dimensional incompressible flow. Area preserv- ing maps with parameters that change in time are used Christian Nolde to model the pure advection, applied numerically with the Universit¨at Augsburg Perron-Frobenius method. Mixing is measured via a nega- [email protected] 324 DS17 Abstracts

Dirk Bl¨omker results show that small perturbations (e.g., 0-12 hours) in Universitat Augsburg sleep onset timing result in shorter sleep episodes compared Germany to large perturbations (e.g., 18-24 hours past the regular [email protected] sleep onset time), as suggested by experimental observa- tions. Moreover, the map allows us to predict the length of the sleep and wake episodes during recovery sleep as PP2 the sleep-wake cycle regains synchrony with the circadian Improving Synchronization Via Adaptive Rewiring rhythm. in Networks of Kuramoto Oscillators Sofia H. Piltz Synchronization of non-identical oscillators coupled University of Michigan through complex networks is an important example of [email protected] collective behavior, and many studies examine how the ar- chitecture of interactions shapes synchronization patterns. Cecilia Diniz Behn Here, we focus on adaptive networks, where the structure Colorado School of Mines of the network changes in response to the node dynamics. Applied Math & Statistics In particular, we use the Kuramoto model to investigate [email protected] how via a local rewiring rule, an initially random network converges to a topology that supports improved synchro- Victoria Booth nization. The adaptation strategy preserves the total University of Michigan number of edges, and depends only on instantaneous, [email protected] pairwise phase differences of neighboring nodes. In the case of binary, undirected networks, a local rule that preserves connections between more desynchronized oscil- PP2 lators, and that breaks and rewires connections between Oscillatory States in Working Memory Model Fa- more in phase oscillators, can improve synchronization. cilitate Activation and Network Capacity Furthermore, in line with results from studies on optimal synchronization, throughout adaptation the Laplacian Working memory is associated with sustained periods of el- spectra and the relationship between its eigenvectors and evated firing rates of neuronal populations. These elevated the intrinsic frequencies undergo specific changes. Finally, firing rates may encode different aspects of a memory or we find that after sufficient adaptation, the resulting different memories altogether, and stimuli must be able network exhibits degree - frequency and frequency - to easily activate and inactivate populations in physiolog- neighbor frequency correlations that have been associated ically relevant timescales. We present a neural field net- with explosive synchronization transitions. By considering work model that incorporates oscillating states under cer- the interplay between structure and dynamics, this work tain parameter regimes. Three timescales (fast excitation, helps elucidate a mechanism through which emergent slow inhibition, and slower excitation) allow the network phenomena can arise in complex systems. to transition from a low baseline firing rate to oscillatory states when external stimuli are applied. The oscillations Lia Papadopoulos facilitate activation and allow for multiple simultaneously University of Pennsylvania active populations in splay-state solutions. By varying the Department of Physics and Astronomy timing of the stimuli we may then select for different net- [email protected] work activation orderings and combinations. We numeri- cally continue these solutions to determine their existence Jason Kim and stability as a function of network parameters such as University of Pennsylvania mutual inhibition. [email protected] Jason E. Pina Danielle S. Bassett University of Pittsburgh Department of Mathematics School of Engineering and Applied Science [email protected] University of Pennsylvania [email protected] PP2 Unraveling the Chaos-Land and Its Organization in PP2 the Rabinovich System A Discontinuous Map for a Human Sleep-Wake Network Model Predicts Recovery from Sleep De- A suite of analytical and computational techniques based privation on symbolic representations of simple and complex dy- namics, is further developed and employed to unravel the Disrupted sleep schedules can cause desynchronization of global organization of bi-parametric structures that under- the sleep-wake cycle and the 24-hour circadian rhythm. lie the emer-gence of chaos in a simplified resonantly cou- Such desynchronization has been suggested to contribute pled wave triplet system, known as the Rabinovich system. to several health issues including diabetes, cardiovascu- Bi-parametric scans reveal the stunning intricacy and in- lar disease, and cancer. In this work, we consider a tramural connections between homoclinic and heteroclinic physiologically-based model for human sleep, namely, a connections, and codimension-2 Bykov T-points and sad- system of differential equations describing the firing rates dle structures, which are the prime organizing centers of of neuronal populations promoting sleep and wake states, complexity of the bifurcation unfolding of the given sys- the circadian rhythm, and the homeostatic sleep drive. To tem. This suite includes Deterministic Chaos Prospector predict recovery from sleep deprivation, we apply a dis- (DCP) to sweep and effectively identify regions of simple continuous map that has been computed from the model (Morse-Smale) and chaotic structurally unstable dynamics and relates the phase of sleep onset (relative to the circa- in the system. Our analysis provides striking new insights dian rhythm) on day n to sleep onset on day n+1. Our into the complex behaviors exhibited by this system, that DS17 Abstracts 325

