76 DS05 Abstracts

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76 DS05 Abstracts

CP1 and the CAP-dependent activation complex. Short-Strand Dna Renaturation by An Active Micro-Mixer David Swigon Department of Mathematics Understanding renaturation of short (40-50 mer long) ss- University of Pittsburgh DNA strands in flow is of interest from both the theoretical [email protected] perspective and from that of experimentalists studying mechanical properties of DNA. Since the size of these strands is less that the persistence length of ssDNA, we model the CP2 strands as rods. Assuming that two rods combine to form Patched Heteroclinic Orbits in a Family of Scalar dsDNA if they are close together and are closely aligned, Wave Equations and considering the dilute limit, we model the process as a reaction-diffusion-advection system in position-orientation We analyze traveling wave solutions of the following family of scalar wave equations: space where the reaction term has a particularly simple 2 2 form. The model contains avenues for control in the form ut + uux = −γα G ∗ ux , (1) of the advective velocity fields and applied external poten- x tials. We employ the model to study the enhancement of for γ>0, where G is the Green’s function of the Helmholtz 2 reaction rate by an active micro-mixer consisting of chan- operator Hu = u−α uxx. We find a novel class of patched nel that is 200 x 100 microns in cross section and is a few heteroclinic orbits which appear to cross a line of singu- hundred microns in length. Two streams containing each larities in the (u,u’) phase plane. As γ increases past the type of ssDNA are introduced at the inlet and are agitated critical value γ = 1, we observe a qualitative change in the by fluid in side-channels (50microns wide) before reaching behavior of trajectories near the singularity line. These the outlet. We perform 2-D as well as 3-D analysis using orbits correspond to shock-like traveling wave solutions of analytical expressions for the underlying velocity field. We (1), which in the zero-α limit, are global weak solutions of also consider another process, where streams of dye and Burgers’ equation. dsDNA are introduced into the micro-mixer. The reaction consists of the dye attaching itself between the strands of Razvan C. Fetecau the DNA, subsequent to which the dye fluoresces and the Stanford University intensity of fluorescence can thus indicate amount of bound Department of Mathematics dye. This process is treated as a special case of the same [email protected] model. Harish S. Bhat Igor Mezic, Thomas John California Institute of Technology University of California, Santa Barbara Control and Dynamical Systems [email protected], [email protected] [email protected]

CP2 CP1 Time Simulation-Based Bifurcation Analysis of A Dynamical Systems Approach to Reconstructing Waves Repetitive Dna In Nonlinearity 16:1257-1275, 2003, we presented a sym- Mathematical methods in biology, especially regarding metry reduction of a PDE which can be used to ”freeze” analysis of DNA sequence information has led to major waves and self-similar solutions. In this presentation, we results in recent years. Still, the task of determining the will review strategies based on time simulation to com- DNA sequence of an organism, genome assembly, remains pute wave solutions, some based on the above symmetry- an expensive and lengthy process. Repetitive regions of reduced PDE. We will also argue that a generalized eigen- DNA are especially difficult to assemble. We develop a value problem is a better way to determine the stability method inspired by ideas from Symbolic Dynamical Sys- since it can filter out the neutral eigenvalues. tems to reconstruct repetitive DNA. We the validity of our method by analyzing two different assemblies of Drosophila Clarence Rowley melanogaster (fruit fly). We conclude by describing ways Princeton University our method can be used to improve genome assembly. Department of Mechanical and Aerospace Engineering [email protected] Suzanne Sindi University of Maryland, College Park Ioannis Kevrekidis [email protected] Dept. of Chemical Engineering Princeton University CP1 [email protected] Mesoscale Modeling of Protein-Dna Complexes Kurt Lust DNA transcription begins with the formation of a complex University of Groningen of DNA and proteins. Recently developed base-pair level Institute of Mathematics and Computing Science theory of DNA elasticity enables construction of structural [email protected] and dynamical models of such complexes that yield new information about the role of the promoter sequence in the mechanism of gene regulation. Presented will be applica- CP2 tions to two complexes important for the regulation of the Traveling Waves for Differential-Difference Equa- Lac operon in E. coli: the LacR-DNA promoter complex tions with Inhomogeneous Diffusion Traveling wave solutions for lattice differential equations DS05 Abstracts 77

are defined by boundary value problems with advances and [email protected] delays on an unbounded domain. Fourier transform techniques and Jacobi operator theory allow one to obtain an- Elbert E. Macau alytic results for a problem of this type with a diffusion LAC - Laboratory for Computing and Applied coefficient that is varied in some interval on the lattice. Mathematics Of particular interest is the issue of propagation failure INPE - Brazilian Institute for Space Research of these traveling waves depending on the inhomogeneous [email protected] diffusion. Tony R. Humphries Celso Grebogi McGill University Instituto de Fisica - IF Mathematics & Statistics Universidade de Sao Paulo/ USP [email protected] [email protected]

Brian E. Moore CP3 Department of Mathematics and Statistics Dynamics of Adaptive Delayed-Feedback Control McGill University Systems [email protected] First, we derive an adaptive control method for discrete Erik Van Vleck time maps, by extending delayed-feedback controls already Department of Mathematics proposed. Then, we study dynamics of adaptive control University of Kansas systems. In particular, we apply our method to the H´enon [email protected] map, and numerically show the following two characteristics: Power law decay of distribution of control times. Almost zero ﬁnite-time Lyapunov exponent. With an an- CP3 alytical treatment, we show the simplest control system Feedback Linearization of Chemostats becomes neutrally stable, which well explains these characteristics. The chemostat is a rare example of a class of biological systems that has real experimental and industrial applications Asaki Saito and also admits a modeling paradigm that yields rigor- Future University - Hakodate ously analyzable mathematical models. We apply the dif- [email protected] ferential geometry-based methodology of nonlinear control to mathematical models of chemostats. We show that by properly choosing control parameters these models can be CP4 made equivalent to linear dynamical systems and suitably The Use of Iterated Function Systems in Cryptog- chosen control objectives can be met via linear methods. raphy and Steganography We stress that this is not an approximation, but actually amounts to an analytic equivalence of the full nonlinear Iterated Function Systems (IFS)has found its applications system with controls to a controllable linear system. in many areas of mathematics and our lives. In this talk we show that one can hide data such as Cryptography Mary Ballyk Keys and messages in the attarctors of IFS such that the New Mexico State U. intended observers or recipients can reterive the message Dept of Mathematical Sciences from the atractor. The algorithm along with several exam- [email protected] ples will be disscussed.

Ernest Barany Mohammad Khadivi New Mexico State University Jackson State University Dept of Mathematical Sciences Department of Mathematics [email protected] [email protected]

CP3 CP4 Geometric Control and Chaotic Synchronization Bounded Nonwandering Sets for Polynomial Maps In this paper, we consider a class of polynomial maps We present a new parameter estimation procedure for non- m m linear systems. Such technique is based on the synchroniza- on R or C which is defined by the assumption that tion between the model and the system whose unknown the delay equations induced by the maps have leading parameter is wanted. Synchronization is accomplished by monomials of single variable. We show that for any map controlling the model to make it follow the system. We use from this class, the nonwandering set is bounded while geometric nonlinear control techniques to design the con- for all unbounded orbits, some kind of monotonicity takes trol system. These techniques allow us to derive necessary place. The class under consideration is proved to contain in and sufficient conditions for synchronization and hence for particular the generalized Hénon maps and the Arneodo- proper parameter estimation. As an example, this proce- Coullet-Tresser maps. dure is used to estimate a parameter of an example serving Mikhail Malkin as a model. Department of Mathematics, Nizhny Novgorod State Ubiratan Freitas University LAC - Laboratory for Computing and Applied Nizhny Novgorod, RUSSIA Mathematics [email protected] INPE - Brazilian Institute for Space Research 78 DS05 Abstracts

Ming-Chia Li is illustrated in several examples. Dept. of Math., National Changhua University of Education Yilian Zhang [email protected] University of South Carolina Aiken [email protected]

CP5 Adrian Nachman Turbulent Spin Dynamics under the Joint Action of University of Toronto the Distant Dipolar Field and Radiation Damping [email protected]

Nonlinear evolution of the microscopic magnetization and experimentally measured net magnetization in high-field CP7 solution magnetic resonance is analyzed. These dynam- Basin Hopping in a Gearbox Model ics arise from the joint action of the distant dipolar field and radiation damping and have been recently used to en- We analyse the consequences of noise in a model for rat- hance sensitivity and contrast in magnetic resonance spec- tling in a single-stage gearbox with a bachlash. One of troscopy and imaging. The dynamics beyond the initial observable effects is the basin hopping, i. e., the switch- transient regime are shown to reach a turbulent phase. The ing between basins of different attractors. This is a conse- observed turbulence may be a manifestation of spatiotem- quence of the intertwined nature of the basins of attraction poral chaos. or the presence of chaotic transients. Susie Huang, Yung-Ya Lin Silvio Souza Department of Chemistry and Biochemistry Institute of Physics University of California, Los Angeles University of Sao paulo [email protected], [email protected] [email protected]

Sandip Datta Antonio Batista Department of Chemistry and Biochemistry Department of Mathematics University of California, Los Angeles (UCLA) State University of Ponta Grossa [email protected] [email protected]

Ibere L. Caldas CP6 Institute of Physics On Some Zero-Preserving Iso-Spectral Flows University of Sao paulo [email protected] I am studying the following general problem: Given a ’structured’ symmetric matrix, (efficiently) find its eigenvalues. In particular, I am studying flows in the space Ricardo Viana of symmetric matrices which preserve eigenvalues (that is, Physics Department are ’iso-spectral’) and preserve structure. The Toda flow Federal University of Parana is an example; this flow is iso-spectral and preserves tridi- viana@fisica.ufpr.br agonal structure. (This flow is closely related to the QR algorithm.) Let A and B be positive semi-definite linear CP7 operators on an inner product space V. Let Inv denote the Moore-Penrose inverse operation. Let A!B := A Inv(A+B) Period Doubling in An Interrupted Machining Pro- B. (This linear operator is called the ’quasi-projection’, cess ’harmonic mean’ or ’parallel sum’ of A and B.) It is known In this lecture a nonlinear delay-differential equation model that the range of A!B is equal to the intersection of the of high-speed milling is studied. The model incorporates range of A and the range of B. We can use this operation the discontinuous ’fly-over’ effect, when the tool leaves the to design structure preserving iso-spectral flows. I shall de- workpiece due to high amplitude vibrations. First, we in- scribe some iso-spectral flows which preserve sparsity - that troduce a numerical method to analyse periodic solutions is, preserve zero entries. I shall show that these flows con- in the periodic delay equation model. We use pseudo- verge. I conjecture that these flows converge to diagonal arclength continuation technique to follow the bifurcation matrices. Numerical experiments support this conjecture. curves of period-two orbits. To check the numerical find- Kenneth Driessel ings we compute the sense of the period doubling bifurca- Colorado State University tion analytically. [email protected] Robert Szalai Massachusetts Institute of Technology CP6 [email protected] A Class of Integrable Dynamical Systems and Their Connection to One-Dimensional Inverse Problem CP7 We present a large class of exact solutions to A-equation Gear Rattle in the Large Stiffness Low Damping introduced by Barry Simon and reveal a new class of arbi- Limit trarily large systems of nonlinear ordinary differential equa- We examine models of gear rattle that take the form of pe- tions, which we show to be C-integrable in the sense of F. riodically forced ODE oscillators which are linear up to the Calogero. Integration scheme is proposed and the approach inclusion of a backlash term that models the meshing force between gears in terms of their relative rotational displace- ment. We use piecewise-smooth dynamical systems theory DS05 Abstracts 79

to develop bifurcation diagrams that demonstrate the co- Erik Bollt existence of a variety of stable rattling behaviors. Indus- Clarkson University trial applications include the control of noise and vibration [email protected] problems in vacuum pumps. Eddie Wilson CP9 University of Bristol, U.K. Virtual Rigid Bodies in the Circular, Re- [email protected] stricted Three-Body Problem: Dynamically- Natural Spacecraft Formations

CP8 The circular, restricted three-body problem provides a Real-Time Construction of Optimized Predictors richer dynamical environment than the two-body problem. from Data Streams There has been a considerable amount of work in recent decades, which takes advantage of equilibrium, stability, A new approach to the construction and optimization of instability, and chaos to develop low energy trajectories. local models is proposed in the context of data streams, Periodic and quasi-period orbits in this realm are exam- that is, unlimited sources of data where retaining and pro- ined to determine if and what types of spacecraft forma- cessing all observations is impractical. A learning data set tions can be maintained with little to no propulsion, so as of limited size optimized in terms of predictive power is to create virtual rigid bodies in space. extracted. Real-time revision of the learning set allows se- lective coverage of regions in state space which contribute Ralph R. Basilio most to reconstructing the underlying dynamical system. University of Southern California [email protected] Leonard Smith OCIAM Paul K. Newton University of Oxford Univ Southern California [email protected] Dept of Aerospace Engineering [email protected] Frank Kwasniok University of Oldenburg, Germany [email protected] CP9 Homoclinic Points Near Resonances in the Re- stricted Three-Body Problem CP8 Testing Causality: Coupling Asymmetry from Bi- Resonance between the periods of planets, satellites, and variate Series Using Surrogate Data other objects is a feature of celestial mechanics. Among the many extra-solar planets discovered recently, many res- Bivariate time series measured in coupled dynamical sys- onances have been observed. This talk will consider reso- tems can be used to infer asymmetry of coupling and thus nant periodic motions in the restricted three-body problem the causality in evolution of the interacting (sub)systems. with the aim of gaining an understanding of instabilities We compare the size and power of asymmetry tests us- caused by resonance. I will outline a demonstration of the ing various types of surrogate data. Real-world applica- existence of homoclinic points near resonances. tions are presented in establishing causality relations in climate evolution (relations between near-decadal oscilla- Divakar Viswanath tory modes of long-term temperature records and North University of Michigan Atlantic Oscillation index) and in the electrophysiological [email protected] data capturing brain-cardiac-respiratory interactions. Milan Palus CP9 Academy of Sciences of the Czech Republic Computation of Low Energy Earth-to-Moon Trans- Institute of Computer Science fers and Their Control by a Novel Method [email protected] We construct a spacecraft transfer with low cost and moderate ﬂight time from the Earth to the Moon. The mo- CP8 tion of the spacecraft is modeled by the planar circular re- On Model Reduction and Nonlinear Parameter Es- stricted three-body problem including a perturbation due timation of Multivariate Time-Series to the solar gravitation. Our approach is to reduce computation of optimal transfers to a nonlinear boundary value Given a multi-variate time-serie, which is obtained from problem. Using a computer software called AUTO, we solve some unknown ODE, we construct an approximating ODE it and continue its solutions numerically to obtain the op- that well constructs the data. In this talk, we will introduce timal transfers. Moreover, we show that the optimal trans- a technique to ﬁt experimental data sets by global modeling fers are unstable and can be stabilized by a novel control and parameter estimation. Also we will give the proof of method. the convergence of the parameter matrix in details. Some numerical results and prospective work will be given in the Kazuyuki Yagasaki end. Gifu University Department of Mechanical and Systems Engineering Chen Yao [email protected] Box 5817 Clarkson University Potsdam, NY 13699 [email protected] 80 DS05 Abstracts

CP10 ferent numbers of loops around the individual component Optimal Routing of a Sailboat in Steady Winds cycles can be stable for the same parameter values, as can combinations of regular and irregular (aperiodic) cycling. We consider a family of novel, insightful, but yet tractable Analytical results for the regular cycling behaviour agree problems regarding the optimal routing of a sailboat in well with numerical simulations, and explain, at least par- the plane, such as a racing yacht, in steady (i.e. time- tially, the occurrence of the irregular cycling. invariant) winds. For the case of non-constant, but smooth wind fields, the minimum-time problem takes on the form Claire M. Postlethwaite, Jonathan Dawes of a Zermelo problem, resulting in a 2-point boundary- DAMTP, University of Cambridge value problem generated from application of the calculus [email protected], of variations and requiring a single iteration of the initial [email protected] boat heading angle. Michael P. Hennessey CP11 School of Engineering Robust Heteroclinic Cycles in the 1D Complex University of St. Thomas Ginzburg-Landau Equation [email protected] A new analysis is undertaken to explain numerical results Jeffrey Jalkio showing previously unobserved stable heteroclinic cycles University of St. Thomas in large parameter regions of the 1D complex Ginzburg- [email protected] Landau equation on a periodic domain. These cycles connect different spatially and temporally inhomogeneous time-periodic solutions. We delineate regions of existence CP10 and stability and indicate rich dynamics when the cycles Early Detection of Anoxic Crises in Coastal La- become unstable, including Shilni’kov-Hopf and blow-out goons bifurcations.

An important and recurrent problem in coastal lagoons is Eddie Wilson anoxia. Anoxic crises occur mainly in summer when tem- University of Bristol, U.K. peratures are high and are triggered off by organic mat- [email protected] ter decomposition. The early detection of anoxic crises is therefore an important aspect in the management of David Lloyd coastal lagoons.In this work techniques from non-linear dy- University of Bristol namical systems theory have been applied to oxygen con- [email protected] centration for estimating in advance the occurrence of an anoxic crises. Alan Champneys University of Bristol Dimitar Marinov Dept. of Engineering Maths European Commission [email protected] Joint Research Centre [email protected] CP11 Francesca Soma A Codimension Two Resonant Bifurcation from a European Commission, Heteroclinic Cycle with Complex Eigenvalues Joint Research Centre [email protected] Robust heteroclinic cycles in systems with symmetry can undergo a variety of bifurcations. The resonant bifurcation Eugenio Gutierrez is usually associated with the birth or death of a nearby European Commission periodic orbit, and as such can occur in a supercritical or Joint Research Centre subcritical manner. We investigate the degenerate case be- [email protected] tween these two for a specific heteroclinic cycle with complex eigenvalues. A complex but ordered structure of further bifurcations of periodic orbits is found around this Jose Manuel Zaldivar codimension two point. European Commission, Joint Research Centre, TP272 [email protected] Claire M. Postlethwaite, Jonathan Dawes DAMTP, University of Cambridge [email protected], CP11 [email protected] Regular and Irregular Cycling Near a Heteroclinic Network CP12 Heteroclinic networks (collections of robust heteroclinic cy- Bifurcation Analysis of a Differential Equations cles sharing common equilibria) appear naturally in sym- Model for Mutualism metric dynamical systems, yet little in general is known about the behaviour of typical nearby trajectories. This We develop from basic principles a two-species differen- talk discusses a specific example of a heteroclinic network tial equations model which exhibits mutualistic population that displays a variety of interesting dynamics. In par- interactions. We vary the intrinsic growth rates of the pop- ticular, trajectories are observed to settle into a series of ulations and perform a bifurcation analysis. The bifurca- excursions around different parts of the network that we tions of primary ecological interest are those which allow call ‘cycling sub-cycles’. Cycling patterns displaying dif- the population(s) to survive, but more interesting bifur- DS05 Abstracts 81

cation phenomena are also observed. The model reduces equations) have solutions which become singular either in to the familiar Lotka-Volterra model locally, but is more finite or infinite time, meaning either that the evolving realistic globally in the case where mutualist interaction is object (map, metric, surface, or function) becomes un- strong. bounded, or that one of its derivatives becomes unbounded. The analysis of the asymptotic behaviour of a solution of Bruce B. Peckham a nonlinear parabolic equation just before it becomes sin- Univ. of Minnesota, Duluth gular is known to be a difficult problem. The main general Dept. of Mathematics and Statistics point of this talk is that this analysis is considerably easier [email protected] in the case where the singularity occurs in infinite time.