are hitherto unexplored. encoding different orientations. We argue that the ob- served patchy cortical activation patterns arise in a sym- Krishna Pusuluri metry breaking from the radial connectivity and that the Georgia State University orientation preference map fixes the spatial phase of such Neuroscience Institute patterns. [email protected] James Rankin Arkady Pikovsky New York University Department of Physics and Astronomy [email protected] University of Potsdam, Germany [email protected] Fr´ed´eric Chavane Institut de Neuroscienes de la Timone (INT) Andrey Shilnikov CNRS & Aix-Marseille University Neuroscience Institute and Department of Mathematics [email protected] Georgia State University [email protected] PP2 Synchronized Bursting from Combined Inhibitory PP2 and Electrical Coupling in a Neuronal Model Kernel-Based Lasso for the Inference of Complex We consider two bursting neurons mutually coupled via Spatial Networks both electrical and inhibitory connections. We report a push-pull synchronization mechanism of the combined cou- With increasing supply of data collected from numerous pling where electrical and inhibitory connections can syn- complex systems, a key question is what patterns and prop- ergistically induce robust synchronization, even though by erties of the underlying systems can be extracted from such itself electrical coupling induces anti-phase spiking and in- data. A particular such problem is to infer the network hibition is repulsive for a majority of initial conditions. structure of interacting components from data, mathemat- The push-pull mechanism is based on the properties of (i) ically formulated as an inverse problem. Given that numer- electrical coupling to stabilize burst- but destabilize spike- ous complex networks in real-life are spatially embedded, synchronization and (ii) inhibition to generally promote we ask, can such spatial constraints be exploited (rather anti-phase bursting but stabilize spike synchrony when ini- than suffered from) to make better inference? An exam- tial conditions are close. The combined action of the two ple is the synaptic connections in human brains, which are couplings uses the best of the two worlds to stabilize syn- mostly short-range and have profound impacts on brains chronized bursting. We also show that the duty cycle of neurological dynamics and functionality. Based on the the self-coupled system that governs synchronization effec- preference of short-range spatial connections, we develop tively controls the stability of synchronized bursting. a kernel-based Lasso framework to infer complex spatial networks. We show by numerical experiments that the Reimbay Reimbayev proposed method improved significantly upon existing net- Department of Mathematics and Statistics work inference techniques. Importantly, such enhancement Georgia State University is achieved even when the exact spatial distribution of the [email protected] embedded edges is unknown, making the method partic- ularly relevant as a computational tool to efficiently and Kevin Daley reliably infer large spatial networks in practice. Georgia State University [email protected] Fernando J. Quevedo 206 Outer Main Street APT301 Potsdam NY, 13676 Igor Belykh [email protected] Department of Mathematics and Statistics Georgia State University [email protected] Erik Bollt Clarkson University [email protected] PP2 Computing Electron Transport Rates Via Classical Jie Sun Periodic Orbits Mathematics Clarkson University Electron transport rates in chaotic atomic systems are com- [email protected] putable from classical periodic orbits. This technique al- lows for replacing a Monte Carlo simulation launching mil- lions of orbits with a sum over tens or hundreds of properly PP2 chosen periodic orbits using a formula called the spectral A Model of Localized States in Visual Cortex determiant. A firm grasp of the structure of the periodic orbits is required to obtain accurate transport rates. We Primary visual cortex features a quasiperiodic orientation apply a technique called homotopic lobe dynamics (HLD) preference map with a regular length scale. Voltage imag- to understand the structure of periodic orbits to compute ing experiments have shown that local, oriented visual the ionization rate in a classically chaotic atomic system, stimuli elicit activation that is orientation-selective and namely the hydrogen atom in strong parallel electric and patchy within the stimulus footprint but non-selective out- magnetic fields. HLD uses information encoded in the in- side the footprint. We study the dynamics of these input- tersections of stable and unstable manifolds of a few or- driven states in a neural field model with a biologically- bits to compute relevant periodic orbits in the system. All motivated radial connectivity profile and sub-populations unstable periodic orbits are computed up to a given pe- 326 DS17 Abstracts