Wendy Graves Joost Hulshof Rainy River Community College VU Amsterdam [email protected] Department of Mathematics [email protected] John Pastor University of Minnesota Duluth Sigurd B. Angenent Dept. of Biology and NRRI University of Wisconsin [email protected] Department of Mathematics [email protected]

CP12 Modeling Population Spread in Heterogeneous En- CP13 vironments Using Integrodifference Equations Multi-Bump, Blowup, Self-Similar Solutions of the Complex Ginzburg-Landau Equation An integrodifference equation model for population dynamics is presented. Dispersal is described by a PDE. Het- In this talk we study, both asymptotically and numerically, erogeneity consists of two types patches, one with a high multi-bump, blowup, self-similar solutions to the complex growth rate and one with a low growth rate, arranged in Ginzburg-Landau equation in the limit of small dissipation. patch-work fashion. In this talk we address two question: Furthermore, a proof of the existence of these multi-bump, (1) can a species persist and (2) at what rate will it spread? blowup solutions will be given. Through the asymptotic To address the first, we perform a linear stability analysis. analysis, we determine the parameter range over which To address the second, we compute a dispersion relation these solutions may exist. Most intriguingly, we determine for traveling wave solutions. a branch of solutions that are not perturbations of solutions to the nonlinear Schrödinger equation, moreover, they are Thomas C. Robbins not monotone but they are stable. Department of Mathematics University of Utah jf Williams [email protected] Leiden University Dept of Mathematics Mark Lewis [email protected] University of Alberta, Canada [email protected] Chris Budd School of Mathematical Sciences University of Bath CP13 [email protected] Blow-Up of Solutions of Degenerate Quasilinear Parabolic Equations Vivi Rottschafer Leiden University Let p and m be real numbers such that p>m>1, and q be Dept of Mathematics a nonnegative real number. We study existence, uniqueness [email protected] and blow-up of the solution of the following degenerate quasilinear parabolic problem q m p CP14 x ut =(u ) + u in (0, 1) × (0,T) , xx Direct Chaotic Flux Formula under General Time- u (x, 0) = u0 (x)in[0, 1] ,u(0,t)=0=u (1,t) for t ∈ (0,T)Periodicity, where u0 (x) is a smooth positive function and u0 (0) = Chaotic flux occurring across heteroclinics under nonsep- u0 (1)=0. arable time-periodic (not necessarily harmonic) perturba- W. Y. Chan tions of area-preserving planar flows is examined. Though Department of Mathematics well-understood phenomenologically, computable flux for- Southeast Missouri State University mulas have been lacking. By directly assessing the un- [email protected] equal lobe areas that are transported via a turnstile mechanism, such is obtained through a bi-infinite modulated summation of coefficients of the Melnikov function. These CP13 are themselves expressible using Fourier transforms. Com- Singularities at Time Equal to Infinity in Equivari- putability under complicated perturbations is illustrated ant Harmonic Map Flow by example.

Many nonlinear parabolic equations in geometry and ap- Sanjeeva Balasuriya plied mathematics (mean curvature flow, Ricci flow, har- University of Sydney monic map flow, the Yang-Mills flow, reaction diffusion School of Mathematics & Statistics 82 DS05 Abstracts

[email protected] injection, we find numerically recurrent oscillations of the mode amplitudes that corresponds to chaotic or periodic attractors. A bifurcation diagram is presented as a func- CP14 tion of the energy injection. Efficient Topological Chaos Embedded in the Blinking Vortex System Antonio Batista Department of Mathematics The particles mixings by blinking vortex system are stud- State University of Ponta Grossa ied. It is well known that the chaotic advection occurs by [email protected] the system due to the homoclinic chaos. A braid is assigned from a well-controlled operation of the system, and such Sergio Lopes braids are classified into three types via Thurston-Nielsen Department of Physics theory. We propose a operation assigned a pseudo-Anosov Federal University of Parana braid, which induces topological chaos, to realize efficient lopes@fisica.ufpr.br and global particles mixing in the whole space. Takashi Sakajo Ibere L. Caldas Hokkaido University Institute of Physics Department of mathematics University of Sao paulo [email protected] [email protected]

Eiko Kin Ricardo Viana Department of mathematics, Kyoto University Physics Department [email protected] Federal University of Parana viana@ﬁsica.ufpr.br

CP14 CP15 Complex Basins in Weakly Dissipative Dynamical Systems Frequency Locking in Extended Systems: the Im- pact of a Turing Mode Chaotic scattering in open Hamiltonian systems under weak dissipation is not only of fundamental interest but A Turing mode in an extended periodically forced oscilla- also important for problems of current concern such as the tory system can change the classical resonance boundaries advection and transport of inertial particles in fluid flows. of a single forced oscillator. Using the normal form equa- Previous work using discrete maps demonstrated that non- tion for forced oscillations, we identify a Hopf-Turing bi- hyperbolic chaotic scattering can be metamorphically in- furcation point around which we perform a weak nonlinear fluenced by weak dissipation in that the algebraic decay of analysis. We show that resonant standing waves can ex- scattering particles becomes exponential. Here we extend ist outside the 2:1 resonance region of uniform oscillations, the result to continuous-time Hamiltonian flows by using and non-resonant mixed-mode oscillations may prevail in- the Henon-Heiles system as a prototype model. We also go side the resonance region. beyond by investigating the basin structure of scattering Arik Yochelis dynamics. We find, in the common case where multiple Department of Chemical Engineering destinations exist for scattering trajectories, Wada basin Technion, Haifa, Israel boundaries are common and they appear to be structurally [email protected] stable under weak dissipation even if the nonhyperbolic scattering dynamics is not. Christian Elphick Ying-Cheng Lai Centro de Fisica No Lineal y Sistemas Complejos de Arizona State University Santiago Department of Mathematics Santiago, Chile [email protected] none

Jesus M. Seoane Ehud Meron Nonlinear Dynamics and Chaos Group. Department of Energy and Environmental Physics Universidad Rey Juan Carlos BIDR, Ben Gurion University, Sede Boker Campus 84990, [email protected] Israel [email protected] Miguel A.F. Sanjuan Nonlinear dynamics and chaos group Aric Hagberg Universidad Rey Juan Carlos Los Alamos National Laboratory [email protected] [email protected]

CP15 CP16 Three-Wave Coupling in the Tokamak Turbulence Dynamics of a Penning-Malmberg Trap

We use the Hasegawa-Mima equation to study the dynamic A grid-free multipole treecode method is used to explore change of the spatial Fourier spectrum of tokamak plasma charged particle dynamics in a Penning-Malmberg trap. turbulence. The energy flow among the three coupled dom- The system is formally equivalent to the 2D incompress- inant spatial modes depends on the low energy injection of sible Euler equations of fluid dynamics. The interaction the stationary tokamak discharges. For a range of energy of the self-induced particle electric field with the confining DS05 Abstracts 83

magnetic field leads to complex dynamics in this system. Academia Sinica Institute of Mathematics Andrew J. Christlieb, Ronert Krasny [email protected] University of Michigan Department of Mathematics [email protected], [email protected] CP18 Bifurcations in a Highway Traffic Model with Drivers’ Reaction-Time Delay CP16 Variational Averaging and the Guiding Center A car-following model of highway traffic involving the Equations for Charged Particle Motion in a Mag- reaction-time delay of drivers is investigated. Bifurca- netic Field tions of the corresponding system of delay differential equations are studied analytically and by numerical continua- We derive governing equations for the average motion of tion techniques. Regions in the parameter space are deter- charged particles in a magnetic field. To this end, we ap- mined, where the equilibrium coexists with periodic solu- ply a novel procedure to average the variational principle. tions corresponding to single or multiple traffic jams. The The resulting equations are equivalent to the guiding cen- low-dimensional dynamics of amalgamation and dispersion ter equations for charged particle motion; this marks an of traffic jams are also investigated. instance where averaging and variational principles commute. Finally, we compare our procedure with others Gabor Orosz for recovering averaged dynamics from multiscale systems, University of Bristol including Whitham averaging and coarse analysis (I. G. [email protected] Kevrekidis). Eddie Wilson Jerrold E. Marsden University of Bristol, U.K. California Inst of Technology [email protected] Dept of Control/Dynamical Syst [email protected] Bernd Krauskopf University of Bristol Jimmy Fung Dept of Eng Mathematics California Institute of Technology [email protected] [email protected]

Harish S. Bhat CP18 California Institute of Technology Periodic Gaits of a Quasi-Passive Dynamic Walking Control and Dynamical Systems Robot with Two Legs and Feet [email protected] We consider a simple walking robot which has two legs and feet and can kick the ground, and show that it can CP17 walk on the horizontal plane and even ascend gentle slopes. Solitons and Electric Conduction in Dissipative 1D Moreover, its energy consumption is very low, compared Lattices with the traditional approaches. To this end, we reduce the problem of obtaining such periodic gaits to a nonlin- The Toda lattice is known to possess soliton (cnoidal wave) ear boundary value problem (BVP) for ODEs, and use a solutions. With appropriate forcing these solitons may be computer software called AUTO to numerically analyze the maintained even in the presence of dissipation, and if cou- nonlinear BVP. pled to electrons via a Coulomb type interaction, can drive a “superconducting” current. Solitons have in fact been Yasuhisa Hasegawa suggested as a possible mechanism for high temperature su- University of Tsukuba perconductivity. We investigate these solitons and soliton- Graduate School of Systems and Information Engineering electron bound states in a simple 1D lattice, describing [email protected] their origin and subsequent bifurcations for different types of interaction potentials and forcing functions. Kazuyuki Yagasaki Gifu University Jeff Porter Department of Mechanical and Systems Engineering Universidad Complutense Madrid [email protected] jport@fluidos.pluri.ucm.es

CP19 CP17 A Flow for Data Clustering Dynamics of a Toda Lattice with Weak Viscosity Data clustering is an increasingly important problem for I will consider a one-dimensional system which consists of engineering, biological, sociological and other sciences. We nonlinear springs with a Toda potential and linear dash- present a dynamical system which evolves on an adjoint pots. In this system, the dash-pots work as the source orbit of the Lie group SO(N) and performs data clustering. of weak viscosity. Thus, in short time scale, the system Our objective is not to provide the most eﬃcient algorithm, behaves like an usual Toda lattice, while in long time scale, but to give an original approach and shed some new light it behaves like a viscous ﬂuid. I will present the equations on this long known problem. of motion, and discuss the dynamics of the system. Mohamed Ali Belabbas Toshio Yoshikawa Harvard University 84 DS05 Abstracts

[email protected] system evaluation. Kevin Wedeward CP19 New Mexico Tech Nonlinear Time Series Prediction Using Local Electrical Engineering Department Modeling: a Comparison of Diﬀerent Approaches [email protected]

Predicting the future evolution of dynamical systems is a Steven Ball major goal in many areas of science. Often the underly- SAIC ing dynamical equations are unknown and only a single- [email protected] channel time series is available. Prediction methodologies based on embedding and attractor reconstruction using lo- Steve Schaffer cal statistical models constructed from the data are well- New Mexico Tech established. A variety of approaches for building local mod- Mathematics Department els from data has been proposed: local polynomial models schaff[email protected] based on nearest neighbors, radial basis function models, cluster-weighted modeling, adaptive local polynomial models. The present study compares/contrasts all these differ- Ernie Barany ent approaches on real observational data from an elec- New Mexico State University tronic circuit. Both best guess and probabilistic prediction Department of Mathematical Sciences is considered. [email protected] Leonard Smith OCIAM CP21 University of Oxford Effects of Degree Correlation on Network Synchro- [email protected] nizability The syncronization of networks of coupled oscillators has Frank Kwasniok drawn recently a lot of interest. It has been shown that the University of Oldenburg, Germany network structure seems to be able to affect the stability [email protected] of the sinchronized state. The scale free nature characteristic of real world networks has been shown to be able to CP19 worsen the synchronizability, as an effect of the strong heterogeneity of the connectivity. In order to explain this phe- Towards a Non-Linear Trading Strategy for Finan- nomenon, the spectrum of the Laplacian of the graph has cial Time Series been widely investigated and a relationship has been found A new trading strategy is proposed. The technique uses between the largest-smallest eigenvalue ratio and synchro- the state space volume evolution and its rate of change as nizability. Lately, it has been discovered that many real indicators. This methodology has been tested off-line using networks are characterized by degree correlation, that is eighteen high-frequency foreign exchange time series with the tendency for vertices to be connected to other vertices and without transaction costs. In our analysis an optimum with similar degrees. A negative correlation (or disassor- mean value of 25% gain may be obtained in those series tativity) is typical of technological networks such as the without transaction costs and an optimum mean value of Internet and the World Wide Web, in which high degree 11% gain assuming 0.2% of costs in each transaction nodes are more likely to be connected to low degree ones. Here we will study the relationship between degree corre- Fernanda Strozzi lation and network sinchronizability and we will show that Carlo Cataneo University (LIUC) negative correlation can have positive effects on network Quantitative Methods Institute sinchronizability. We will discuss the effects of the network [email protected] assortativity properties on the synchronization dynamics. Francesco Sorrentino Jose Manuel Zaldivar Universita’ di Napoli Federico II European Commission, Joint Research Centre, TP272 [email protected] [email protected] Mario Di Bernardo CP21 University of Bristol Simulation-Based Methodology and Tools for Anal- Dept of Engineering Mathematic ysis of Electric Power Networks [email protected]

We present a computer-based methodology for studying large-scale electric power networks. An efficient and ver- CP21 satile matlab code forms the foundation of the assessment The Fermi-Dirac Distribution and Traffic in Com- procedures providing dynamic simulation and analysis ca- plex Networks pabilities. The methodology integrates a variety of techniques based on geometric nonlinear control theory, pa- We propose an idealized model for traffic in a network, in rameter estimation methods, and other dynamical system which many particles move randomly from node to node, analysis tools. These strategies were developed to address following the network’s links, and it is assumed that at model development and validation, vulnerability assess- most one particle can occupy any given node. We show ment and mitigation, long-term planning, and real-time that the particles behave like free fermions, and their statistical properties are given by the corresponding Fermi-Dirac distribution. We use this to obtain analytical expressions DS05 Abstracts 85

for dynamical quantities of interest. CP22 Oscillatory Pattern Formation with a Conserved Alessandro Moura Quantity Institute of Physics Universidade de Sao Paulo The influence of a conserved quantity on an oscillatory [email protected] pattern-forming instability is examined in one space dimension. Amplitude equations are derived which are generic for systems with a pseudoscalar conserved quantity (e.g., ro- CP22 tating convection, magnetoconvection) but are also appli- Reentrant Hexagon Patterns in Non-Boussinesq cable to systems with a scalar conserved quantity. The sta- Convection bility properties of travelling and standing waves are analysed and new long-wavelength instabilities are reported. The classical scenario for non-Boussinesq convection con- These instabilities give rise to modulated waves and highly sists of hexagonal patterns at onset and a transition to localised pulse-like solutions. roll patterns somewhat above onset. We investigate the strongly nonlinear regime numerically for various fluids and David M. Winterbottom find quite generally in the roll regime a second bifurca- University of Nottingham tion, which restabilizes the hexagons. For stronger non- [email protected] Boussinesq effects the two transitions merge, eliminating the instability of hexagons to rolls altogether. This scenario is captured qualitatively in an extension of the usual CP23 amplitude equations. Bifurcations on the Sphere with Inhomogeneity

Hermann Riecke We study the steady-state pattern formation problem with Applied Mathematics spherical symmetry and inhomogeneity. Group theory re- Northwestern University veals the most general normal form for a system in one [email protected] representation of O(3), subject to an explicit symmetry breaking that belongs to another representation. For the Santiago Madruga morphogenesis problem we ﬁnd one viable scenario for the Northwestern University formation of the antero-posterior and dorso-ventral axes: Applied Mathematics that with both the system and the inhomogeneity belong- [email protected] ing to the l = 1 representation.

Werner Pesch Timothy K. Callahan U. Bayreuth Arizona State University Theoretische Physik Department of Mathematics and Statistics [email protected] [email protected]

CP22 CP23 Spatio-Temporal Chaos of Hexagon Patterns in Ro- Hopf Bifurcation from Rotating Waves on the tating Non-Boussinesq Convection Sphere

In rotating systems the transition from hexagonal convec- Spiral waves are spatio-temporal patterns that have been tion patterns to rolls is replaced by a Hopf-bifurcation to observed in numerous physical situations, ranging from oscillating hexagons. For strongly non-Boussinesq convec- Belousov-Zhabotinsky chemical reactions to cardiac tissue. tion in water we ﬁnd that the Hopf-bifurcation can be- The global geometry of the heart is closer to a sphere than come subcritical. The ensuing bistability between oscil- to a plane. Therefore, it would be important to study spiral lating and steady hexagons leads to spatio-temporal chaos wave dynamics in a context where the symmetries are those characterized by chaotically evolving domains of oscillating of the sphere instead of the plane. Also, rigidly rotating hexagons amidst steady hexagons. Since these hexagonal spiral waves are rotating waves. In this talk, we consider patterns are defect-free, the spatio-temporally chaotic dy- one-parameter dependent reaction-diﬀusion systems on the sphere of radius r, which are equivariant under the group namics might be captured by a quintic complex Ginzburg- SO Landau equation. (3) of all rigid rotations:

Hermann Riecke ∂u 2 (t, x)=D∆u(t, x)+F (u(t, x),λ)onrS , (2) Applied Mathematics ∂t Northwestern University [email protected] u =(u1, u2, ..., uN ) with N ≥ 1, D is the diﬀusion matrix and F =(F1, F2, ..., FN ) is a suﬃciently smooth Santiago Madruga function. Northwestern University Applied Mathematics For λ = 0 we consider a relative equilibrium SO(3)u0 with I [email protected] trivial isotropy Σu0 = 3 consisting of rotating waves. By Hopf bifurcation from rotating waves, we generically get Werner Pesch modulated rotating waves. We show that there exists a U. Bayreuth decomposition of these modulated rotating waves in two parts: a primary frequency vector part, eX(λ)t and a 2π - Theoretische Physik |ωλ| [email protected] periodic part, B(t, λ). We present a way of constructing X(λ) and B(t, λ) using the BCH formula in so(3) and the solution Z(t, λ) ∈ so(3) to the initial value problem on 86 DS05 Abstracts

some interval [0, 2π ], with the integer n>0 indepen- Results and Numerics n|ωλ| dent of λ: We study the dynamics of diﬀusing particles in one space dimension with annihilation on collision and random nucleation (creation of particles). A new method of analysis −→ cos |Z| −−−−−→ 1 1 2 2 G Z˙ = I3 + Z + 2 − Z X (t, λ), permits exact calculation of the density of particles. With 2 |Z| 2 sin |Z| |Z| 2 paired nucleation at suﬃciently small initial separation the Z(0) = O3. nucleation rate is proportional to the square of the steady (3) state density. With unpaired nucleation, the nucleation This decomposition allows us to show that the quasi- rate is proportional to the cube of the steady state density. periodic meandering of the modulated rotating waves obtained by Hopf bifurcation is possible and, in fact, that Grant Lythe there are three types of motions for the tips of these modu- Department of Applied Maths lated rotating waves. These types of motions are visualized University of Leeds using Maple. [email protected]

In the case of a resonant Hopf bifurcation from a rotating wave in two-parameter dependent SO(3)-equivariant CP25 reaction-diffusion systems on the sphere, we obtain a Perturbing Flows for Optimum Chaotic Flux branch of modulated rotating waves with primary fre- Motivated by optimizing mixing in microfluidic devices, the quency vectors orthogonal to the frequency vector of the 0 rotating wave undergoing Hopf bifurcation. nature of the time-periodic perturbation subject to a C bound which would generate the maximum chaotic flux Adela Comanici across a heteroclinic separatrix is investigated. Though an University of Houston ill-posed optimization problem, it is shown that choosing [email protected] the perturbation as close as possible to a certain (unphysi- cal) flow generates the best mixing. An expression for this maximum chaotic flux approachable at each frequency of CP23 perturbation is obtained and analyzed. Tissue Dynamics - Modeling and Experiment Sanjeeva Balasuriya Early detection of cancer is critical to improved success in University of Sydney treatment. Much progress has been made in understanding School of Mathematics & Statistics the molecular and genetic underpinnings of cancer. Less is [email protected] known about how cells, as collective populations, progress towards a malignant state. This talk will review some of the models that treat tissues as dynamical systems, with a CP25 focus on epithelial tissues such as the lining of the mouth. Hamiltonian Theory of Mixing in a Microfluid De- Our goal is to design experiments that explore dynamical vice and pattern forming aspects of tissues as they continually remodel themselves, both in healthy and diseased states. Mixing of fluids in microchannels cannot rely on turbu- Progess in our experimental characterization will be de- lence since the flow takes place at extremly low Reynolds scribed. numbers. Various active and passive devices have been developed to induce mixing in microfluid flow devices. We describe here a model of an active mixer where a transverse Randall Tagg 2 University of Colorado at Denver periodic flow interacts with the main flow in R . We de- Department of Physics velop a Hamiltonian description of the dynamics. We prove [email protected] nonintegrability of the system, describe adiabatic invariants, show the existence of KAM-tori, and examine some specific solutions. CP24 Singularly Perturbed Cluster Growth in Aggrega- Mikhail V. Deryabin tion Kinetics Faculty of Mathematics and Mechanics Moscow State University In the classical Becker–Doring model of aggregation ki- [email protected] netics, each cluster assumes “critical size” relative to its surface monomer concentration. This matching between Poul G. Hjorth cluster size and surface concentration seems energetically Technical Univ of Denmark unstable. However, careful analysis reveals the stability of Department of Mathematics the matching. Cluster growth is described by a singularly [email protected] perturbed diffusion BVP. Using different time-scales, the matching can be shown to be stable. This analysis is one step in a larger ongoing singular perturbation analysis of CP25 aggregation kinetics. Describing Homoclinic Tangles Through Homo- topic Lobe Dynamics Yossi Farjoun UC Berkeley Homoclinic tangles are basic geometric structures that or- [email protected] ganize the dynamics of invertible maps on the plane. The topology of such tangles provides an approach to understanding the qualitative dynamics of transport and escape CP24 phenomena. Since the original introduction of tangles by Diffusion-Limited Reaction in 1 Dimension: Exact Poincaré, a number of researchers have characterized spe- DS05 Abstracts 87

cific classes of tangles. However, there seems to be a lack Department of Physics and Astronomy of general techniques applicable to the wide variety of tan- Francis Marion University, Florence SC 29501 gles realized by maps. To address this issue, we present a [email protected] new symbolic technique for describing tangles. This technique, called homotopic lobe dynamics, allows a homoclinic Eugenia Kalnay tangle to be described up to arbitrarily long times, or al- Department of Meteorology ternatively, to arbitrarily small scales. University of Maryland [email protected] John Delos College of William and Mary [email protected] CP26 A Filtration Algorithm for Computing Lyapunov Kevin A. Mitchell Exponents Univ of California, Merced School of Natural Sciences Recently, Ott and Yorke described the construction of “en- [email protected] veloping manifolds” to answer the following question: If A is an arbitrary compact set in Rn, what is the smallest integer d such that, given x ∈ A, there is a compact C1 CP26 manifold of dimension d that contains all points of A that Stalking: Aggressive Shadowing of a Noisy Trajec- lie in some neighborhood of x? In this talk, we describe tory how to apply this machinery to the computation of Lya- punov exponents from chaotic time series. In particular, Shadowing theory addresses the question of how trajecto- we discuss how spurious exponents can be identified and ries of a low-resolution model L compare with solutions of eliminated. a high-resolution model H. For typical H, no trajectory of L remains close to an H trajectory for all time. There- Eric J. Kostelich fore, we examine perturbed trajectories of L. We find such Arizona State University pseudo-trajectories of L can remain close to H (for a rea- Department of Mathematics sonable definition of ’close’) even when the perturbations [email protected] are much smaller than the difference between L and H. Pseudo-trajectories of a low-resolution model L shadow an H trajectory for orders of magnitude longer than true tra- CP27 jectories of L. We discuss applications of stalking trajecto- Dispersive States in Propagative Systems with Pe- ries to weather forecasting. riodic Structure

James A. Yorke I will analyze the weakly nonlinear dynamics of a propaga- University of Maryland tive spatially extended system with spatial periodic struc- Departments of Math and Physics ture. Light propagation in fiber Bragg gratings and Bose- [email protected] Einstein condensates in optical lattices are two examples of systems with spatial periodic structure that have re- Chris M. Danforth cently received very much attention. The nonlinear reso- Applied Mathematics and Scientific Computation nant dynamics of these systems is dominated by two main University of Maryland effects: the counter-propagating transport (that is coupled [email protected] through the periodic structure) and the dispersion, which has been systematically neglected in previous studies based on the nonlinear coupled mode equations description. I CP26 will show that dispersion gives rise to new instabilities, Convex Error Growth Patterns in a High Dimen- complex spatio-temporal patterns and new localized struc- sional Chaotic Dynamical System for Weather Pre- tures, which cannot be detected if dispersion effects are not diction considered. I will also present some numerical simulations to illustrate these essentially dispersive states. The error growth, that is the growth of the distance between two almost identical solutions of a high dimensional Carlos Martel chaotic dynamical system, is studied and compared to Universidad Politecnica de Madrid those with lower dimensions. Typically, the exponential [email protected] growth rate plateaus and decreases concavely in amplitude- wise asymptotic to a log-linear line. In contrast, we find that in a higher dimensional system, the error growth rate CP27 decreases as if a convex function in amplitude-wise and Convection Induced Spatial Nonlinear Phase Mod- close to two log-linear lines. ulations in Optical Systems

John Harlim We show that the combined action of diffraction and Department of Mathematics convection (walk-off) leads to a nonlinear phase modu- University of Maryland lation in degenerate optical parametric oscillators. Near [email protected] threshold, this mechanism is described analytically by a Ginzburg-Landau amplitude equation where the nonlinear Brian R. Hunt, James Yorke self-coupling coefficient depends on the convection term. University of Maryland This enable us to characterize the induced asymmetry in [email protected], [email protected] the generated travelling waves. The predictions are in excellent agreement with the solutions of the governing equa- Michael Oczkowski 88 DS05 Abstracts

tions. ral scales. In practice, various hierarchical modeling techniques have been explored. However, the concurrent cou- Roberta Zambrini pling which is more relevant and ubiquitous in biological Department of Physics systems has not yet been fully understood. In this talk, University of Strathclyde a mathematical framework of concurrent multi-scale and [email protected] multi-physics modeling of biological systems is laid out. In particular, two specific aspects of this modeling strategy Céline Durniak will be elaborated. Firstly, the validity of the newly de- PhLAM - Universite de Lille (France) veloped immersed continuum method (ICM) will be exam- [email protected] ined via a meshless finite element solid model coupled with the surround fluid. This new modeling method enables a Maxi San Miguel unified treatment of flexible structures or solids moving in Instituto Mediterr aqueous environment and the direct coupling with mass ’aneo de Estudios Avanzados IMEDEA and heat transfer equations. Secondly, a finite tempera- (CSIC-UIB) ture bridging scale approach is presented to link macro- [email protected] scopic computational mechanics with localized refinements at micro-or nano-scale using molecular dynamics. Finally, Majid Taki future work spanning the quantum and continuum length scales will be proposed. PhLAM - Universite de Lille (France) [email protected] Sheldon Wang Department of Mathematical Sciences New Jersey Institute of Technology CP28 [email protected] 3D Modeling and Surface Oscillation of a Levitated Droplet CP29 The effect of the surface oscillations for a levitated (EML) Structural Stability of Boundary Equilibria droplet is considered in this study. The droplet is modelled as a three dimensional system with lumped masses In recent years, much research effort in applied science and elastic springs. The expression for the spring elas- and engineering has focussed on dynamical systems which tic constants in a global stiffness matrix and the surface are nonsmooth or discontinuous. This lecture is concerned oscillations modes are presented. It is shown that adjust- with the structural stability of boundary equilibria in non- ments to the magnetic field, enable the levitated specimen smooth dynamical systems. Namely, we study the struc- to approach to a spherical shape and improve the ability tural stability under parameter variations of equilibria ly- to control the stability. Computations were performed for ing on discontinuity boundaries in phase space dividing re- droplets of aluminum and copper. gions where the system under investigation is smooth. We show that it is possible to give a set of conditions to account Mihai Dupac, Davig Beale, Ruel Overfelt for the possible dynamical scenarios that can be observed. Auburn University In particular, we are interested in isolating the branches [email protected], [email protected], of solutions originating from a boundary equilibrium point [email protected] under parameter variations. Mario Di Bernardo CP28 University of Bristol Geometrical Formulation of Multilevel Mesoscopic Dept of Engineering Mathematic Dynamics and Thermodynamics [email protected] To interpret multilevel experimental observations of complex fluids we need a multilevel (multiscale) models. A CP29 general framework for such models is provided by contact Dynamics of Stochastic Nonsmooth Systems geometry. The mesoscopic time evolution is the time evolution preserving the contact structure of mesoscopic thermo- Many power electronics systems are nonsmooth (eg dynamics. Particular realizations of such abstract dynam- DC/DC boost and buck converters). These systems can ical system include, among others, classical and extended undergo changes in behaviour under parameter variation nonequilibrium thermodynamics and the Boltzmann ki- that are not present in smooth systems called grazing bi- netic theory. In this talk the abstract mesoscopic dynamics furcations. We highlight the dramatic effect that noise can is illustrated in the context of multilevel modelling of tur- have on these bifurcations (including advance or smooth- bulent flows. ing of the bifurcation) and conclude by showing excellent agreement between experimental results and a stochastic Miroslav Grmela model for the DC/DC boost converter. Ecole Polytechnique de Montreal [email protected] John Hogan Bristol Centre for Applied Nonlinear Mathematics Department of Engineering Mathematics, University of CP28 Bristol A Concurrent Multi-Scale and Multi-Physics Mod- [email protected] eling of Biological Systems

A major challenge in the modeling of biological systems is CP29 the accurate description of physical, chemical, and biolog- Continuous and Discontinuous Grazing Bifurca- ical phenomena over a wide range of spatial and tempo- DS05 Abstracts 89

tions in Quasi-Periodic Oscillations Survival

Grazing bifurcations, occuring due to tangential contact We consider the tent map as the prototype of a chaotic between a system attractor and state-space discontinu- system with escapes. We show analytically that a small, ities, may cause dramatic changes in system dynamics. bounded, but carefully chosen perturbation added to the Although its well known that grazing bifurcations of pe- system can trap forever an orbit close to the chaotic saddle, riodic system attractors can be either continuous or dis- even in presence of noise of larger, although bounded, am- continuous, this work reveals that grazing bifurcations of plitude. This problem is focused as a two-person, mathe- quasi-periodic attractors are never strictly discontinuous in matical game between two players called “the protagonist” nature. Numerical results are presented for a 2D torus in and “the adversary”. The protagonist’s goal is to survive. 4D state space, where continuous and apparently discon- He can lose but cannot win; the best he can do is sur- tinuous bifurcations arise for various parameter values. vive to play another round, struggling ad infinitum. In absence of actions by either player, the dynamics diverge, Phanikrishna Thota leaving a relatively safe region, and we say the protagonist Virginia Polytechnic Institute and State University loses. What makes survival difficult is that the adversary [email protected] is allowed stronger “actions” than the protagonist. What makes survival possible is (i) the background dynamics (the Harry Dankowicz tent map here) are chaotic; and (ii) the protagonist knows Virginia Polytechnic Institute and State University the action of the adversary in choosing his response and is Engineering Science and Mechanics permitted to choose the initial point x0 of the game. We [email protected] use the “slope 3” tent map in an example of this problem. We show that it is possible for the protagonist to survive. CP30 Jacobo Aguirre Dynamics of Conflicting Decision-Making Groups Universidad Rey Juan Carlos, Spain [email protected] We present a model of the interaction of two rival leader- ship groups engaged in conflict. Each group is represented Miguel A. Sanjuan as a hierarchical social network and modeled by a set of Nonlinear Dynamics and Chaos Group coupled nonlinear differential equations which evolve each Universidad Rey Juan Carlos members policy position with respect to individual prefer- [email protected] ence, group influence, and the policy of the opposing group. The model is based on social and cognitive psychology the- Francesco d’Ovidio ories of attitude change, group dynamics, and conflict and IMEDEA (CSIC-UIB) represents an alternative approach to the two-level game Universitat Illes Balears, E-07071 Palma de Mallorca, problem which seeks to account for the interaction between Spain domestic and foreign policies in negotiations. [email protected] Michael Gabbay Information Systems Labs CP31 [email protected] Bifurcations in Integrable and Near-Integrable Hamiltonian Systems CP30 Topological properties of bifurcations in low-dimensional Scaling, Renormalization, and Universality in Hamiltonian systems are investigated. We give a list Combinatorial Games: the Geometry of Chomp of generic possible singularities appearing in the energy- Combinatorial games, which include chess, go, checkers, momentum bifurcation diagrams of a certain class of inte- Nim, Chomp, and dots-and-boxes, have both captivated grable and near-integrable systems and provide the classi- and challenged mathematicians, computer scientists, and fication of the corresponding bifurcations. Also, the topo- players alike. Using Chomp as an archetype, we report on logical description of energy surfaces at bifurcation states a new geometrical approach which unveils some unexpected is given. parallels between combinatorial games and key ideas from Milena Radnovic physics and dynamical systems, most notably notions of Faculty of Mathematics and Computer Science scaling, renormalization, universality, and chaotic attrac- The Weizmann Institute of Science, Israel tors. [email protected] Adam S. Landsberg Claremont McKenna College, Pitzer College, Scripps Vered Rom-Kedar College Weizmann Institute [email protected] of Science [email protected] Eric Friedman ORIE CP31 Cornell University [email protected] Hierarchy of Bifurcations in the Truncated and Forced NLS Model.

CP30 We analyze the truncated forced non-linear Schr¨odinger (NLS) model using a novel framework in which a hierarchy Control of Chaotic Transients: Yorke’s Game of of bifurcations is constructed. Consequently, we provide insights regarding its global structure and the type of in- 90 DS05 Abstracts

stabilities which appear in it. In particular, we show that a Clarkson University parabolic resonance mechanism of instability arises in the [email protected] relevant parameter regime of this model. The analogous behavior in the NLS pde is discussed. Joe Skufca Dept. Mathematics Eli Shlizerman US Naval Academy The Weizmann Institute of Science [email protected] Department of Computer Science and Applied Mathematics [email protected] Dan Stillwell Virginia Tech [email protected] Vered Rom-Kedar Weizmann Institute of Science CP32 [email protected] Measuring Network Structure and Its Eﬀect on Network Dynamics.

CP31 Disease spread over contact networks is one example of Data Assimilation and Hamiltonian Systems dynamics on networks. We use simple models of disease spread (SI or SIR) to investigate how the spread depends Weather systems can, in part, be described by Hamilto- on characteristics of the network structure at the site of nian mechanics. Numerical weather prediction models use initial infection. The notion of a backbone of the network data assimilation to estimate the optimal initial conditions is discussed and attempts made to identify it. Eﬀects of by combining observations with dynamical constraints us- inoculation of this backbone on the spread suggests a po- ing control theory. We investigate whether the geomet- tentially fruitful prevention strategy. ric features inherent in Hamiltonian systems can be exploited when implementing data assimilation. We use the Daniel A. Schult restricted three body problem as a test bed for studying Colgate University constraints in control problems with a Hamiltonian struc- Dept of Mathematics ture. [email protected] Laura R. Watkinson University of Reading CP32 [email protected] Dynamics of Epidemics on Correlated Networks

Mike Cullen A major and unexplained difference between social net- Met Office works and biological/technological networks is the positive mike.cullen@metoffice.gov.uk assortativity (degree-degree correlation) of the former and the negative assortativity of the latter. We will discuss Amos Lawless the structural constraints between clustering and degree- University of Reading degree correlation and the resulting effects on dynamics of [email protected] epidemics and reaction-diffusion processes on networks. Aric Hagberg, Pieter Swart Ian Roulstone Los Alamos National Laboratory University of Surrey [email protected], [email protected] [email protected]

Nancy K. Nichols CP33 University of Reading Localised Convection Cells in the Presence of a Department of Mathematics Vertical Magnetic Field [email protected] Thermal convection in a horizontal fluid layer heated uniformly from below usually produces a spatially periodic ar- CP32 ray of convection cells of roughly equal amplitudes. In the Proof of Communication and Synchronization in presence of a large scale vertical magnetic field, convection Disconnected But Fast Moving-Neighborhood Net- may instead occur in vigorous isolated cells, separated by works regions of strong magnetic field. Convection in this form may be responsible for the appearance of bright umbral We model a time-varying ad-hoc network of moving agents dots in the centres of sunspots. This talk will discuss a which move according to a dynamical system. Instanta- model for two-dimensional localised solutions of this kind neously, the network is disconnected, but nonetheless the in which the amplitude of convection is not assumed to be moving averaged graph Laplacian allows for communica- small. The model reflects the physics of the interaction be- tion in the form of synchronization to occur. In this sense, tween field and flow, and elucidates the approximate region we have generalized the master-stability function formal- of parameter space over which these solutions exist. ism to time varying networks. We prove that fast enough varying connections relative to information flowing across Jonathan Dawes the network compensates competing time scales between DAMTP, University of Cambridge agents and oscillation. [email protected] Erik Bollt DS05 Abstracts 91

CP33 CP34 Boundary Eﬀects and the Onset of Taylor Vortices Stochastic Resonance in Vision: Models and Data

The onset of spatially periodic vortex states in the Taylor– Stochastic resonance (SR) is a phenomenon through which Couette flow between rotating cylinders occurs at the value small amounts of additive noise enhance the performance of of Reynolds number predicted by local bifurcation theory. a signal processing system. This paradoxical enhancement However, the symmetry breaking induced by the top and is to be expected whenever the system has nonlinearities. bottom plates means that the true situation should be a Recently the phenomenon has found exciting applications disconnected pitchfork. This leads to an apparent contra- in sensory physiology, where nonlinearities in biological diction: why should Taylor vortices set in so sharply at the sensory transducers make SR ubiquitous. We report on re- Reynolds number predicted by the symmetric theory, given cent experimental and modelling studies that demonstrate large symmetry-breaking effects caused by the boundary the extent to which the human visual system is adapted to conditions? We offer a generic explanation, based on a use SR. Swift–Hohenberg pattern formation model that shares the same qualitative features as the Taylor–Couette flow. Carole L. Tham, Mark Muldoon The University of Manchester Alan Champneys [email protected], [email protected] University of Bristol Dept. of Engineering Maths [email protected] CP34 Waves in Turtle Visual Cortex Have Three Com- Alastair M. Rucklidge ponents Department of Applied Mathematics University of Leeds Visual stimuli evoke propagating waves in turtle visual cor- [email protected] tex . A large-scale model was used to characterize the activity of cortical neurons in response to simulated light flashes. Pyramidal cell firing shows three components: an CP33 initial depolarization, a primary propagating wave and a Thermocapillary Controlled Rupture of Thin Liq- secondary wave. The transitions from the initial depo- uid Sheets larization to primary waves and from primary to the secondary waves are controlled by separate populations of in- We study the rupture process of a fluid sheet induced by a terneurons. (Supported by the NSF CRCNS program) spatially distributed initial temperature profile. For vari- cose disturbances, analysis of a long-wave approximation Philip Ulinski (yielding a coupled system of nonlinear evolution equa- The University of Chicago, Chicago tions) shows that film rupture can be controlled by careful Department of Organismal Biology and Anatomy selection of the Marangoni number. We extend these ideas [email protected] to sinuous disturbances, discuss the implications of the results to cylindrical jets and also comment on important CP35 industrial applications of the work. Multiple Equilibria in Biochemical Reaction Net- Burt S. Tilley works and the Jacobian Criterion Franklin W. Olin College of Engineering [email protected] The Jacobian criterion examines the capacity for multiple equilibria of high dimensional dynamical systems given by some classes of functions, for example polynomial func- Mark Bowen tions. In particular, we generate methods that allow us to School of Mathematical Sciences decide whether a given biochemical reaction network has University of Nottingham the capacity for multiple equilibria, and to identify sub- [email protected] networks that give rise to multiple equilibria. Also, we discuss implications for the interpretation of experiments CP34 in cell biology. This is joint work with Martin Feinberg. Spontaneous Activity in the Visual Cortex Gheorghe Craciun Mathematical Biosciences Institute, Ohio State University I will present experimental results (Grinvald et al.) illus- [email protected] trating the spontaneous background activity in the visual cortex. Then I will present our model for the visual cortex, and briefly discuss the computational problems associated CP35 with this model, and our numerical method. I will use our Constrained Hybrid Monte-Carlo and Free Energy model to explain this spontaneous bacground activity, and Calculation present predictions of the model. If I have time, I’ll go over other scale-up techniques, and the numerical methods We address the problem of computing the potential of associated with them. mean force (free energy) for a prescribed set of curvilin- ear coordinates in a molecular system. The approach gives Aaditya Rangan concise geometrical insight into the different contributions Courant Institute of Mathematical Sciences to the mean force and we provide a hybrid Monte-Carlo New York University based algorithm to compute the mean force using con- [email protected] strained simulations. Carsten Hartmann Free University Berlin 92 DS05 Abstracts