riod, and the ionization rate computed from periodic orbits Between Hyperbolic Equilibria for ODEs converges exponentially to the true value as a function of the period used. We then use periodic orbit continuation In this talk we present a computer-assisted procedure for to accurately compute the ionization rate when the field proving the existence of transverse heteroclinic orbits be- strengths are varied over a range of parameter where the tween hyperbolic equilibria for polynomial vectorfields. structure of the periodic orbits remains constant, and no The idea is to compute high-order Taylor approximations periodic orbits are born or lost in a bifurcation. Then, by of local charts on the (un)stable manifolds by using the using the same periodic orbits and computing their Maslov Parameterization Method and to use Chebyshev series to indices, semiclassical spectral determinant is used to com- parameterize the orbit in between. The existence of a hete- pute semiclassical resonances for the system. roclinic orbit can then be analyzed by setting up an appro- priate fixed-point problem amenable to computer-assisted Sulimon Sattari analysis. In addition, we explain how this scheme can be University of California, Merced used to perform validated continuation. This is joint work School of Natural Sciences with Jan Bouwe van den Berg and Christian Reinhardt. [email protected] Ray Sheombarsing PP2 VU University Amsterdam [email protected] Droplet Motion for the Stochastic Cahn-Hilliard Equation PP2 We study the stochastic Cahn-Hilliard equation, which is a model describing the phase seperation and subse- The Effect of Latent Confounding Processes on the quent coarsening of binary alloys. The infinite dimensional Estimation of the Strength of Casual Influences in stochastic dynamics is given by the motion along a finite Chain-Type Networks dimensional slow manifold. In particular, we consider the dynamics of bubble solutions that is almost spherical inter- Reliable recognition of casual interactions between pro- faces. We derive the effective equation on the slow manifold cesses is an issue prevalent in applications of dynamical and show stochastic stability. systems. When the structure of a network is not a priori known it is almost impossible to observe and measure all Alexander Schindler components of a system. Missing components could lead to Universit¨at Augsburg the appearance of false - or spurious - interactions. The aim [email protected] of this study is to demonstrate the effect of missing com- ponents of a network on the inferred strength of a spurious Dirk Bl¨omker interaction. In cases where a hidden process - or latent Universitat Augsburg confounder - is influencing a network and a spurious inter- Germany action appears, it is not possible to rely on estimates of the [email protected] strength of this link as corresponding estimation methods are influenced by the latent confounder. This study demon- strates how a latent confounder affects the inferred strength PP2 of a causal connection between two processes based on vec- Model of Dendronotus Iris Swim Cpg Circuit tor autoregressive models. In summary, while it is possi- ble to measure the strength of directed causal influences We propose a detailed model to study robust network between processes, the estimation of strength can be con- bursting activity recorded in the swim central pattern gen- founded if components of a system are not observed. Con- erator of the sea slug Dendronotus Iris. The neural circuit sequently, unless a network inferred from data is free from consists of calibrated Hodgkin-Huxley type model form- latent confounders, any inferred causal connection could ing two pairs of half-center oscillators, which are coupled be spurious. As we will show, the estimated strength of a by inhibitory and excitatory alpha and dynamic synapses, spurious link is influenced by a latent confounder and can- along with gap junctions. We also discuss a mathematical not be relied upon. This is true of various measures of the abstraction of such a network. strength of Granger causal influences in which any delayed interaction is treated as casual. Andrey Shilnikov Neuroscience Institute and Department of Mathematics Helen Shiells Georgia State University University of Aberdeen [email protected] [email protected]