Institute of Mathematics CP36 [email protected] Elliptical Instability at Large Aspect Ratio

Elliptical instability is a universal mechanism by which pla- CP35 nar flows with elliptical streamlines become unstable to Pattern Formation and Surface Design for Het- three dimensional perturbations. The system is governed erogenous Reactions on Structured Surfaces by two parameters µ and which are aspect ratios of the domain. Previous work has largely focused on near ax- Reaction diffusion models for heterogenous catalytic re- isymmetry cases ( ≈ 0) using perturbation theory. Using actions on structured surfaces are considered. Space- symmetries, we develop a non-perturbative approach that dependent coefficients represent different types of materials allows us to obtain exact description of the boundaries of and the spatiotemporal patterns of the system depend on instability wedges in the µ– plane. This analysis reveals this surface structure. The surface design is optimized, i.e. a complex structure of the parameter plane, including a a certain amount of an expensive catalyst material is dis- bifurcation point at large aspect ratio where a seemingly tributed on a cheaper carrier material such that the output infinite number of wedge boundaries intersect. of the chemical reaction is maximized while considering the dynamical properties of the system. Norman Lebovitz University of Chicago Jens Starke Department of Mathematics University of Heidelberg [email protected] Institute of Applied Mathematics [email protected] Lay May Yeap University of Arizona CP36 Department of Mathematics [email protected] Low Order Modeling, Dynamics, and Control of Flow Separation CP37 We present efforts to derive low order models, analyze dynamics, and synthesize controllers for a class of fluid sepa- Explicit Periodic Solutions in a Model of a Relay ration problems dominated by coherent vortex structures. Controller with Delay and Forcing Due to POD’s inability to resolve patterns not present in We use a combination of numerical and analytical meth- the flow when the basis function were derived, we pursue ods to find and construct solutions of a model of relay a different, Lagrangian approach. The resulting systems control, formulated as a piecewise-constant delay differ- of nonlinear ODEs captures key features of the flows. The ential equation (DDE). Numerical solutions of a related structure of the ODEs are unusual, but amenable to anal- equation, where the discontinuities of the original DDE are ysis and control synthesis. smoothed, are used to guide the construction of explicit so- Brianno Coller lutions of the original DDE. Conversely, the construction Northern Illinois University of explicit solutions provides starting data for numerical [email protected] continuation of the smoothed equation. David A. Barton, Eddie Wilson CP36 University of Bristol, U.K. [email protected], [email protected] Adiabatic Invariance and Geometric Angle in Two- Dimensional Fluids Bernd Krauskopf We examine the response of two-dimensional, incompress- University of Bristol ible and inviscid fluids to slow deformations of the bound- Dept of Eng Mathematics ary of their domain. Two issues are addressed using per- [email protected] turbation techniques: (i) the determination of the (leading- order) Eulerian flow fields, which are shown to depend only on the shape of the fluid domain at any given time, and CP37 (ii) the determination of Lagrangian particle trajectories, Locating Periodic Orbits in High-Dimensional Sys- which depend on the successive shapes taken by the do- tems by Stabilising Transformations main. We emphasise the similarity between this problem and some familiar finite-dimensional analogues, such as the An algorithm for detecting unstable periodic orbits pendulum with varying length, and in particular the con- (UPO’s) that combines a modified semi-implicit Euler nection between (ii) and the Hannay–Berry (geometric) an- method and a set of stabilising transformations has had gles. We present explicit results for nearly axisymmetric considerable success in low dimensional chaotic systems. flows in slightly deformed discs. Applying the same ideas to higher dimensional systems is non-trivial due to the rapid increase in the number of trans- Jacques Vanneste formations. We have proposed a much smaller set of trans- School of Mathematics formations based on the stability properties of the UPO’s University of Edinburgh and successfully applied it to systems with dimension up [email protected] to six. In this talk we explore the possibility of extending this approach to high-dimensional flows representing Djoko Wiroseotisno PDE’s such as, for example, the Kuramoto-Sivashinski sys- Department of Mathematical Sciences tem. University of Durham Jonathan J. Crofts, Ruslan L. Davidchack [email protected] University of Leicester DS05 Abstracts 93

[email protected], [email protected] CP38 Bifurcations of Competing Modes in Laser Arrays

CP37 We study dynamics of laser arrays in terms of composite- Approximating Limit Cycles of An Autonomous cavity modes for the entire coupled-laser structure. We Delay Differential Equation find that essentially isolated micro-cavity semiconductor lasers may drive each other chaotic. As the optical iso- This talk presents an existence theorem for periodic so- lation between these lasers reaches practically attainable lutions of autonomous delay differential equations that limits, instead of approaching independent operation, the states: If an approximate frequency and periodic solu- lasers exhibit mutually induced chaotic oscillations. The tion is computed that satisfies a noncriticality condition, multimode equations are analyzed with bifurcation contin- then there is an exact frequency and periodic solution in uation techniques to reveal the formation of a gap in the a neighborhood of the approximate frequency and periodic lockband that is gradually occupied by instabilities of limit solution with a specific numerical bound. To obtain this cycles. numerical bound requires computing several parameters. Algorithms for computing these parameters and an appli- Sebastian Wieczorek cation will be given. Semiconductor Material and Device Sciences Sandia National Laboratories David E. Gilsinn [email protected] Mathematical and Computational Sciences Division National Institute of Standards and Technology Weng Chow [email protected] Sandia National Labs [email protected] CP38 Bifurcation Analysis of Dimensionless Equations CP39 Modelling Semiconductor Lasers Linear Stability of Viscoelastic Shear Flows

The Yamada equations are ordinary differential equations We present a linear stability analysis of a viscoelastic governing the dynamics of semiconductor lasers. It is pos- Oldroyd-B fluid in a plane Couette geometry. For vortices sible to use chaotic self-pulsating lasers to mask a signal aligned with the mean flow we find that the amplitudes of in a chaotic communication system. We optimise param- vortices and streaks decay. At the same time, non-normal eters via a symbolic and numerical bifurcation analysis of amplification of streaks leads to a transient growth, such dimensionless equations derived from the Yamada equa- that the maximum value of the kinetic energy associated tions and other models, including terms representing the with the perturbation exceeds its initial value by several modulation required to induce chaos, spontaneous emis- orders of magnitude. For Newtonian fluids the maximum sion, diffusion and radiative/non-radiative recombination amplification occurs when the width of the vortices is of the effects. The level of chaos is quantified via Lyapunov ex- order of the channel height. We find that for viscoelastic ponents. fluids the optimal vortex width is significantly wider and Gareth Roberts that the efficiency of the non-normal self sustaining process University of Wales Bangor, United Kingdom is reduced. % Juergen Buehrle [email protected] Philipps-University Marburg [email protected] Ricki P. Walker University of Wales Bangor, United Kingdom Bruno Eckhardt [email protected] Philipps Universität Marburg Fachbereich Physik CP38 [email protected] Pulsating External Cavity Semiconductor Lasers CP39 We investigate numerically and analytically a simple ODE model that captures key characteristics of external cavity Interactions Between Polymer Dynamics and Self- semiconductor lasers. This model has the same set of fixed Sustaining Coherent Flow Structures: a Model for points than the ones derived from the celebrated Lang and Turbulent Drag Reduction Kobayashi single mode single delay rate equations. Pul- Traveling-wave solutions to the Navier-Stokes equations sating phenomena originally found numerically at the level have been found that capture the dominant structure of of the Lang and Kobayashi model are investigated analyti- the near-wall buffer region of turbulence. We describe the cally in the context of the simple ODE model coupled with effect of viscoelasticity on these states, aiming to better the observation that the free running semiconductor laser understand the mechanism(s) of drag reduction by poly- is a nearly conservative oscillator. mer additives. The changes to the velocity fields mirror Vassilios Kovanis, Tamas Wiandt those experimentally observed in turbulent flows of poly- Rochester Institute of Technology mer solutions: drag decreases, streamwise velocity fluctua- [email protected], [email protected] tions increase while wall-normal fluctuations decrease, and smaller wavelength structures are suppressed. Wei Li Dept. of Chemical Engineering University of Wisconsin-Madison 94 DS05 Abstracts

[email protected] which exhibit temporal chaos and spatial coherence.

Michael D. Graham Elbert E. Macau Dept. of Chemical and Biological Engineering LAC - Laboratory for Computing and Applied University of Wisconsin-Madison Mathematics [email protected] INPE - Brazilian Institute for Space Research [email protected]

CP39 Erico Rempel Discrete Element Modeling of Shear Bands in Soils DGE INPE - Brazilian Institute for Space Research Soils and other granular materials can be modeled either as [email protected] a continuum or as discrete particles. The second approach is known as molecular dynamics or as a discrete/distinct element method (DEM). We simulate soil under several test CP40 scenarios with a discrete element method, with the goal Homoclinic Snaking in Reversible Systems of determining which constitutive laws are realistic without sacrificing computational speed. Of particular interest We study the unfolding of heteroclinic cycles between equi- is which constitutive laws can generate observable shear libria in reversible systems of ODEs. It is shown that if bands, even under continuous boundary conditions. one of the equilibria is a saddle-focus, snaking curves of homoclinic orbits can be found. Along a snaking curve Robert E. Wieman infinitely many fold bifurcations of homoclinic orbits oc- University of Bristol cur; the corresponding solutions spread out and develop Department of Engineering Mathematics more and more oscillations about their centre. The analy- [email protected] sis is illustrated by computations for a system of Boussinesq equations. CP40 Thomas Wagenknecht Homoclinic Orbits in the Chua’s Circuit Bristol Laboratory for Advanced Dynamics Engineering [email protected] We describe the complex scenario formed by the homoclinic orbits, to the saddle-focus points, in the parameter space Juergen Knobloch of the Chua’s circuit. In particular, we present a scaling TU Ilmenau law that gives the ratio between bifurcation parameters of Institute of Mathematics different nearby orbits. The results to be presented are [email protected] valid for a large class of dynamical systems.

Ibere L. Caldas CP41 Institute of Physics Dynamics Under Random Boundary Conditions University of Sao paulo [email protected] Scientific and engineering systems are usually subject to uncertainty or random influence. Often, the noise acts on Rene Medrano-T nonlinear systems at physical boundary. Randomness can Institute of Physics have delicate impact on the overall evolution of such sys- Unuversity of Sao Paulo tems. Taking stochastic effects into account is of central [email protected] importance for the development of mathematical models of complex systems under uncertainty. The speaker presents Murilo Baptista recent results on techniques for understanding dynamical Institut fuer Physik Am Neuen Palais impact of random boundary conditions. Universitaet Potsdam [email protected] Jinqiao Duan Illinois Institute of Technology [email protected] CP40 Chaotic Saddles and The Kuramoto-Sivashinsky CP41 Equation Boundary Conditions For A Substrate Binding To This work presents a methodology to study the role played An Enzyme by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. We seek to construct boundary conditions for the diffusion Our methodology is applied to the Kuramoto-Sivashinsky equation in one dimension corresponding to absorption at equation, a reaction-diffusion partial differential equation. an endpoint with the constraint that only one diffuser can occupy the absorbed site at a time. Consequently, we con- We describes a novel technique tah uses the stable mani- N fold of a chaotic saddle to characterize the homoclinic tan- sider a set of diffusers which are independent except for gency responsible for an interior crisis, a chaotic transition the constraint at the absorbing boundary. This problem that resultis in the elargement of a chaotic attractor. The potentially has applications to many systems on a molec- numerical techniques explained here are important to im- ular scale, including the problem of a substrate binding to prove the understanding of the connections between low- an enzyme. dimensional chaotic systems and spatiotemporal systems Mark F. Schumaker Department of Mathematics DS05 Abstracts 95

Washington State University nology are related. Moreover, we demonstrate that this [email protected] graph is ’scale-free’ with a high clustering coefficient, showing some kind of preferential attachment. In addition, the presence of Universities is important in the transfer CP42 of knowledge, despite they represent only a small fraction Molecular Vibrations and the Rotating Eckart of all participants. Frame Miguel A. Sanjuan By changing its shape while conserving angular momen- Nonlinear Dynamics and Chaos Group tum, a polyatomic molecule can return to its initial shape Universidad Rey Juan Carlos with a different orientation (as a “falling cat” or a diver can [email protected] do). Classical molecular dynamics is described by a Hamil- tonian dynamical system. Using geometric mechanics, the Juan Almendral net angle of overall rotation is explicitly described in terms Nonlinear Dynamics and Chaos Group of “internal” coordinates, including Jacobi coordinates, in- Universidad Rey Juan Carlos, Madrid, Spain teratomic distances, and generalized Eckart coordinates. [email protected] The net angle of overall rotation is the sum of a dynamic phase plus a geometric phase. The latter is also described Joao Gama in terms of a gauge potential or, when available, molecular Department of Physics rotational constants. Universidade de Aveiro, Aveiro, Portugal Florence J. Lin joaogo@fis.ua.pt University of Southern California [email protected] Luis Lopez Laboratory of Distributed Algorithmics Universidad Rey Juan Carlos, Madrid, Spain CP42 [email protected] Symmetry and Stability in Hamiltonian Systems Jose Mendes Given a periodic solution to a Hamiltonian system with Department of Physics special forward or time-reversal symmetry, useful reduc- Universidade de Aveiro, Aveiro, Portugal tions exist which greatly simplify the associated linear sys- jfmendes@fis.ua.pt tem about the periodic orbit. This has nice applications when computing the Floquet multipliers to determine linear stability. We demonstrate these techniques on the fa- CP43 mous figure-eight orbit of the three-body problem, reducing A Local Method for Detecting Community Struc- stability calculations down to a 2 × 2 matrix whose entries ture in Networks. depend on approximating the linearized system over the first twelfth of the orbit. Other applications to the n-body We propose a novel method of community detection that is problem will be presented. both computationally less complex than many competing techniques, O(n), while possessing more physical signifi- Gareth E. Roberts cance to a member of, for example, a social network. In Dept. of Mathematics and C.S. addition, this method is truly local: a community can be College of the Holy Cross detected within a network without requiring knowledge of [email protected] the entire network, by a directed crawler. Several networks, including the famous Zachary Karate Club, are analyzed. CP42 Erik Bollt Motion of Unstable N-Ring Vortex Points on a Clarkson University Sphere with a Background Flow [email protected]

We consider the polygonal ring configuration of N identi- James P. Bagrow cal vortex points, say N-ring, on a sphere in the presense Clarkson University of a background flow. Starting with the linear stability Department of Physics analysis, we investigate unstable and recurrent motions of [email protected] the perturbed N-ring from the viewpoint of the theory of dynamical system using a projection method. CP43 Takashi Sakajo Hokkaido University Statistics of Cycles: How Loopy Is Your Network? Department of mathematics We study the distribution of cycles of length h in large net- [email protected] works and find it to be an excellent ergodic estimator, even in the extreme inhomogeneous case of scale-free networks. CP43 The distribution is sharply peaked around a characteristic cycle length h∗∼N α. We present a Monte-Carlo sam- A Complex Network of Scientific Collaborations of pling algorithm for approximately locating h∗. We show the V European Framework Program that for small random scale-free networks of degree expo- λ α / − λ α A collaboration network among companies and European nent , =1(1 ), and grows as the net becomes Universities of the V Framework Program is thoroughly an- larger. alyzed. When this network is represented by a graph, we Erik Bollt, Joseph Kirk, Daniel ben-Avraham, obtain valuable information about how science and tech- 96 DS05 Abstracts

Hernan D. Rozenfeld neous as well as spatially inhomogeneous dynamical noise Clarkson University - ubiquituous in real systems - is investigated.We find that [email protected], [email protected], the collapse process is delayed in the presence of small- [email protected], [email protected] amplitude inhomogeneous dynamical noise, but advanced by spatially homogeneous noise and spatially inhomogeneous large-amplitude noise. CP44 Pattern Selection for Impulsively-Forced Faraday Sumire Kobayashi Waves Dartmouth College Department of Physics We consider parametrically-excited standing waves on a [email protected] fluid subjected to a periodic sequence of delta-function im- pulses, an idealization of the Faraday system for which Renate A. Wackerbauer waves are excited by sinusoidal modulation of gravity. Us- University of Alaska Fairbanks ing the Zhang-Vinals model of weakly-viscous Faraday Department of Physics waves, we construct an explicit stroboscopic map that de- ff[email protected] termines the weakly nonlinear evolution of a class of two- dimensional wave patterns. We compare our pattern selection results with those obtained numerically for sinusoidal CP45 and multi-frequency forcing. Metastable States and Modelling of Persistant Neural Activity Anne J. Catlla Northwestern University Starting from the Kolmogorov equation, we describe dy- [email protected] namics of the mean and the variance of neural activity. Bi- furcation analysis shows that a metastable state (the mean Mary C. Silber is bounded and the variance grows to infinity) exists in the Northwestern University system. Basing on this study, we simulate a neural net- Dept. of Engineering Sciences and Applied Mathematics work of integrate-and-fire elements with noise and apply [email protected] the result of simulations to describe the experimental data on persistent activity in the brain. CP44 Roman M. Borisyuk Analysis of the Nonlinear Behavior of the Forced University of Plymouth Response of Combustion Systems Centre for Theoretical and Computational Neuroscience [email protected] We present results from analysis of data from two combustion experiments conducted in which harmonic modulation of fuel flow was employed as a means to improve CP45 combustion stability. We investigated the observed non- N-Bump Solutions of Amari-Type Equation linear interaction between the driven oscillations and the natural acoustic modes of the combustion chamber. The We explore a partial integro-differential equation (Amari, dynamic properties of the forced oscillations were analyzed Troy et.al.) that models pattern formation in neuronal networks: with time delay embedding techniques. For the low pressure combustion experiment, fluctuations of the attractor ∞ ∂u(x, t) and a boundary crisis occurred near instability. For the −u x, t ω x−y f u y, t dy s x, t h. ∂t = ( )+ ( ) ( ( )) + ( )+ high pressure experiment, various regimes of quasiperiod- −∞ icity, multistability, various modes of synchronization, and chaotic behavior were observed as the forcing frequency The main objective has been to establish the existence and was varied. stability of N-bump stationary solutions. Our focus has been to extend existing results for the equal width bump Nikolai F. Rulkov case, establish stability of those solutions, and characterize Information Systems Labs a class of mexican-hat coupling functions that allow N- [email protected] bump solutions. Fernanda Botelho, James Jamison, Angie Murdock Michael L. Larsen University of Memphis Information Systems Labs. [email protected], [email protected], jmur- [email protected] [email protected]

CP44 CP45 When Noise Inﬂuences the Collapse of Spatiotem- Dynamics of Fluctuation-Driven Neuronal Net- poral Chaos works

Wave-induced spatiotemporal chaos in the Gray-Scott Temporal fluctuations in the synaptic coupling between model, an excitable medium based on cubic autocataly- neurons naturally arise in neuronal networks and can sis, is transient with an exponential increase of the aver- have a significant effect on the dynamics of the network. age transient time with medium size. The collapse of this These fluctuations are especially important when the mean diffusion-sustained spatiotemporal chaos is initiated by in- synaptic drive of individual neurons are not sufficient to trinsic statistical spatial correlations that force the medium evoke spiking response. To highlight the importance of into the asymptotically stable steady state. The robustness these fluctuations, we contrast the dynamics in these net- of the collapse process in the presence of spatially homoge- works with mean-driven networks where the mean synap- DS05 Abstracts 97

tic input into each neuron is sufficiently strong to drive CP47 spiking. We find that mean-driven networks tend to be Unsteady Fluid Flow Separation by the Method of hysteretic and harder to control and that the activation of Averaging certain neuroreceptors (e.g., NMDA or AMPA) can determine the nature of the network dynamics. We use the method of averaging to improve recent separation criteria for two-dimensional unsteady fluid flows with Louis Tao no-slip boundaries. Our results apply to general compress- Department of Mathematical Sciences ible flows that admit a well-defined asymptotic average. New Jersey Institute of Technology Such flows include periodic and quasiperiodic flows, as well [email protected] as aperiodic flows with a mean component. As an example, we predict and verify unsteady separation location and angle in variants of an oscillating separation bubble model. CP46 Application of General Models in Model Aggrega- George Haller tion Massachusetts Institute of Technology Department of Mechanical Engineering The modeling of complex systems often leads to models [email protected] with high numbers of state variables and parameters. Such models are difficult to study. Moreover, the high number of Mustafa Sabri Kilic parameters means that many experimental measurements Department of Mathematics are necessary for tuning and validation. It is therefore of- Massachusetts Institute of Technology ten desired to reduce the complexity of a given model. We [email protected] present a new approach for model aggregation which is based on the application of general models. In contrast to others this approach conserves the local bifurcation struc- Anatoly Neishtadt ture of the system. Space Research Institu Russian Academy of Sciences Ulrike Feudel [email protected] ICBM Theoretical Physics/Complex Systems Carl von Ossietzky Universitaet Oldenburg [email protected] CP47 Experimental Investigations of a Dynamical Sys- Thilo Gross tems Approach to Unsteady Separation Fachbreich Physik We present the results of a study concerning the applica- Universität Potsdam tion of a new approach to unsteady flow separation in an [email protected] experimental setting. The flow geometry used is the ‘rotor- oscillator’ flow, which allows for investigation of separation Dirk Stiefs under carefully-controlled unsteady flow conditions. The ICBM, Theoretical Physics/Complex Systems location and geometry of the separating material spike are Carl von Ossietzky University Oldenburg, Germany reported for qualitatively different unsteady flows, and the [email protected] results related to recent theoretical work concerning fixed and moving separation. CP46 George Haller Reduced Atmospheric Models: Proper Basis Func- Massachusetts Institute of Technology tions, Dimensionality, Stochastic Modeling of Fast- Department of Mechanical Engineering Evolving Modes [email protected]

The construction of reduced atmospheric models, that is, Thomas Peacock, Raul Coral models that explicitly deal only with a limited number of MIT essential degrees of freedom while keeping as much realism [email protected], [email protected] as possible has attracted some attention in recent years. In the present paper, nonlinear reduced models of large-scale atmospheric dynamics are derived using a quasigeostrophic CP47 three-level spectral model with realistic variability as dy- Kinematic Theory of Unsteady Separation in Three namical framework. The study focuses on three issues: (i) Dimensional Flows finding appropriate basis functions for efficiently spanning the dynamics and a comparison between different choices of Exact quantitative characterization of separation in three basis functions; (ii) the minimal dimension of the reduced dimensional unsteady flows over no-slip boundaries has model necessary to faithfully simulate certain aspects of the been an outstanding problem. We attribute separation long-term behavior of the full spectral model; (iii) the ques- to the existence of non-hyperbolic invariant manifolds at- tion of whether the influence of unresolved fast-evolving tached to the boundary. With this viewpoint we derive nec- modes onto the resolved slowly-evolving large-scale modes essary and sufficient conditions for existence of these mani- can be modeled by stochastic terms. folds and their shape using: Normally hyperbolic invariant manifolds, Distinguished asymptotic behavior of material Frank Kwasniok surfaces/lines, Averaging and Finite Time Invariant Man- University of Oldenburg, Germany ifolds. We verify these conditions on different examples. [email protected] George Haller Massachusetts Institute of Technology Department of Mechanical Engineering 98 DS05 Abstracts

[email protected] Based on these results we present a set oriented approach for the approximation of invariant manifolds and illustrate Amit Surana, Olivier Grunberg the methods by several examples. Department of Mechanical Engineering Massachusetts Institute of Technology Michael Dellnitz, Kathrin Padberg [email protected], [email protected] University of Paderborn, Germany [email protected], [email protected]

CP48 CP49 Symmetry of Attractors and Frobenius-Perron Op- erator Pump Coupled Lasers with Delay

The application of group representation theory for ana- We considered two lasers coupled by modifying the pump lyzing the Frobenius-Perron (F-P) operator of equivariant signal of one laser by the light-intensity fluctuations of the maps (both continuous maps and homeomorphisms) is dis- other. The optical-to-electronic feedback can lead to a cussed. We consider applications to admissible symmetry significant time delay in the coupling signal. We inves- types of attractors, symmetry increasing bifurcations of at- tigate how the coupling strength and delay time affect the tractors (via collisions of attractors) as well as – from the synchronization of the two lasers and observe a surprising computational point of view – to the efficient discretiza- weak-coupling resonance effect. We compare our results to tion of the F-P operator in symmetry adapted co-ordinates, an experimental system of two cross-coupled semiconduc- computations of invariant measures, and to the detection tor lasers. of bifurcations of attractors. Thomas W. Carr Mirko Hessel Southern Methodist University Institute for Mathematics Department of Mathematics University of Paderborn [email protected] [email protected] Ira B. Schwartz Prashant G. Mehta Naval Research Laboratory United Technologies Research Center Nonlinear Dynamical Sysytems Section [email protected] [email protected]

Michael Dellnitz CP49 Institute for Mathematics Delay Differential Equations Modeling Lasers University of Paderborn [email protected] Lasers subject to optical or opto-electronic feedback exhibit pulsating outputs that are described in terms of two or three delay differential equations. Bifurcations to peri- CP48 odic and quasi-periodic oscillations, as well as the isolated Accurate Computation of Invariant Densities branching of steady and periodic solutions can be determined analytically. A series of basic problems for the sim- Invariant densities, when they exist, describe the statistical ple and multiple Hopf bifurcations are reviewed and their behaviour of a dynamical system. Ulam’s method (involv- relevance for other areas of science and engineering is dis- ing a discretization of a transfer operator) can be used to cussed. approximate such densities numerically, but convergence can be very slow when the density has complicated struc- Thomas Erneux ture (for example, in the logistic family of one-dimensional Université Libre de Bruxelles maps). In this talk I’ll present a strategy for automatic par- Optique Nonlinéaire Théorique tition selection in Ulam’s method which accelerates conver- [email protected] gence and yields surprisingly sharp images. The behaviour of the algorithm also suggests a phenomenological test for chaos. CP49 Dynamics of Fiber Laser Arrays Rua Murray Department of Mathematics Recent experiments show that fiber lasers can be synchro- University of Waikato nized simply by coupling them through an optical waveg- [email protected] uide near the output end. These results can lead to dramatic improvements in technology, yet how these lasers synchronize is not understood. We propose a theoretical CP48 model based on an iterated map with a particular sym- A Set Oriented Approach for the Approximation of metry. Our model captures key qualitative features seen Invariant Manifolds Using Finite-Time Lyapunov in the experiments and provides an insight into underlying Exponents dynamics. Invariant manifolds determine the geometric skeleton of a Kurt Wiesenfeld dynamical system and play a crucial role in transport and Georgia Tech mixing processes. In nonautonomous systems, often sta- Department of Physics tistical methods such as finite-time Lyapunov exponents (FTLE) are used to pinpoint the geometrical structures Slaven Peles of interest, exploiting that the stable manifolds of hyper- Georgia Institute of Technology bolic objects are typically local maximizers of the FTLE. DS05 Abstracts 99

[email protected] [email protected]

Jeﬀrey Rogers Mike Friswell, Nick Lieven HRL Laboratories LLC Aerospace Engineering jeﬀ@hrl.com University of Bristol [email protected], [email protected] CP50 Localized Elastic Instabilities and WKB-type CP51 methods Parasitic Solutions of a Discrete Rod Model

In this this work we show how localised responses arise in Analysis of a discrete model of the unshearable and inex- the context of non-homogeneous linear bifurcation equa- tensible Kirchoff-rod, consisting of straight rods and elas- tions for several structural models; The WKB method with tic joints will be presented. The discretization makes the turning points is employed to locate localized instability rod shape non-unique, and causes parasitic solutions dur- patterns, and then numerical simulations are performed ing the computation of the equilibrium branches of twisted to establish the correctness of our asymptotic analysis. In rings. Those parasitic solutions do not satisfy the closing contrast to similar works on structural localization, the ap- conditions. We will present an efficient way of labeling the proach we have taken seems to be more natural in dealing joints, what makes the rod shape unique, and allow us the with localization, as it is based on the concept of turn- computation of equilibrium branches. ing points and which in solid mechanics have a natural interpretation as points which are higly stressed. This cor- Robert Nemeth responds to the physical intuition that localization is di- Department of Theoretical and Applied Mechanics rectly related to those parts of the structure experiencing Cornell University the most severe deformation. [email protected] Ciprian D. Coman University of Glasgow, Department of Mathematics CP51 [email protected] Stability of a Whirling Conducting Rod in the Presence of a Magnetic Field: a Model for a Space Tether CP50 On Control As a Facet of Mechanics We study whirling instabilities of a conducting rod whirling in a magnetic field by performing a (largely numerical) con- The topic of control may be considered naturally in the tinuation and bifurcation analysis. Attention is also paid context of the physical theories of the potential and dis- to exact finite-length helical solutions for which exact re- sipation. In this setting, the control design process may sults can be obtained. The results are relevant for the be thought of as the compilation of potential reconstruc- Short Electrodynamic Tether (SET) prototype of the Eu- tion and dissipation remodeling. In fact, so far as sim- ropean Space Agency, which is designed to use the earth’s ple mechanical systems are concerned, the commonly used magnetic field, rather than chemical fuel, for thrust and techniques are precisely of this character. In this talk this drag. philosophy and its extensions are addressed. Gert van der Heijden Gregory P. Hicks Centre for Nonlinear Dynamics NRC/AFRL-VSSV University College London [email protected] [email protected]

Juan Valverde CP51 Department of Mechanical and Materials Engineering Bifurcations and Transient Response of a Dynamic University of Seville Balancer for Rotating Machines [email protected] We present a detailed analysis of the bifurcations and transient response of a dynamic balancer for rotating machines. CP52 Continuation techniques are employed to compute curves Numerical Study of Stochastic Reaction-Diffusion of equilibria and their bifurcations as parameters are var- Equations Subject to Internal Noise ied. A stable balanced state is identified for a broad range of physical parameters. However, complex unbalanced We address the problem of implementing numerical al- states including limit-cycles and chaotic motion are also gorithms to study models of reaction-diffusion problems found. Transient response and sensitivity of the balanced subject to internal noise. The schemes proposed conserve state under perturbation are analysed using pseudospectra the nonnegativity of the solutions and allow us to inves- techniques. tigate the effect of small perturbations in the propagation of fronts in reaction-diffusion problems. We also study the Alan Champneys development of instabilities due to the discretenes of par- University of Bristol ticles and the statistical properties of front interfaces in Dept. of Engineering Maths higher dimensions. [email protected] Esteban Moro Kirk Green Departamento de Matemáticas University of Bristol Universidad Carlos III de Madrid Bristol Laboratory for Advanced Dynamics Engineering [email protected] 100 DS05 Abstracts

CP52 pig action potential shape can produce large electrotonic Determination of Barriers in Bistable Stochastic effects that suppress alternans despite slope¿1 .As in ex- Dynamical Systems A Graph Theoretic Approach periments we obtain fibrillation by an extra stimulus but not by fast pacing. Understanding global stochastic dynamics depends on determination of pseudo-barriers with transport only due to Richard Gray stochastic basin hopping. This is particularly true for Department of Biomedical Engineering multi-stable systems, the key being to find closest ap- University of Alabama at Birmingham proaches between stable and unstable manifolds of the [email protected] basins. We take a graph theoretic approach, by approximately projecting the Frobenius-Perron operator onto basis Steven Evans functions, and then determining communities correspond- Beth Israel Medical Center ing to basins, We develop techniques and language from [email protected] recent advances in random graph theory. Harold M. Hastings Naratip Santitissadeekorn Hofstra University Clarkson University Dept. of Physics - CHPHB 102 [email protected] [email protected]

CP53 Elizabeth M. Cherry A Model for Human Ventricular Action Potentials: Hofstra University Insights into Brugada Syndrome Mechanisms Physics Dept [email protected] Using published experimental data, we have developed cell models of the Brugada syndrome, a genetic disease that Flavio Fenton can cause sudden cardiac death without underlying struc- Beth Israel Medical Center & Hofstra University tural defects. Because Brugada syndrome affects cells in New York, NY different regions of the ventricles differently, we also have [email protected] developed models of different cell types present in human ventricles as well as a spatial representation of their distribution through the ventricular wall. In this talk we present CP53 potential mechanisms by which Brugada syndrome may Calcium Alternans and Intracellular Calcium Cy- lead to lethal arrhythmias. cling in Cardiac Cells

Victor Perez-Garcia Calcium (Ca2+) alternans in cardiac cells are beat-to-beat Universidad de Castilla-La Mancha alternations in the amplitudes of the systolic Ca2+ tran- [email protected] sient. We will talk about the mechanisms by which intracellular Ca2+ cycling induces Ca2+ alternans. First, Steven J. Evans we set up a discrete time Ca2+ movement model with Beth Israel Medical Center two compartments: cytoplasm and sarcoplasmic reticulum New York, NY (SR). The model shows that reduced SR Ca2+ release can [email protected] increase the SR content. When the SR content reaches a threshold, CICR becomes unstable and Ca2+ alternans Flavio Fenton are exposed. A primary mechanism for this instability is Beth Israel Medical Center & Hofstra University the steep, nonlinear SR Ca2+ release function. Second, New York, NY we use a CICR model by Keizer and Smith (1998) to ex- [email protected] plore the relation between Ca2+ waves and Ca2+ alternans. We applied a local periodic Ca2+ stimulation with varying pacing interval (PI). When the total Ca2+ is high Elizabeth M. Cherry (Ca2+ overload), the local Ca2+ stimulation with the large Hofstra University PI induces Ca2+ waves (1:1 rhythm). As PI is reduced, Hempstead, NY propagated Ca2+ waves alternate with waves that fail to [email protected] propagate, giving a 2:2 rhythm. This results from slowed recovery from refractoriness of the SR Ca2+ channel at the Alfonso Bueno reduced PI. Department of Mathematics Universidad de Castilla-La Mancha James P. Keener [email protected] University of Utah [email protected]

CP53 Young-Seon Lee Fibrillation without Alternans in Porcine Ventri- Department of Mathematics cles: Theory and Numerical Simulations University of Utah [email protected] Despite simple restitution theory that predicts electrical alternans and ﬁbrillation for restitution curve slope¿1, porcine ventricles exhibit steep restitution without alter- CP54 nans. With numerical simulations using a mathematical The Impact of Behavior Changes on the Spread of model for cell electrophysiology and a 3D anatomical model of porcine ventricles, we explain how speciﬁc properties of DS05 Abstracts 101

a Smallpox Epidemic scopically simulating cell motility via membrane extensions and retractions-we have built a computational model of in We propose and analyze a mathematical model to consider vitro endothelial cell cultures. The model quantitatively the impact that behavior changes can have spread of small- reproduces in vitro vasculogenesis and subsequent in vitro pox. We assume that some individuals will lower their daily remodeling, and also reproduces aspects of sprouting angio- contact activity rates once an epidemic has started and genesis. Our models predict that cell polarization, through demonstrate that the spread of the disease is highly sen- the elongation of the endothelial cells, and the rate of dif- sitive to how rapidly the population reduces their contact fusion of a morphogen are key to correct spatiotemporal activity rates. We analyze the eﬀectiveness of mass vacci- in silico replication of vascular patterning and subsequent nation and quarantining the infected population through remodeling. contact tracing. Roeland Merks Sara Del Valle Indiana University Los Alamos National Laboratory Biocomplexity Institute Los Alamos National Laboratory [email protected] [email protected]

Carlos Castillo-Chavez CP55 Department of Mathematics and Statistics Stochastic Phase Resetting of Coupled Ensembles Arizona State University of Phase Oscillators Stimulated at Different Times. [email protected] We study the transient resynchronization dynamics of two Mac Hyman weakly interacting ensembles of noisy phase oscillators af- Los Alamos National Laboratory ter being phase reset at different times. Different coupling [email protected] strengths and distribution of eigenfrequencies are studied. We analyze the different types of mechanisms in which these ensembles reorganize to resynchronize. We demon- Herb Hethcote strate the impact of eigenfrequency distribution and noise University of Iowa amplitude on the resynchronization behavior and its con- [email protected] sequences for the design of therapeutic brain stimulation techniques. CP54 Jorge N. Brea Multiscale Stochastic Approaches for Spatiotempo- Center of Neurodynamics, University of Missouri Saint ral Disease Spread Louis. [email protected] Richard Jordan Health and Enironmental Security Group Peter A. Tass Dynaimcs Technology, Inc. Institute of Medicine (MEG) [email protected] Research Centre Juelich [email protected]

CP54 Frank E. Moss Cell-Centered Computational Modeling of De Center for Neurodynamics Novo and Sprouting Blood Vessel Growth Univ. of Missouri - St Louis [email protected] An essential question for developmental biology is how cells shapes and behaviors drive their assembly into tis- Kevin Dolan sues. Experimental studies have extensively analyzed en- Institute of Medicine, Research Center Juelich, Germany. dothelial cell behavior during blood vessel development. [email protected] We integrate such experimental data using a synthetic approach which constructs computer models mimicking endothelial cell behavior. In the first stages of blood ves- CP55 sel growth endothelial cells form a network-like structure, Adaptive Synchronization of Coupled Nonlinear called the primary capillary plexus. The plexus loses un- Oscillator Arrays Subject to External Forcing derused branches, expands and remodels by associating with additional cell types, transforming itself into a vas- We present results on the response of a system of coupled cular tree. Capillary plexi can form either through vascu- van der Pol oscillators to inhomogeneous periodic external logenesis, the assembly of disconnected endothelial cells, or forcing. We describe a scheme to make the Arnold Tongue through angiogenesis, the sprouting or subdivision of ex- adaptive with regard to variations in the external forcing isting blood vessels. Many different computational models amplitude and frequency by changing the array parame- have attempted to explain and describe both vasculogene- ters. The method is also applied to the same system with sis or angiogenesis. Since vasculogenesis and angiogenesis added external noise. This has application to signal pro- are closely related processes regulated by the same genetic cessing using nonlinear oscillator arrays. machinery, a plausible mechanism must explain both. Our strategy for elucidating how endothelial cells form vascular Lesley Ann Low networks and how new vessels sprout from existing ves- Information Systems Laboratories, Inc. sels, is to focus on individual cell behavior. We aim to [email protected] reconstruct the minimal set of cell behaviors that suffices for vascular patterning. Using the Cellular Potts model- a cellular-automaton-based Monte-Carlo technique meso- 102 DS05 Abstracts