Deniz Alacam Marco Thiel Georgia State University University of Aberdeen Mathematics Department Scotland [email protected] [email protected]

Jack Scully Claude Wischik Georgia State University University of Aberdeen [email protected] [email protected]

Bj¨orn Schelter PP2 Institute for Complex Systems and Mathematical Biology Validated Computations for Heteroclinic Orbits University of Aberdeen, UK DS17 Abstracts 327

[email protected] [email protected], [email protected]

PP2 PP2 Coherent Structures in Axially Symmetric Landau- Bounds for Generalised Lyapunov Exponents for Lifshitz-Gilbert-Slonczewski Equation Random Products of Shears

The Landau-Lifshitz-Gilbert-Slonczewski equation de- We give lower and upper bounds on both the Lyapunov scribes magnetization dynamics in a ferromagnetic thin exponent and generalised Lyapunov exponents for the film in the presence of an applied field and a spin polarized random product of positive and negative shear matrices. current. In the case of axial symmetry and with focus on These types of random products arise in applications in- one space dimension, we study the existence and stability volving randomized stirring devices. The bounds, ob- of wavetrains, especially absolute and convective stability, tained by considering invariant cones in tangent space, and of domain wall-type coherent structures whose pro- give excellent accuracy compared to standard and gen- files asymptote to wavetrains or the constant up-/down- eral bounds, and are increasingly accurate with increasing magnetizations. The coexistence and interaction of wave- shear. Bounds on generalised exponents are useful for test- trains and/or constant magnetization states is further eval- ing numerical methods, since these exponents are difficult uated. For certain polarization the Slonczewski term can to compute. be removed which allows for a more complete charateri- Rob Sturman zation, including soliton-type solutions. Decisive for the University of Leeds solution structure is the polarization parameter as well as [email protected] size of anisotropy compared with the difference of field in- tensity and current intensity normalized by the damping. Jean-Luc Thiffeault Lars Siemer Dept. of Mathematics University of Bremen University of Wisconsin - Madison Faculty Mathematics and Computer Science [email protected] [email protected] PP2 Jens Rademacher Mean Field Reduction for Expanding Maps Cou- University of Bremen pled on Heterogeneous Networks Faculty of Mathematics and Computer Science [email protected] We consider expanding maps coupled on heterogeneous networks, meaning that each map is considered as sitting Christof Melcher on one node of a graph and is perturbed by interactions RWTH Aachen University with adjacent nodes. Loosely speaking, if xi(t)isthestate [email protected] of the i-th node at time t n PP2 xi(t +1)=fi(xi(t)) + Aij h(xi(t),xj (t)) j=1 System Parameter Variations from Attractor De- formation where fi is the uncoupled dynamics, h(xi,xj )istheinterac- tion term between nodes i and j,andAij ∈{0, 1} selects In damage detection and sensing applications, repeated which interactions are present according to the edges of observations of system dynamics are often used to assess the network. We prove that under certain hypotheses for if system parameters are changing over longer periods of a large set of initial conditions, the dynamics of any highly time. The process of determining parameter changes by connected node is given by examining dynamics is usually difficult for chaotic systems. Our research compares two methods for quantifying param- xH (t +1)=f˜(xH (t)) + ξ(t) eter changes in chaotic systems by analyzing Poincare sec- tions. In the first method, every point on a given section is where f˜ is a map of the node’s coordinate only that con- assigned an indexed bin and a histogram for each bin is con- sider the mean interaction coming from the rest of the net- structed based on the surrounding density. Bins from dif- work, and ξ(t)isafluctuationtermproventoremainsmall ferent Poincare sections (corresponding to systems having for a long time depending on the heterogeneity of the net- different parameters) are matched by assigning a cost based work. This reduction provides explanation for synchroni- on the differences between histograms. By comparing the sation and other type of behaviours. locations of matched bins on their respective Poincare sec- tions, an estimate of the parameter change can be made. Matteo Tanzi In the second method, attractor comparison is realized by Imperial College London calculating the earth movers distance involved in moving [email protected] points on one Poincare section to create another Poincare section. Location comparison again allows for an estimate of the parameter change to be made. Several variations of PP2 these two techniques are presented using data from simu- Bifurcation of Coherent Structures in Nonlocally lated and physical systems. Particular attention is paid to Coupled System the potential for detecting multiple parameter changes in thepresenceofnoise. Motivated by models for neural fields, we study the exis- tence of pulses bifurcating from a spatially homogeneous Andrew R. Sloboda, Trung Tran state in nonlocally coupled systems of equations. More Bucknell University specifically, we look at equations of the form U + K ∗ U = 328 DS17 Abstracts