CP55 develop in such a case. Catalytic reactions with alkali met- Phase Synchronization and Coherence Analysis: als as promoters represent systems of this type as has been Sensitivity and Specificity shown with Rh(110)/K where in the O2+H2 reaction K+O islands of macroscopic dimensions form. In the excitable When applying multivariate time series analysis techniques system NO+H2 on Rh(110) the presence of potassium leads to empirical data, conclusions about the underlying dy- to pulses transporting the alkali metal. The conditions for namics generating the data are of particular interest. reactive phase separation are not at all restrictive and it is Therefore, not only sensitive analysis techniques for investi- expected that such a mechanism applies to a broad class gating direct problems but also specific analysis techniques of systems. We also discuss new aspects in the mathe- are desired to avoid erroneous conclusions in inverse prob- matical modeling of such systems and present numerical lems. We illustrate the problem of missing specificity of simulations. phase synchronization and coherence analysis. We propose a methodology to increase specificity and demonstrate its Ronald Imbihl performance using two dynamic model systems. Institut für Physikalische Chemie und Elektrochemie University of Hannover, Germany Jens Timmer [email protected] University of Freiburg Department of Physics [email protected] MS1 Traveling Waves in Rapidly Varying Heterogenous Matthias Winterhalder Media Center for Data Analysis and Modelling University of Freiburg, Germany Traveling waves can become pinned or modulated in re- [email protected] action diffusion equations in unbounded cylindrical heterogenous media. The presence of small scale structures in the medium can cause failure of propagation (’pinning’) Björn Schelter so that rather than having travelling waves one has sta- Freiburg Center for Data Analysis and Modeling tionary, spatially localized solutions. We use a descrip- University of Freiburg tion of the elliptic equation on the unbounded domain as [email protected] an (ill-posed) evolution equation together with exponential averaging techniques to show that pinning occurs only in Juergen Kurths very small ranges of parameters. With similar techniques, Physics Department it is also possible to describe the periodic modulation of University of Potsdam travelling waves i.e., travelling wave solutions can be de- [email protected] scribed by a spatially homogenous equation and exponentially small remainders. This is partly joint work with H. Uecker, G. Schneider, and C.E. Wayne. MS1 Fluctuation-Induced Pattern Formation in a Sur- Karsten Matthies face Reaction Institut für Mathematik I Free University Berlin, Germany Nucleation, pulse formation and subsequent propagation [email protected] failure have been observed in the CO oxidation on Pt(110) at intermediate pressures (≈ 10−2 mbar). This phenomenon can be reproduced with a stochastic model in- MS1 cluding temperature effects, in which nucleation occurs via Corners in Front Propagation fluctuations, whereas the subsequent behaviour follows essentially the deterministic path. Conditions for the obser- We investigate corners of interfaces in anisotropic systems. vation of stochastic effects in the pattern formation during Starting from a stable planar front in a general reaction- CO oxidation are discussed. diffusion system, we show existence of almost planar interior and exterior corners. When the interface propagation Jens Starke is unstable in some directions, we show that small steps University of Heidelberg in the interface may persist. Our assumptions are based Institute of Applied Mathematics on physical properties of interfaces such as linear and non- [email protected] linear dispersion, rather than properties of the modeling equations such as variational or comparison principles. We Markus Eiswirth also comment on corners in pattern forming fronts and in- Fritz-Haber-Institute, Berlin terfaces between planar wave trains. [email protected] Arnd Scheel Christian Reichert University of Minnesota University of Heidelberg School of Mathematics [email protected] [email protected], [email protected]

MS1 MS2 Reactive Phase Separation on Catalytic Surfaces Multiscale Modeling and Numerical Methods for Sorting of Particles in Heterogeneous Devices Energetic interactions between the reacting species in a reaction-diﬀusion system may lead to a reactive phase sep- The problem of separation of large biomolecules such as aration. In a Turing-like instability periodic patterns will DNA and proteins is of high interest for biological research and biomedical application. In this presentation DS05 Abstracts 103

we discuss a multiscale modeling approach for the sepa- paper, we develop a novel formulation based on a locally- ration of particles in heterogeneous devices such as micro conservative variational multiscale method for the pressure arrays. These transport phenomena are modeled with mul- equation, and a front-tracking technique for the solution of tiscale advection-diffusion equations. This new modeling the system of transport equations along streamlines. The approach allows to investigate theoretically and numeri- proposed method captures the fine-scale heterogeneity on cally the macroscopic behaviour of the micro array and a coarse grid, and provides an accurate and efficient simu- its effects on the injected particles, The numerical simula- lation technique for miscible and immiscible flow in porous tion of this problem, present a number of challenges due to media. the multiple scales involved. We will also discuss numerical methods for the efficient solution of such type of multiscale Ruben Juanes equations. Department of Petroleum Engineering Stanford University Assyr Abdulle [email protected] University of Basel Department of Mathematics [email protected] MS3 Evolution of Pattern Complexity in the Cahn- Hilliard Theory of Phase Separation MS2 Multiscale Modeling of an Underground Nuclear Phase separation processes in compound materials can pro- Waste Site duce intriguing and complicated patterns. Yet, characterizing the geometry of these patterns quantitatively can be An underground nuclear waste repository consist of nuclear quite challenging. In this paper we use computational al- waste packages stored in excavated vaults, all connected by gebraic topology to obtain such a characterization. Our drifts and galleries or tunnels which are backfilled after the method is illustrated for the complex microstructures ob- packages storage. Usually, for safety reasons, the entire served during spinodal decomposition and early coarsening repository site is embedded in a low permeability layer. in both the deterministic Cahn-Hilliard theory, as well as On the one hand, the model of a repository site should in- in the stochastic Cahn-Hilliard-Cook model. While both clude the multi scale geometry, the large variations of the models produce microstructures that are qualitatively sim- geology and the coupling of the different phenomena. . But ilar to the ones observed experimentally, our topological on the other hand far field simulations for performance as- characterization points to significant differences. sessments cannot use such detailed models. In the case of a high number of leaking packages with a possible damaged Marcio Gameiro zone, we present the results of mathematical homogeniza- School of Mathematics tion and asymptotical methods, leading to scaled up but Georgia Institute of Technology accurate and physical macroscopic model. Both proofs of [email protected] accuracy and numerical simulations demonstrate the quality of the scaling up. Thomas Wanner George Mason University Alain Bourgeat Department of Mathematical Sciences Equipe MCS [email protected] Université Lyon1 [email protected] Konstantin Mischaikow Department of Mathematics MS2 Georgia Tech [email protected] Numerical Homogenization of Nonlinear Parabolic Differential Equations and its Applications MS3 The numerical homogenization methods presented in this talk are designed to compute homogenized solutions. I Topological Feature Extraction in Cubical Grids will describe numerical homogenization methods that we Cubical sets are a convenient geometric structure for rep- proposed recently and their relation to some other multi- resenting, among others, information contained in digital scale methods. Convergence of these methods for nonlinear images. Moreover, the graph of a map can also be repre- parabolic equations will be discussed. Numerical examples sented by a cubical set in the product space. Homology is and applications will be considered. an algebraic tool for extracting information about specific Yalchin Efendiev features of an image or a multidimensional structure. In Dept of Mathematics this talk we present three stages of that feature extraction: Texas A&M University Obtaining a cubical set, generating its chain complex and [email protected] computing its homology. Tomasz Kaczynski MS2 Universite de Sherbrooke Departement de mathematiques Variational Multiscale Methods and [email protected] Front-Tracking Techniques for Multiphase Flow in Porous Media MS3 Multiscale phenomena are ubiquitous to flow and transport in porous media. The problem can be expressed in Topological Characterization of Spatial-Temporal terms of a pressure equation (almost elliptic) and a system of transport equations (almost hyperbolic). In this 104 DS05 Abstracts

Chaos LL(G) Equations

It is well established both numerically and experimen- The Landau-Lifshitz and Landau-Lifshitz Gilbert equa- tally that nonlinear systems involving diffusion, chemo- tions are the basic evolution equations in micromagnetics, taxis, and/or convection mechanisms can generate com- a continuum model for magnetic behavior in ferromagnetic plicated time-dependent patterns. Since this phenomenon materials. In the setting where the magnetic behavior is is global in nature, obtaining a quantitative mathematical determined by the Dirichlet energy, these equations are a characterization that to some extent records or preserves hybrid Sch¨rodinger map flow and harmonic map heat flow the geometric structures of the complex patterns is dif- into the unit sphere S2. The question of singularity for- ficult. We illustrate a technique aimed at this problem. mation for finite energy data for these equations is open, More precisely, using algebraic topology, in particular ho- but the search is motivated by the fact that under a suit- mology, we can measure Lyapunov exponents that imply able transformation, these equations are reminiscent of the the existence of spatial-temporal chaos and suggest a ten- cubic nonlinear Schrödinger equation for which singular tative step towards the classification and/or identification solutions abound. Analytical attempts have met impasses of patterns within a particular system. but have yielded insight which underlies current numerical investigations (in collaboration with S, Bartels). William D. Kalies Florida Atlantic University Joy Ko Department of Mathematical Sciences Brown University [email protected] [email protected]

MS3 MS4 Computing Homology of Maps Schrödinger Maps and the Landau-Lifshitz-Maxwell Equation An algorithm computing homology of maps is needed in rigorous qualitative numerical analysis of dynamical sys- A special case of the geometric PDEs known as Schrödinger tems based on topological tools such as Conley Index, maps, the Landau-Lifshitz equation is a nonlinear PDE Fixed Point Index or Degree Theory. The typical rigorous that describes the magnetic moment, or spin, of a ferro- information one can extract from an implicitly given map magnetic material. Since the spin contributes a magnetic are upper estimates of images of sets in the map, usually field, we couple the above to Maxwell’s equations govern- given in the form of a multivalued representation. In the ing electromagnetic fields. Using geometric ideas from our lecture we review two algorithms for computing homology previous work on Schrödinger maps, along with energy es- of continuous maps based on multivalued representation timates for the Landau-Lifshitz-Maxwell system, we prove and cubical homology. The first algorithm utilizes a direct local existence. However, long-term behavior is an open construction of a chain selector of the multivalued represen- question. tation. The other algorithm is based on projections from the graph of the multivalued map onto the domain and the Helena McGahagan set of values. We also briefly mention a third approach, University of California at Santa Barbara based on some ideas of Cech homology. [email protected]

Marian Mrozek Jalal Shatah Jagiellonian University New York University Institute of Computer Science [email protected] [email protected]

MS4 MS4 Anti-ferromagnetic Chains and Schrodinger Maps Schroedinger Maps Near Harmonic Maps with Strong Potentials

The Schroedinger map equation is a basic model in ferro- In the Heisenberg model, the continuum limit of the evo- magnetism, as well as a geometric (and hence nonlinear) lution of the spin vectors on a ferromagnetic lattice is version of the linear Schroedinger equation. It is an open described by Landau-Lifshitz equation. While the anti- question whether ﬁnite energy solutions are global, or blow ferromagnetic chains are also described by the Heisenberg up in ﬁnite time. Here we consider equivariant Shroedinger model, we show rigorously that the continuum limit is the maps from 2+1-dimensional space-time into the 2-sphere, σ model, i.e. the wave map targeted on the unit sphere. with energy close to the energy of an equivariant harmonic This result is proved in a general setting of Schroding maps map. We show that solutions stay close to harmonic maps with strong potentials. before blow up (if any), and that they blow up if and only if the length scale of the nearest harmonic map goes to zero. Chongchun Zeng This is joint work with Kyungkeun Kang and Tai-Peng University of Virginia Tsai at UBC. Department of Mathematics [email protected] Stephen Gustafson, Kyungkeun Kang, Tai-Peng Tsai Department of Mathematics Jalal Shatah University of British Columbia New York University [email protected], [email protected], [email protected] [email protected]

MS4 Numerical Study of Singular Solutions to the DS05 Abstracts 105

MS5 MS5 Self-Organization in Microbial Colonies A Nonlocal Continuum Model for Localized Bio- logical Aggregations In nature, microorganisms must often cope with hostile environmental conditions. To do so they have developed We construct and study a nonlinear, nonlocal continuum sophisticated cooperative behavior and communication ca- model for the movement of biological populations whose pabilities. Utilizing these capabilities, the colonies develop members experience long-range social attraction and short- complex spatio-temporal patterns. We present a wealth range dispersal. Using phase plane analysis, energy meth- of beautiful patterns formed during colony development of ods, and numerical computations, we study the dynamics, various microorganisms. Invoking ideas from pattern for- pattern selection, and steady states. The model displays mation innon-living systems and using “generic” modeling coarsening behavior, and has localized, clump-like steady we are able to account for the salient features of the obser- solutions with key characteristics observed in natural bi- vations. ological aggregations, namely sharp boundaries and constant internal population density. Eshel Ben-Jacob Tel-Aviv University Mark Lewis [email protected] University of Alberta, Canada [email protected] Herbert Levine Univ. of Cal. at San Diego Chad M. Topaz Department of Physics UCLA [email protected] Department of Mathematics [email protected]

MS5 Andrea Bertozzi Motion of Biological Organisms as Interacting Self- UCLA Department of Mathematics propelled Particles [email protected] Organisms, from cells to humans, move and interact. To understand the emerging order, models consisting of self- MS6 propelled particles can be studied. Early results showed The Effect of Feedback on the Pacemaker Unit of that local alignment can order the motion of the whole a CPG group, even in the presence of noise. Slightly more intel- ligent particles can describe regularities in pedestrian mo- In certain CPGs, pacemaker units have inhibitory synapses tion, such as spontaneous lane formation or panic. Motion onto other neurons which display short-term synaptic plas- and interaction between tissue cells is required for the for- ticity. In this talk, we discuss the ramifications of synap- mation of anatomical structures, as seen in the example of tic feedback onto the pacemaker. Feedback changes the the vascular network. frequency of the pacemaker, which, in turn, affects the strength of its synapses. Using geometric singular pertur- Andras Czirok, Illes Farkas bation theory, we analyze how feedback inhibition affects Department of Biological Physics the phase relationship between neurons in the pyloric net- Eotvos University work CPG. [email protected], fi[email protected] Amitabha K. Bose Tamas Vicsek New Jersey Inst of Technology Department of Biological Physics Department of Mathematical Sciences Eotvos University [email protected] [email protected] MS6 MS5 Stability and Variability of Central Pattern Gener- Modelling Daphnia Swarming ation

We propose a self-propelled particle model for the swarm- What is the target activity of homeostatic regulation of ing of Daphnia, which takes into account propulsion of the network performance? The pyloric rhythm of the lob- particles, mutual avoidance of close encounters and attrac- ster stomatogastric ganglion shows substantial animal-to- tion to a center. Various key parameters are identified in animal variability in frequency but the phase relationships order to arrive at a phase diagram for qualitatively differ- between different neurons within the circuit are relatively ent steady-state motions. We find that a vortex is formed tightly constrained. Individual networks with similar mean only in a finite range of propulsions, and analyze its tran- motor output can behave differently on a cycle-to-cycle ba- sitions to other states. Hydrodynamic interaction between sis, and model networks show that different combinations the particles can stabilize the vortex and change its velocity of intrinsic and synaptic conductances can yield similar ac- profile. tivity. Bruno Eckhardt Dirk Bucher Philipps Universität Marburg Brandeis University Fachbereich Physik Volen Center for Complex Systems [email protected] [email protected] 106 DS05 Abstracts

MS6 in a special case of two-dimensional stratified flow, the non- Regulation of Bursting Activity in a Neuronal linear steady streamfunction satisfies the linear Helmholtz Model equation. Surprisingly, it is only recently that very careful flow simulations for a series of two or three mountains Bursting activity of neurons is an oscillatory activity con- found that Long’s solutions could be unstable. A numeri- sisting of intervals of repetitive spiking separated by in- cal linear stability analysis, which uses an Arnoldi approach tervals of quiescence. Neurons which are capable of gen- for the large sparse finite-difference system, produces un- erating bursting activity endogenously play an important stable eigenmodes which compare well with the simulated role in Central Pattern Generators. Burst duration, in- instability in the hydrostatic parameter regime. A spectral terburst interval and spike frequency are crucial temporal analysis of the eigenmodes suggests that triad resonances characteristics of bursting activity and thus have to be reg- play a key role in the dynamics of these flows. ulated.Application of the bifurcation theory of dynamical systems suggests new mechanisms of how bursting activity Youngsuk Lee can be generated by neurons and its temporal characteris- Mathematics Department, Simon Fraser University tics can be regulated. The work is supported by NIH NS Burnaby, BC, Canada 43098. [email protected] Gennady S. Cymbalyuk Georgia State University MS7 Department of Physics and Astronomy Effects of Stratification on the Variability of the [email protected] Double-gyre Wind-driven Ocean Circulation

Andrey Shilnikov The double gyre wind driven ocean circulation exhibits a Georgia State University rich variety of behaviors, including variability on timescales Department of Mathematics and Statistics relevant to climate dynamics. While much is known about [email protected] the role forcing and dissipation have in determining the variability of the flow, the effects of stratification have been less thoroughly studied. In the two layer model, there are MS6 two stratification parameters, which can be interpreted as Electrically Coupling Distinct Neurons the depth and strength of the thermocline. Both of these parameters independently affect the variability of the flow; We developed a multi-compartment model of two intrinsi- increasing the strength or decreasing the depth of the ther- cally distinct, electrically coupled neurons, inspired by the mocline each lead to chaotic variability, but the routes to pacemaker group of the crustacean pyloric network. We chaos are distinct. use the model to illustrate several neuron and network behaviors that arise from electrically coupling very different Cavendish McKay oscillators. We explore conditions under which a small, Department of Mathematics intrinsically bursting neuron can drive a large, tonic spik- University of Wisconsin ing neuron to burst in-phase with it, and show that current [email protected] segregation significantly enhances their ability to burst synchronously. MS7 Eve Marder Floquet Instability & Triad Resonance in a Strati- Brandeis University fied Fluid [email protected] An important fluid mechanical property of the Earth’s atmosphere, is that its density is stably-stratified. Unlike Farzan Nadim the case of unstable stratification (light fluid under heavy) New Jersey Institute of Technology where convective motions can arise, a stably-stratified fluid [email protected] permits buoyancy oscillations known as gravity waves. These motions result from localized vertical disturbances Pascale Rabbah of heavier air upward (or lighter air downward) which are Rutgers University at Newark then subject to the restoring force of gravity. Such waves [email protected] are commonly experienced as in-fight turbulence when an aircraft flies through oscillatory fields of vertically moving Cristina Soto-Trevino air. In the absence of viscosity, the stratified fluid equations New Jersey Institute of Technology admit an exact, nonlinear plane wave solution. Although it Department of Mathematical Sciences was established in 1976 that these finite-amplitude waves [email protected] are parametrically unstable, there is much about the nonlinearity of this system which is poorly understood. A stability analysis of this wave reveals a simple connection be- MS7 tween the Floquet eigenvalues of the linearized problem Instability of Winds over Mountainous Terrain with the resonant traces of triad interaction theory.

In a density-stratiﬁed atmosphere, winds over mountains David J. Muraki generate waves due to oscillations of vertical motion and Department of Mathematics buoyancy. These wave disturbances are a signiﬁcant contri- Simon Fraser University bution to the variations of cloudiness and precipitation in [email protected] alpine regions. They are also an important loss mechanism in the energy and momentum budgets for the global climate. R. Long (1953) made the remarkable discovery that, DS05 Abstracts 107

MS8 early stages of growth of vertebrate embryos is responsible High Efficiency Mixing using the Shear Super- for determining the left-right axis, with the heart on the posiyion Micromixer left of the body, the liver on the right, and so on. The role of physics, in particular, of fluid dynamics, in the process is We present new experimental results showing fast and high one of the important questions that remain to be answered. efficiency mixing procedures for an active shear superposition micromixer. The mixing process is studied numer- Idan Tuval ically and experimentally using flow visualizations tech- Instituto Mediterraneo de Estudios Avanzados niques. The numerical simulations are performed for 3-D CSIC - Universidad de las Islas Baleares, Spain flows using Fluent. We quantify, numerically and exper- [email protected] imentally, the degree of mixing achived using the Mixing Variance Coefficient. We show that single channel mixing can be very good when the amplitude and frequency are MS9 chosen carefully. Almost Invariant Sets for Complex Systems Igor Mezic, Frederic Bottausci Over the last years set-oriented numerical methods have University of California, Santa Barbara been developed for the approximation of statistical char- [email protected], [email protected] acteristics of dynamical systems. These methods allow to approximate not just invariant measures but also almost invariant sets. These subsets define a macroscopic structure in state space for the underlying dynamical process. Here MS8 we discuss the most recent numerical approaches for identi- Chaotic Advection in Three-Dimensional Unsteady fying almost invariant sets. The algorithms are illustrated Incompressible Laminar Flow by several challenging examples such as the approximation of chemical conformations for molecules. We review results regarding an experimentally realizable three-dimensional time-dependent nonturbulent fluid flow Mirko Hessel-von Molo that displays the phenomenon of global diffusion of passive- Institute for Mathematics scalar particles at arbitrarily small values of the nonin- University of Paderborn tegrable perturbation. This type of chaotic advection, [email protected] termed resonance-induced dispersion, is generic for a large class of flows. Robert Preis Institute for Mathematics Julyan Cartwright [email protected] Laboratorio de Estudios Cristalograficos University of Granada Michael Dellnitz [email protected] Institute for Mathematics University of Paderborn MS8 [email protected] Designing chaotic microflows in spherical volumes MS9 Achieving thorough chaotic mixing inside small spherical volumes, such as liquid microdroplets suspended in a liquid Air Force Challenges in Networked Dynamical Sys- substrate, has proved to be quite challenging due to the in- tems herently high symmetry of the problem. This talk will provide an overview of the important results in this area and Sharon Heise describe some recent theoretical developments and exper- Air Force Office Of Scientific Research imental approaches to manipulating the microflow inside [email protected] liquid droplets.