N(U; μ), where N encodes nonlinear terms, K an even ma- Sarah Iams trix convolution kernel. Assuming the presence of neutral Harvard University modes, that is, solutions of the form u ∼ exp(ix)tothe Cambridge, MA linear part, we show under appropriate assumptions on the [email protected] nonlinearity and the unfolding in μ that pulses bifurcate. Such an analysis is carried out using center manifold reduc-  Mary Silber tion, when coupling is local, say, K = δ . Here, we rely on University of Chicago functional analytic methods using predictors from formal Dept. of Engineering Sciences and Applied Mathematics expansions and correctors obtained after preconditioning [email protected] the nonlinear system.

Tianyu Tao PP2 University of Minnesota [email protected] Goal-Oriented Network Design for Pairwise Com- parison Ranking

Arnd Scheel Inferring the rankings of a set of objects such as documents University of Minnesota or images is closely related to problems such as designing School of Mathematics and optimization of search engines and recommendation [email protected] systems. In many applications, direct ranking or scoring may be challenging and unreliable. Alternatively, pairwise comparisons are typically easier to obtain and more ro- PP2 bust, especially for human decision makings. Consider a A New Partitioned Method for Viscous Fsi given collection of items with some unknown but intrin- sic ordering, a Pairwise Comparison Network (PCN) is a Fluid-structure interactions (FSI) are a common class of directed network that encodes information about the pair- problems in several applications, including flow problems wise comparison data/results, which can either be passively in biological and engineering applications. In this presenta- obtained or actively designed. In contrast to previous work tion we consider a particular class of FSI algorithms, parti- which only considers single-shot experiments, we propose a tioned methods, that loosely couple the fluid and structure two-round design of goal-oriented PCNs which allows for ef- solvers along a well defined interface. A common difficulty ficient additional data collection that yield much enhanced in partitioned FSI algorithms is the added mass instability: ranking inference. In particular, we present three such net- we demonstrate some new results which show how one can work designs using only partial observations of pairwise overcome the added mass instability with a careful choice comparison data. Adopting the PageRank algorithm orig- of boundary conditions and discretization with the finite inally developed for search engines, we infer rankings from element method. the designed PCNs and validated their effectiveness against David Wells alternative methods. In addition, we present analytical re- Rensselaer Polytechnic Institute sults for special types of PCNs and discuss potential ex- [email protected] tensions and challenges. Shandeepa D. Wickramasinghe PP2 Mathematics Clarkson University Connecting Vegetation Models with Data: the Role [email protected] of Topography, on Multiple Scales, for Water Har- vesting in Channeled Vegetation Patterns Christino Tamon In semi-arid climates, regions of self-organized striped Clarkson University patterned vegetation can arise. Soil water availability [email protected] is thought to influence stripe properties such as spac- ing, width, and curvature. Here, focusing on channelized Jie Sun stripes, we compare local (catchment area) and non-local Mathematics (upslope area) computational methods for estimating the Clarkson University effect of soil water presence on stripe characteristics. Satel- [email protected] lite image and SRTM topography data from a number of channels found at sites in Somalia and Western Australia are used to test the approaches. The goal of this investi- PP2 gation is to help identify pattern properties from data that Spectra and Turing Instabilities for Reaction Dif- might provide good validity tests for the deterministic veg- fusion Systems with Anomalous Diffusion etation models. We consider activator-inhibitor systems with fractional dif- Lucien D. Werner fusion, in particular subdiffusion, which renders the equa- Northwestern University tion nonlocal in time. On the one hand, we consider the [email protected] perturbation from regular to subdiffusion near Turing in- stabilities and show that spectra converge locally in the Punit Gandhi wavenumber, while in contrast to regular diffusion, large Mathematical Biosciences Institute, Ohio State University wavenumbers have slow decay. On the other hand, we [email protected] study the change in nature of pattern forming instabilities when the anomalous diffusion exponents of activator and Karna V. Gowda inhibitor is further from being regular. In addition to the Northwestern University Turing instability threshold, we find a threshold for large [email protected] wavenumber instabilities. Finally, we discuss bifurcations DS17 Abstracts 329

associated with marginally stable wavenumbers. Northwestern University [email protected] Jichen Yang University of Bremen Faculty of Mathematics and Computer Science PP2 [email protected] Stability Properties of Flying Snakes During Tran- sient Glides Jens Rademacher University of Bremen Flying snakes of the genus Chrysopelea turn their body into [email protected] a morphing wing and perform controlled aerial glides us- ing three-dimensional undulation. Here, we investigate the stability properties of snake gliding flight using reduced- PP2 order modeling, phase space analysis, and by measuring Dynamics of a Class I Calcium Model body undulation waveforms in glide trajectories originat- ing from 8.5 m. The empirical waveforms are then used Oscillation in the concentration of free cytosolic calcium as input to a dynamic model of snake flight to investi- plays a crucial role in very many cell types, controlling gate the coupling of aerodynamic and inertial moments on processes such as cell growth, muscle contraction and gene flight stability as the simulated glider progresses through expression. It is thus important that we understand the the horizontal-vertical velocity space. We find that iner- mechanisms underlying such oscillations. We have devel- tial moments dominate about the yaw axis, a feature that oped a simple mathematical model for certain types of cal- could be exploited for maneuvering, and which also may cium oscillations, and conjecture that the model is applica- be useful as a control modality for bioinspired engineered ble to a variety of cell types; validity of the model has been gliders. confirmed by experimental data from three different cell types. We are thus interested in understanding the dynam- Isaac Yeaton ics behind this model. This poster shows recent progress Virginia Tech on understanding the role that the existence of multiple [email protected] timescales has on determining the dynamics of the model. Gary K. Nave Xueshan Yang, Vivien Kirk Virginia Tech University of Auckland Department of Biomedical Engineering and Mechanics [email protected], [email protected] [email protected]