Michael Schatz MS9 Center for Nonlinear Science & School of Physics Modeling, Analysis, and Design of Large Dynami- Georgia Institute of Technology cal Systems in Industry [email protected]

Roman Grigoriev Clas Jacobson Georgia Institute of Technology United Technologies Center for Nonlinear Science & School of Physics Research Center [email protected] [email protected]

Vivek Sharma MS9 Georgia Institute of Technology [email protected] Representation of Dynamical Systems: Application to Building Systems

MS8 The Frobenius-Perron (F-P) operator construction for graph partitioning will be discussed, with applications to Right Hand - Left Hand model reduction and dynamic analysis of the Building sys- Experimental work in developmental biology has recently tems models. We describe mathematical problems associ- shown in mice that fluid flow driven by rotating cilia in the ated with meeting objectives for efficiency, security, people egress, and network control solutions in integrated building 108 DS05 Abstracts

systems design. We model these problems as convection- dence Intervals diffusion equations on a (building) graph. While the original graph is complex, we use the F-P operator construction along with coarse spatio-temporal scales for any control Matthew Kennel objective to obtain effective (low-dimensional) representa- Institute For Nonlinear Science tion of dynamics on the graph. Additionally, application of University of California, San Diego, group representation theory to take advantage of symme- [email protected] tries in the problem, and hierarchical model development to obtain hierarchical control solutions will be discussed. MS10 We illustrate the approach by means of examples drawn from efficiency and security problems in buildings. Symbolic Dynamics of Coupled Map Lattices Prashant G. Mehta We show that coupled lattices of unimodal maps admit a United Technologies Research Center simple generating partition which can be analyzed using [email protected] the methods of symbolic dynamics. These methods have been very successful at rigorously defining and classifying chaotic motion in one and two dimensional maps. Cou- MS10 pled Map Lattices (CMLs) are the first known example of Symbolic Dynamics in Stochastic Dynamical Sys- a high-dimensional system that can be treated this way. tems: A Way to Define Their Markov Partitions Much like the one-dimensional case, we find a generalized Gray ordering, a set of maximal sequences, and rules for Symbolic dynamics are an important method for topolog- determining the admissibility of symbol sequences. A spe- ical description of deterministic dynamical systems, but cific case utilizing coupled tent maps is presented. they rely on well choosing a generating partition. In particular, a Markov partition allows for a sophic shift of fi- Shawn Pethel nite type. Despite the ubiquity of at least random per- U.S. Army RDECOM turbations, there is much less work to allow symbolic dy- [email protected] namical description of a stochastic dynamical systems. We generalize the definitions of Markov partitions, through a MS11 Galerkins method approach, to allow for stochastic systems. Symmetry in Circulant Multiple Agent Systems Erik Bollt We study the dynamics of a collection of point-mass agents evolving under linear pursuit control laws. It is shown that Clarkson University C [email protected] a sufficient condition for the planar symmetry groups m and Dm to be invariant under the agent dynamics is that the closed-loop dynamics be circulant. Necessary condi- Karol Zyckowski tions on the agent dynamics are also obtained. Next we Instytut Fizyki im. Smoluchowskiego examine the stability of the finite symmetry groups in the Uniwersytet Jagiello{ } plane under circulant dynamics, extending prior work by ’n ski, Poland Richardson [T. Richardson. Stable polygons of cyclic pur- [email protected] suit. Annals of Mathematics and Artificial Intelligence. vol. 31, pp. 147-172, 2001] and Bruckstein et. al. [A. MS10 M. Bruckstein, G. Sapiro, and D. Shaked. Evolutions of planar polygons, International Journal of Pattern Recogni- Unstable Recurrent Patterns in tion and Artificial Intelligence, vol. 9, no. 6, pp. 991-1014, Kuramoto-Sivashinsky Dynamics 1995] on the stability of the cyclic group Cm for agents in We test the “recurrent patterns” description of turbulence “cyclic pursuit”. Some other aspects such as determination on a Kuramoto-Sivashinsky model, deploying a new vari- of collisions are explored. ational method that yields a large number of numerical Joshua Marshall unstable spatiotemporally periodic solutions. For a small University of Toronto but turbulent system, the attracting set appears surpris- [email protected] ingly thin. Its backbone are several Smale horseshoe repellers, well approximated by local return maps, each with good symbolic dynamics. Global dynamics appears decom- Mireille E. Broucke posable into chaotic dynamics within such local repellers, University of Toronto interspersed by infrequent transitions between them. Department of Electrical Comp. Engineering [email protected] Predrag Cvitanovic Department of Physics and Center for Nonlinear Science Georgia Institute of Technology MS11 [email protected] Analysis and Design Tools for Distributed Motion Coordination

Yuehng Lan This talk presents novel mathematical tools useful to study Physics, Georgia Tech the motion of mobile autonomous agents. First, motion [email protected] coordination tasks are encoded into aggregate cost functions from Geometric Optimization. Second, the limited MS10 communication between agents is modeled via proximity graphs from Computational Geometry. Third, algorithms Estimating Entropy Rates with Bayesian Conﬁ- correctness is established via LaSalle invariance principles for non-deterministic systems in discrete and continuous DS05 Abstracts 109

time. Finally, these tools are applied in motion coordina- Yale University tion examples such as deployment, rendezvous, and ﬂock- [email protected] ing. Brian Anderson, Ming Cao Francesco Bullo Yale University Mechanical & Environmental Engineering tba, [email protected] University of California at Santa Barbara [email protected] MS12 Sonia Martinez Manipulation of Self Aggregation Patterns and University of California Santa Barbara Waves in a Reaction-Diﬀusion-System by Optimal [email protected] Boundary Control Strategies

Jorge Cortes We show numerically how spatiotemporal behaviour like University of California Santa Cruz pattern formation in a two component nonlinear reaction [email protected] diffusion model of bacterial chemotaxis, described by a system of two coupled quasilinear PDEs, can be externally controlled. We formulate the control goal as an objective MS11 functional and apply numerical optimization for the solu- From Nonsmooth Analysis and Geometric Opti- tion of the resulting control problem. Due to model insuffi- mization to Distributed Coordination Algorithms ciencies and measurement noise feedback control strategies can be applied to feed model answers in real-time back In this talk, we discuss dynamical systems that solve disk- to the optimization process. Nonlinear model predictive covering, sphere-packing and maximum-visibility prob- control (NMPC) for partial differential equations (PDE) lems. We present aggregate cost functions from geomet- is deemed to be crucial for feedback control of distributed ric optimization that encode various notions of network parameter systems. We demonstrate the application of an deployment. We design and analyze a collection of dis- efficient nonlinear model predictive control (NMPC) algo- tributed control laws that are related to nonsmooth gradi- rithm. ent systems. The resulting dynamical systems promise to be of use in robotic coordination problems for networked Ulrich Brandt-Pollmann robots; in this setting, the distributed control laws cor- Interdisciplinary Center for Scientific respond to remarkably simple local interactions between Computing the robots. We formally discuss to what extent the pro- [email protected] posed coordination algorithms are spatially distributed. The technical approach relies on concepts from computa- Dirk Lebiedz tional geometry, nonsmooth analysis, and the dynamical Interdisciplinary Center for Scientific Computing system approach to algorithms. University of Heidelberg, Germany [email protected] Francesco Bullo Mechanical & Environmental Engineering University of California at Santa Barbara MS12 [email protected] Localized Feedback Control of Pattern Forming Systems: Successes and Failures Jorge Cortes Department of Applied Mathematics and Statistics This talk will concentrate on simple pattern form- University of California, Santa Cruz ing systems described by one-dimensional partial differ- [email protected] ential equations such as Ginzburg-Landau, Kuramoto- Sivashinsky or Swift-Hohenberg equation. I will show how pattern formation in these systems can be suppressed (or MS11 another pattern imposed) using feedback applied at one or Asynchronous Coordination of Multi-Agent Sys- both boundaries. I will also show that such control breaks tems down when the size of the system increases due to exponentially growing transient amplification of noise. This paper is concerned with the coordination of a group of n>1 mobile autonomous agents which all move in Roman Grigoriev the plane with the same speed but with different head- Georgia Institute of Technology ings. Each agent changes its heading from time to time Center for Nonlinear Science & School of Physics to a new value equal to the average of its present heading [email protected] and the headings of its current “neighbors”. Although all agents use the same rule, individual headings are updated Andreas Handel asynchronously. By appealing to the concept of “analytic Emory University synchronization”, it is shown that under mild connectivity [email protected] assumptions of the underlying directed graph characterizing neighbor relationships, the local update rules under consideration can cause all agents to eventually move in the MS12 same direction despite the absence of centralized coordina- Feedback Control of Morphological Instabilities tion and despite the fact that each agent’s set of neighbors change with time as the system evolves. Feedback control of morphological instabilities of crystal- lization fronts in directional solidification is considered: (i) Stephen Morse global control of the long-wave monotonic instability in sys- Department of Electrical Engineering tems with small segregation coefficient and (ii) local con- 110 DS05 Abstracts

trol of the short-wave oscillatory instability in systems with Michael F. Schatz, Jennifer Rieser kinetic undercooling. In case (i) it is shown that global Center for Nonlinear Science and School of Physics feedback control can prevent the finite-time blow-up and Georgia Institute of Technology lead to the formation of localized structures. In case (ii) it [email protected], [email protected] is found that the local feedback control can suppress the instability but the control efficiency depends on the delay. MS13 Vadim Panfilov Stability of Pulses in Mixed Type Equations University of Nevada at Reno panfi[email protected] Christopher Jones Alexander Nepomnyashchy Department of Mathematics Technion UNC, Chapel Hill Israel Institute of Technology [email protected] [email protected] MS13 Alexander A. Golovin Nucleation of localised patterns in the 2D Swift- Department of Engineering Sciences and Applied Hohenberg equation Mathematics Northwestern University We study the nucleation of localised patterns in 2- [email protected] dimensions using a variety of rigorous, formal and numerical techniques. We show that in a generic pattern for- Tatiana Savin mation equation, namely the Swift-Hohenberg equation, Northwestern University localised patterns are formed by a heteroclinic connection [email protected] from the trivial state to the cellular pattern. We present a priori estimates on the attractor and direct variational Valeria Gubareva methods are used to show rigorous existence of small am- Department of Mathematics plitude pulses in 2-dimensions. Using asymptotics we find Technion - Israel Institute of Technology an estimate for the heteroclinic connection to form in the [email protected] case that the cellular pattern consists of squares. Finally, we use numerical methods to explore the bifurcation sequence of the nucleation of localised patterns. MS12 Controlling Synchronization in Oscillator Ensem- David Lloyd, Alan Champneys bles University of Bristol [email protected], [email protected] We consider a possibility to suppress synchronous collective dynamics in ensembles of globally coupled oscillators. We show that addition of a delayed feedback of the mean MS13 field allows us to stabilize the asynchronous state irrespec- A Mountain Pass and Gradient Flow for Stable So- tively on the particular way of the influence. Theory for lutions in Cylinder Buckling the Kuramoto model is presented. We also present numerical modelling of neural ensembles and discuss possible A classical problem in structural engineering is the predic- applications in neuroscience. tion of the load-carrying capacity of an axially compressed cylindrical shell. We obtain a single-dimple solution as a Arkady Pikovsky, Michael Rosenblum mountain-pass point which is unstable, in the sense that Department of Physics there are directions in state space in which the energy de- University of Potsdam, Germany creases. In one direction the dimple roughly shrinks and [email protected], mros(-)agnld.uni- disappears, and in the other direction it grows and multi- potsdam.de plies into a periodic array of dimples. This gives insight into scaling properties and prediction of failure loads.

MS12 Jiri Horak Optical Manipulation and Control of Micro-Scale Universit Fluid Flow ”at K ”oln We describe experiments using an all-optical approach to [email protected] controlling microscale fluid flow. Surface-tension gradients are induced in a fluid by the thermal absorption of Mark Peletier light. The gradients can drive flow on unstructured liquid Technische Universiteit Eindhoven or solid substrates; moreover the flow can be altered and [email protected] controlled by spatially and temporally varying the light intensity. Optical control permits the flow to be dynami- Gabriel J. Lord cally reprogrammed, suggesting an approach for construct- Heriot-Watt University ing fluidic devices analogous to microcomputing’s CPU. [email protected] Roman Grigoriev Georgia Institute of Technology MS13 Center for Nonlinear Science & School of Physics Numerical Evaluation of the Evans Function by [email protected] DS05 Abstracts 111

Magnus Integration MS14 Braid-Theoretic Methods for Parabolic PDEs We use Magnus methods to compute the Evans function for spectral problems as arise when determining the linear stability of travelling wave solutions to reaction-diffusion and Robert W. Ghrist related partial differential equations. In a typical applica- Department of Mathematics tion scenario, we need to repeatedly sample the solution University of Illinois, Urbana-Champaign to a system of linear non-autonomous ordinary differential [email protected] equations for different values of one or more parameters as we detect and locate the zeros of the Evans function in the right half of the complex plane. In this situation, MS14 a substantial portion of the computational effort—the nu- Classification of Strange Attractors in Three Di- merical evaluation of the iterated integrals which appear in mensions the Magnus series—can be performed independent of the parameters and hence needs to be done only once. More It is finally possible to classify low-dimensional strange at- importantly, for any given tolerance Magnus integrators tractors. There are four levels of structure in this classification: (1) basis sets of orbits; (2) branched manifolds; possess lower bounds on the step size which are uniform 3 across large regions of parameter space and which can be (3) bounding tori; and (4) embeddings into R . All four estimated a priori. We demonstrate, analytically as well levels involve links of knots in very powerful ways. We de- as through numerical experiment, that these features ren- scribe how singularities form the backbone of stretching der Magnus integrators extremely robust and, depending and squeezing processes that generate chaotic behavior. on the regime of interest, efficient in comparison with stan- We conclude with a brief description of all the covers of dard ODE solvers. Authors: NAIRO D. APARICIO, SI- a universal image dynamical system, the horseshoe. MON J.A. MALHAM and MARCEL OLIVER Robert Gilmore Simon Malham Physics Dept., Drexel Univ. Herriott-Watt University Philadelphia, PA Department of Mathematics [email protected] [email protected] MS14 MS13 Topological Analysis of Experimental Data : Knot Blowing-Up Exact Solutions of Long-Wave Unsta- Holders and Topological Entropies ble Thin Film Equations Knots and links formed by unstable periodic orbits in a Long-wave unstable thin film equations strange attractor carry signatures of the stretching and squeezing mechanisms that organize it. We review how this property has been harnessed to design a robust method for analyzing experimental chaotic data. We also report recent n m experiments in a nonstationary system where the knot type ht =(h hxxx)x − B(h hx)x of an orbit has been used as an unambiguous signature of chaos. Possible extensions to higher dimensions will also be discussed. are a fourth-order analogue of the the semilinear heat equa- Marc Lefranc tion. A ”reaction” term destabilizes a ”diffusion” term, al- PhLAM/Université Lille I, lowing for a competition between effects. This competition France admits a variety of steady states and temporal behaviors, [email protected] depending on whether the equation is subcritical, critical, or supercritical. In the critical case, Witelski, Bertozzi, and Bernoff propose a critical mass, Mc, and demonstrate MS14 that if the mass of the initial data is less than Mc then Analysis of Low Dimensional Dynamics: Algo- finite-time blow-up is impossible. Their computations and rithms Based on the Conley Index asymptotics case suggest that (generically) if the initial mass is larger than Mc the resulting solution demonstrates The Conley index theory allows one to conclude the exis- selfsimilar blowup focussed at points. Slepcev and I con- tence of a variety of invariant sets including fixed points, sider the critical case and construct exact solutions with periodic, heteroclinic and homoclinic orbits, and symbolic compact support that blow up in a selfsimilar manner. dynamics. However, to apply the theory requires the con- Their mass can be arbitrarily close to the critical mass struction of an isolating neighborhood and the computa- proposed by Witelski et al., proving the sharpness of the tion of the associated index. I will discuss algrorithms that critical mass. In addition, Slepcev has proven the linear perform these constructions and computations. stability/instability of such solutions. Konstantin Mischaikow Dejan Slepcev Department of Mathematics UCLA Georgia Tech Mathematics Department [email protected] [email protected]

Mary Pugh MS14 University of Toronto Perestroikas of Strange Attractors in Three- Department of Mathematics [email protected] 112 DS05 Abstracts

Dimensional Phase Space MS15 Semi-Strong Pulse Interaction Strange attractors can exhibit bifurcations just as periodic orbits in these attractors can exhibit bifurcations. We de- Pulse-pulse interactions play central roles in a variety of scribe two classes of large-scale bifurcations that strange pattern formation phenomena, including self-replication. attractors can undergo. For each we provide a mechanism. From the analytical point of view, pulse interactions can be These bifurcations are illustrated in a simple class of three- distinguished into three types: weak interactions, in which dimensional dynamical systems. the pulses are assumed to be suﬃciently far apart, the fully strong interactions, and the intermediate concept of semi- Tsvetelin D. Tsankov strong interactions, that has been introduced in the context Physics Department of singularly perturbed systems, in which only some com- Bryn Mawr College, Bryn Mawr, PA ponents of the pulse interact weakly. Recently, methods [email protected] based on constructing (approximate) invariant manifolds and/or renormalization techniques have been developed to MS15 rigorously study weak pulse interactions. In this talk these methods will be extended, so that they can be applied to Stability of a Stripe in the Two-Dimensional Gray- semi-strong pulse interactions. This talk is based on joint Scott Model work with Arjen Doelman (Amsterdam) and Keith Promis- We analyse the stability of a stripe solution of the Gray- low (Michigan). Scott model in the weakly-coupled regime. We study Arjen Doelman three diﬀerent types of instabilities: a splitting instability, CWI Amsterdam, the Netherlands whereby a stripe self-replicates into two parallel stripes; a [email protected] breakup instability, where a stripe breaks up into spots; and a zigzag instability, whereby a stripe develops a wavy perturbation in the transversal direction. We derive ex- Tasso J. Kaper plicit thresholds for all three types of instability. Some Boston University open problems will be discussed. This is a joint work with Department of Mathematics Michael J. Ward and Juncheng Wei. [email protected] Juncheng Wei Keith Promislow Department of Mathematics Michigan State University Chinese University of Hong Kong [email protected] [email protected]