James Sneyd John Socha University of Auckland Virginia Tech Department of Mathematics [email protected] [email protected] Shane D. Ross Virginia Tech PP2 Engineering Mechanics program Political Elections and Third Parties: A Mathe- [email protected] matical Model and Empirical Evidence Despite a large portion of the US public thinking third PP2 political parties are necessary, third parties receive little Pattern Formation in Marsh Ecosystems Mod- representation in US politics. We formulate a differential eled Through the Interaction of Marsh Vegetation, equation system to model party positions and vote share Mussels and Sediment in political elections, with focus on the role of third par- ties. Our model is different from many previous models Spatial patterning in multi-species communities can be in our key hypothesis — voters “satisfice”: they accept a critical to ensuring their proper function and survival, mak- party that is good enough and do not obsess over other ing the investigation of self-organization in ecology cru- options. This is motivated by psychological research on cial for understanding the underlying interactions in var- decision making, especially with incomplete information. ious ecosystem communities and their ability to adapt to Our model has two major implications. First, for a 2-party emerging environmental changes. In this presentation, we system, the possibility of additional parties can significant focus on finger-like projections consisting of marsh vegeta- change its behavior. Second, having less parties, or higher tion,musselsandaccumulatedsedimentobservedonthe barriers for minor parties, can lead to more extremist vote marsh shorelines of the York River. Although we consider distributions in elections. We use empirical data of party this specific location, similar structures may exist in other positions in the US and worldwide to validate our model, marsh communities. We propose a system of reaction- and find strong support. diffusion equations with nonlocal interactions to model the formation of these aggregations through interactions be- Vicky Chuqiao Yang,DanielM.Abrams tween marsh vegetation, mussels and sediment. The model Northwestern University presents a phenomenological approach, focused on the self- [email protected], [email protected] organization as a result of interactions between species. We analytically and numerically study the three-species sys- Georgia Kernell tem (marsh vegetation-mussels-sediment), as well as the University of California, Los Angeles mussel-free subsystem, to gain understanding into the pos- [email protected] sible dynamics. In addition, snapshots of the system are analyzed along an erosion gradient to investigate the sys- Adilson E. Motter tem’s transition to a degraded state with increasing envi- 330 DS17 Abstracts

ronmental stresses. from the Hamilton’s equations, the analytical equilibria are determined and linearization of the equations of mo- Sofya Zaytseva tion about the saddle is obtained. After computing the The College of William and Mary eigenvalues and eigenvectors of the coefficient matrix as- [email protected] sociated with the linearization, the symplectic transforma- tion is given which puts the Hamiltonian into normal form Leah Shaw and simplifies the equations, allowing us to use the concep- College of William and Mary tual framework known as tube dynamics. The flow in the [email protected] equilibrium region of phase space as well as the invariant manifold tubes in position space are discussed. Also, we Junping Shi account for the addition of damping in the tube dynam- College of William and Mary ics framework, which leads to a richer set of behaviors in Department of Mathematics transition dynamics than previously explored. [email protected] Jun Zhong Dept. of Biomedical Engineering & Mechanics, Virginia Rom Lipcius Tech Virginia Institute of Marina Science [email protected] [email protected] L. N. Virgin PP2 Dept. of Mechanical Engineering, Duke University [email protected] A Parameter Estimation Method Using Linear Re- sponse Statistics Shane D. Ross This paper presents a new parameter estimation method Virginia Tech for Ito diffusion such that the resulting model predict the Engineering Mechanics program equilibrium statistics as well as the sensitivities of the un- [email protected] derlying system under external disturbances. Our formu- lation does not require the knowledge of the underlying system, however we assume that the linear response statis- tics can be computed via the fluctuation-dissipation theory. The main idea is to fit the model to a finite set of ”essen- tial” statistics that is sufficient to approximate the linear response operators. In a series of test problems, includ- ing a simplified turbulence model and a Langevin dynam- ics model, we will show the consistency of the proposed method in the sense that if we use the proposed method to estimate the parameters of the underlying model, then we must obtain the true parameters.

He Zhang Department of Mathematics The Pennsylvania State University [email protected]

John Harlim Pennsylvania State University [email protected]

Xiantao Li Department of Mathematics Pennsylvania State University [email protected]

PP2 Geometry of Transition Dynamics for a Buckled Beam

We present a dynamic buckling model for a moderately buckled beam. The analytical model is established by the Euler-Bernoulli beam model with a two-mode truncation. Using the Legendre transformation, the two-mode Hamil- tonian, which is the sum of a kinetic energy term and a po- tential energy term is obtained from the Lagrangian. The form of the potential energy—two stable wells connected by rank-1 saddle points—shows an analogy with resonance transitions in celestial mechanics or molecular reconfigu- rations in chemistry, where here transition corresponds to switching between two stable beam configurations. Then,