Theodore Kolokolnikov MS15 Free University of Brussels Pulse Splitting on Growing Domains [email protected] Pattern formation on growing domains has attracted interest in the past few years due to its biological applica- Michael Ward tions, e.g. for the patterning of animal skins. A recurrent Department of Mathematics phenomenon is the repeated splitting of patterns during University of British Columbia growth, in particular for Turing patterns. We present fur- [email protected] ther analysis of this, also for excitation pulses, in terms of stability and bifurcation considerations with focus on the MS15 onset of splitting. This is joint work with Michael Ward. Can Weak Interaction Cause Annihilation? Michael Ward Department of Mathematics We discuss about annihilation-repulsion criterion for the University of British Columbia scattering dynamics of particle-like patterns in dissipative [email protected] systems. Hidden saddles, called ”scattors”, essentially control the input-output relation during the scattering process. For non-fusion type of scattering, a scattor with codim 3 Jens Rademacher singularity consisting of saddle-node, Hopf and drift insta- University of British Columbia bilities is responsible for the onset of annihilation-repulsion Department of Mathematics boundary in a parameter space. This implies that annihi- [email protected] lation occurs for arbitrary small velocity of traveling patterns. MS15 Yasumasa Nishiura Competition and Oscillatory Instabilities of Spikes RIES, Hokkaido University in the Semi-Strong Interaction Limit Japan [email protected] The dynamics and instability mechanisms of one-spike and two-spike solutions to the Gierer-Meinhardt and Gray- Scott models are analyzed on a bounded one-dimensional Kei-Ichi Ueda domain. For each of these non-variational two-component RIMS, Kyoto University systems, the semi-strong spike-interaction limit, where the [email protected] ratio of the two diffusion coefficients is asymptotically large, is analyzed. In this limit, differential equations for Takashi Teramoto the slow time-dependent motion of the locations of the Chitose Institute of Science and Technology spikes are derived. The stability of these solutions on a fast [email protected] time scale is studied by a spectral analysis of certain non- DS05 Abstracts 113

local eigenvalue problems that involve the instantaneous as direct numerical simulations of reaction diﬀusion sys- spike locations. From these nonlocal eigenvalue problems tems, we study several aspects of the dynamics of twisted it is shown that eigenvalues can enter into the unstable scroll waves: the propagation of pulses appearing from a right half-plane along either the real axis or through a Hopf secondary instability, the build-up of twist in systems with bifurcation, leading to either a competition instability or spatially varying excitability, and the eﬀect of conduction a synchronous oscillatory instability of the spike pattern. anisotropy on the formation of twist. The important feature of these instabilities are that they can be dynamic in nature, and so can be triggered at some Vincent Hakim point during the slow evolution of a spike pattern that LPS, Ecole Normale Superieure is initially stable at time t=0. The asymptotic theory is [email protected] compared with full numerical results. Competition and synchronous oscillatory instabilities are also shown to oc- Blas Echebarria cur for k-spike equilibria in a one-dimensional domain, and Universitat Politecnica de Catalunya in a two-dimensional spatial domain. [email protected] Juncheng Wei Department of Mathematics MS16 Chinese University of Hong Kong Contributions of Cellular and Structural Hetero- [email protected] geneity to Rabbit Ventricular Arrhythmia

Theodore Kolokolnikov Analysis of cardiac electrophysiology in intact myocar- Free University of Brussels dial tissue is made more complex by the presence of cel- [email protected] lular and structural heterogeneities, as well as a number of interacting biophysical processes, including neuro- Michael Ward hormonal control. We present here computational mod- Department of Mathematics els that integrate detailed cellular systems models of car- University of British Columbia diomyocyte excitation-contraction coupling and its regu- [email protected] lation into anatomically detailed 3D continuum models of the rabbit ventricles. These models may prove useful tools in elucidating the mechanisms of arrhythmogenesis. Wentao Sun Department of Mathematics Andrew McCulloch Simon Fraser University University of California, San Diego [email protected] Department of Bioengineering [email protected] MS15 Sarah Healy Stripes and Wriggled Stripes in Reaction-Diffusion University of California San Diego Systems Department of Bioengineering We consider the existence and stability of perfect and wrig- [email protected] gled stripes for two reaction-diffusion systems: First one is Gierer-Meinhardt system with saturation: MS16 a2 A Tridomain Model of Discrete 3D Cardiac Tissue a 2 a − a t = ∆ + 2 h(1 + ka ) Most models used to study the cardiac electrodynamics assume the tissue structure as being either a continuous mon- 2 τht = D∆h − h + a odomain or a bidomain. For many diseased states, how- k> ever, the cell coupling is heterogeneous, impacting wave- where 0 is a fixed paramter; The second one is di-block front propagation. We have recently developed a tridomain model model of cardiac tissue in which each cell is a discrete three 2 u u − u3 γ − −1 u − a Constant dimensional object, enclosed by a membrane and embed- ∆ + + ( ∆) ( )= ded in a discrete interstitial domain. We present simulation studies comparing the model with traditional models. Both equations are considered in a perfect rectangle R = [0, 1] × [0, 1]. The stability of N− stripes is completely de- Sarah Roberts, Jeroen Stinstra, John Pormann termined by the small eigenvalues which we have computed Duke University explicitly. Our results show that the stripes can be stable [email protected], [email protected], [email protected] in some cases. Juncheng Wei Craig Henriquez Department of Mathematics Dept of Biomedical Engineering Chinese University of Hong Kong Duke University [email protected] [email protected]

MS16 MS16 Reduced Model for Twist Induced Instabilities of Computational Infrastructure for Biomedical Mod- Scroll Waves eling, Simulations, and Visualizations

We present a reduced model for the dynamics of the mean Computational infrastructures provide the framework in ﬁlament and twist of a scroll wave. Using the model, as well which computing can support a particular application. A 114 DS05 Abstracts

good infrastructure can be critical to a project as it de- ing to model thermal fluctuations. Developing numerical termines how a program (or set of programs) looks, feels, methods which capture the dynamics of the system well and is amenable to extension and change. Problem solving is made difficult by the range of time scales associated environments (PSEs) are a form of computational infras- with the degrees of freedom of the fluid and elastic struc- tructure that seeks to provide an integrated set of tools for tures. We show how a numerical method which achieves a particular application area. In this talk, I discuss our re- long time steps can be developed by approximating the cent research and development of component-based PSEs fastest degrees of freedom of the system using an appropri- for biomedical computing. ate stochastic model. A statistical analysis of the behavior of particles, polymers, and membrane-like sheets will be Chris Johnson presented for the method. We will further discuss applica- University of Utah tions of the method to biological and microfluidic systems. Department of Computer Science [email protected] Peter R. Kramer Rensselaer Polytechnic Institute Department of Mathematical Sciences MS16 [email protected] Ventricular Fibrillation Activation Patterns on the Surface of the Swine Heart Paul Atzberger Rensselaer Polytechnic Institute We recorded ventricular fibrillation (VF) activation pat- [email protected] terns from the surface of Langendorff-perfused swine hearts using a new panoramic optical mapping system that allows almost all of the epicardium to be mapped with 1-2 mm MS17 resolution at 750 fps. Individual phase singularities did Nonlinear Proper Orthogonal Decomposition for not persist for more than about 500 ms; however, unin- Molecular Systems terrupted epicardial wavefronts could be tracked for entire mapped episodes (4 s). This indicates that VF mainte- We consider the method of proper orthogonal decomposi- nance can be explained by multiple-wavelet epicardial reen- tion (POD) on functions of Cartesian molecular data, e.g., try. on torsion angles. By pulling back the nonlinear POD information thus obtained to the original model equations we Matthew Kay, James Gladden can decompose the system in a nontrivial way, and derive Department of Biomedical Engineering a reduced model. If we moreover replace truncated modes University of Alabama at Birmingham by an appropriate stochastic process the reduced model [email protected], [email protected] captures the essential features of the original dynamics. Gregory Walcott Christof Schuette Department of Medicine University of Berlin University of Alabama at Birmingham Germany [email protected] [email protected]

Jack Rogers Peter R. Kramer University of Alabama at Birmingham Rensselaer Polytechnic Institute [email protected] Department of Mathematical Sciences [email protected]

MS16 Carsten Hartmann Complex Electrical Dynamics in a Simulated Heart Free University Berlin Slice Institute of Mathematics [email protected] Cardiac tissue engineering has opened up the possibility of studying electrical wave-front dynamics in complex but controlled substrates. An in silico method is presented that MS17 mimics the structural development of these cultures from Stochastic Mode Reduction with Metastability in protein patterning, through cell spreading to intercellular Biomolecular Modeling coupling. The electrical behavior of these in silico will help design experimental studies and interpret results. One approach to accelerating biomolecular simulations is through a mode reduction in which only certain “slow” de- Joseph Tranquillo grees of freedom of interest are simulated, and the influence Duke University of the remaining “fast” variables are incorporated through [email protected] effective deterministic and stochastic terms. Metastability is a prevalent feature in biomolecular systems, and we illus- MS17 trate, through detailed analysis of a simple model problem, various ways in which it can affect the effective equations Multiscale Modeling of Biological Systems obtained from a mode reduction procedure.

The mechanics associated with many biological processes Christof Schuette can be modeled to a first approximation by elastic struc- University of Berlin tures which interact with a fluid. At small length scales Germany thermal fluctuations play a significant role in system dy- [email protected] namics. In this talk we discuss how to extend the immersed boundary method of Peskin using stochastic forc- Peter R. Kramer DS05 Abstracts 115

Rensselaer Polytechnic Institute attainable with zero-net mass flux controls in a channel Department of Mathematical Sciences flow and the maximum Reynolds number for stabilizabil- [email protected] ity of vortex shedding in the flow past a cylinder. This talk will summarize the first result established of this kind for Jessika Walter, Wilhelm Huisinga the Navier-Stokes equation. Free University of Berlin Institute of Mathematics Thomas Bewley [email protected], [email protected] University of California, San Diego Mechanical and Aerospace [email protected]. Carsten Hartmann Free University Berlin Institute of Mathematics MS18 [email protected] Robust Stability and Control of Wall Bounded Tur- bulence

MS17 In this talk the limitations of traditional hydrodynamic sta- The Stochastic Spectral Dynamics of Bending and bility theory and control are shown and a framework for ro- Tumbling bust flow stability and control is formulated. A host of new techniques like gramians, singular values, operator norms, Traditional models of wormlike chains in shear flows at fi- etc. are introduced to understand the role of various kinds nite temperature approximate the equation of motion via of uncertainty. An interesting feature of this framework is finite difference discretization (bead and rod models). We the close interplay between theory and computations. It is introduce here a new method based on a spectral represen- shown that a subset of Navier-Stokes equations are glob- tation in terms of the natural eigenfunctions. This formu- ally, non-nonlinearly stable for all Reynolds number. Yet, lation separates tumbling and bending dynamics, clearly invoking this new theory, it is shown that these equations showing their interrelation, naturally orders the bending produce structures and features as seen in our experiments. dynamics according to the characteristic decay rate of its Feedback control results are presented. modes, and displays coupling among bending modes in a general flow. This hierarchy naturally yields a low dimen- John Doyle sional stochastic dynamical system which is amenable to Caltech analytic treatment (recovers and extends previous numer- Department of Control and Dynamical Systems ical results on stability of rods in shear) and which leads [email protected] to a fast and efficient numerical method for studying the stochastic nonlinear dynamics of semiflexible polymers in Kumar M. Bobba general flows. University of Massachusetts Chris Wiggins Mechanical and Industrial Engineering Dept. Applied Physics and Mathematics [email protected] Columbia University [email protected] MS18 Panel Discussion Alberto Montessi, Matteo Pasquali Rice University The speakers will address some of the key unsolved prob- [email protected], [email protected] lems in theoretical and computational robust fluids stability and control arena. The panel will also try to identify some of our current limitations or “bottlenecks” from MS18 the mathematical and simulation side towards reaching the Stability and Robustness of Channel Flows with above goals—in an effort to speculate what can be expected Corrugated and Flexible Walls in the coming years. The relevance of these developments to experimental fluid mechanics and technological applica- We study wall bounded turbulent flows with either corru- tions of fluid flow in aerospace and mechanical engineering gated or flexible walls. Numerical and experimental studies will be touched upon. have shown that both types of walls produce reductions in skin-friction drag. However, no systematic theory of the Kumar M. Bobba dynamical effects of such walls exist. We study the effect University of Massachusetts of such walls on robust stability and noise amplification of Mechanical and Industrial Engineering the underlying linearized Navier-Stokes equations. [email protected] Bassam A. Bamieh UC Santa Barbara MS18 [email protected] Stability, Turbulence and Direct Numerical Simu- lation MS18 Fundamental Limitations in the Control of Fluid Robert Moser Systems University of Illinois [email protected] A new area for the mathematical machinery for Navier- Stokes systems is in the characterization of the fundamental performance and stabilization limitations in represen- MS18 tative flow control problems, such as the minimum drag Control of Swept Attachment-Line Boundary Lay- 116 DS05 Abstracts

ers [email protected]

Swept attachment-line boundary layers form at the leading edge of any blunt body moving through a fluid at a MS19 non-zero yaw angle. Using an optimization scheme based Spiral Waves in Mammalian Brain - Experiments on a variational approach, we investigate disturbances favored by the nonhomogeneous mean flow and assess their We report stable rotating spiral waves in mammalian tan- temporal evolution and growth mechanism. The same op- gential brain slices visualized by voltage-sensitive dye imag- timization scheme is then used to apply an optimal blowing. Spiral waves occurred spontaneously and alternated ing/suction strategy to minimize the rise of previously with plane, ring, and irregular waves. Spiral rotation rates identified instabilities. were about 10 turns per second, and the rotation was linked to the oscillations in a one-cycle-one-rotation manner. A Peter Schmid small slowly drifting phase singularity occurred at the cen- University of Washington, Seattle ter of the spirals. Such spiral waves may provide a spatial Department of Applied Mathematics framework to organize cortical oscillations. [email protected] Steven J. Schiff George Mason University MS19 The Krasnow Institute Interplay Between Excitation and Inhibition Dur- sschiff@gmu.edu ing Seizures Jian-Young Wu The dynamics of seizures in the CA1 region of the hip- Georgetown University pocampus is largely sculpted by the behavior of interneu- Physiology and Biophysics rons. Excessive inhibitory activity leads to a widespread [email protected] failure of inhibition as these cells are transiently forced into depolarization block. The loss of effective inhibitory action releases a significant excitatory response. The mechanisms Xiaoying Huang for these network interactions are probed through novel Georgetown University analysis of experimental data along with simulations using [email protected] networks of conductance based neurons. John R. Cressman MS19 krasnow institute Modulating Neuronal Synchronization with Elec- george mason university trical Fields - Theory [email protected] Two heterogeneous model neurons were synaptically coupled and embedded within a resistive array, thus allowing MS19 the neurons to interact both chemically and electrically. Controlling Wave Propagation in Cortex Theory An applied electric field was found to be effective in con- and Experiment trolling the transition of synchrony between these neurons. A simple phase oscillator reduction was successful in qual- We experimentally confirmed predictions that modulation itatively reproducing these results. These findings suggest of neuronal threshold with electrical fields can speed up, a larger scale model in which the effects of electric fields slow down, and even block traveling waves in neocortical on seizure activity may be simulated. slices. The predictions are based on a Wilson-Cowan type integrodifferential equation model of propagating neocor- Paul So tical activity. Wave propagation could be modified quickly George Mason University and reversibly. To the best of our knowledge, this is the The Krasnow Institute first example of direct modulation of threshold to control [email protected] wave propagation in a neural system. Bruce Gluckman MS19 George Mason University Spiral Waves in Brain Theory [email protected] We investigate wave formation in a two dimensional Wilson-Cowan type model of neural media. Two dimen- MS19 sional waves that are observed in the model include rotat- How Synaptic Dynamics May Organize Seizures ing spirals, ring shaped plane waves, and periodic waves. All of these have experimental counterparts recently dis- The dynamics of seizures involves the imbalance between covered in the occipital cortex of the rat. Movies will be excitatory and inhibitory cell populations. Recent experi- shown which compare both model and experimental re- ments have observed periodic interactions between distinct sults. neuron populations. I will give a bifurcation analysis for a biophysical model for a two population network. The key William Troy dynamical mechanisms are a Hopf bifurcation due to dy- University of Pittsburgh namical synapses, and periodic fast-slow dynamics due to Department of Mathematics an increase of ion concentration during elevated firing. We [email protected] also give predictions for the event propagation. Evelyn Sander MS20 George Mason University Practicalities of Implementing a Bistatic Link Us- DS05 Abstracts 117

ing Chaotic Signals bﬂ[email protected]

The practicalities of transmitting a continuous chaotic Gabriel Thomas waveform from a drifting sonar projector to a drifting re- Electrical and Computer Engineering ceiver are discussed. These include the trade-off between [email protected] transducer characteristics, target properties and signal characteristics, the need to communicate a signal replica to the remote receiver for matched filtering and the effect Berenice Verdin of multipaths. Experimental results from a sea trial are Electrical and Computer Engineering used to illustrate the arguments. University of Texas at El Paso [email protected] P. R. Atkins University of Birmingham Dept. of Electronic, Electrical and Computer Engineering MS20 [email protected] Chaotic Waveforms for Radar Applications

MS20 Kim Scheff US Naval Research Lab Testing Chaotic Waveforms for Radar scheff@radar.nrl.navy.mil Bandwidth increases distance resolution in a radar waveform, but not all broad band waveforms are good for radar. MS20 We look at waveforms from several chaotic systems, including several Rossler like systems and a recently published Robust Chaotic Signal Detection in Dispersive hyperchaotic system. We plot ambiguity diagrams for Channels these waveforms, and compare to bandpass filtered noise Radar, sonar and communication systems conventionally or a linear FM chirp. Finally, using results from a wave- use coherent detection to improve reliability but this re- form that was actually transmitted, we show that one may quires accurate synchronisation between transmitter and identify which one of several chaotic signals was transmit- receiver. Identical chaotic synchronisation has been pro- ted. posed and demonstrated, but performance is poor in noisy Thomas Carroll and distorting channels. This presentation will discuss less US Naval Research Lab restrictive definitions of synchronisation, and demonstrate [email protected] the impact of typical radio and underwater channels on chaotic signals and synchronisation performance. Further, processing techniques to improve detection performance MS20 will be analysed and discussed. Bifurcation, Bursting, Detection and Classification Christopher Williams The use of chaotic signals provides opportunities for novel University of Bristol detection and classification schemes in radar and sonar. [email protected] In particular when there are non-linear interactions, bifurcations may occur. Some situations will be given. With MS21 linear systems, bursting can occur in chaotic echoes under conditions which will be described. The use of bursting to Delay Effects in Applications: A Quasi-Historical provide an indication of the presence or type of target will Introduction be discussed. Time delays (or lags or retardations) have been recognized Alan Fenwick for quite some time to play a rôle in the dynamics of numer- QinetiQ ous regulatory systems, particularly those with feedback. UPS As an introduction to this minisymposium, whose goal is [email protected] to illustrate current knowledge of the analysis of delayed systems in applications from a dynamical point of view, we present, with a historical perspective, examples from MS20 decades and centuries past to illustrate how the explicit Parameter Selection of Discrete Chaotic Maps for incorporation of time delays in the modeling of biological Improved FM Ambiguity and physical systems yields a more faithful mathematical representation. The importance of the modeling procedure We recently proposed the use of first-order chaotic maps to itself will not be underestimated. In addition, and perhaps construct wideband FM signals for radar imaging. In par- more importantly, we also argue that contemporary tech- ticular we showed that an FM signal based on the Bernoulli niques from dynamical systems theory can be incorporated map exhibited a near optimum ambiguity surface. In sub- in the analysis of these models. sequent analyses we have observed that the Bernoulli map can yield highly correlated samples. The degree of cor- Jacques Belair relation depends on the choice of the B parameter of the University of Montreal chaotic map. We show that only specific values of B yield Department of Mathematics uncorrelated samples, which improve the characteristics of [email protected] the chaotic FM signal’s ambiguity surface. Benjamin C. Flores MS21 University of Texas at El Paso Restrictions on Dynamics in Biological Delay Mod- Electrical and Computer Engineering 118 DS05 Abstracts

els at Double Hopf Bifurcation Afdeling Natuurkunde en Sterrenkunde [email protected] This talk will be concerned with the restrictions on dynamics at double Hopf bifurcations for a delay model of a motor control task experiment with Parkinsonian patients MS21 and the delay model of the pupil light reﬂex. I will be Dynamics of Systems with Delayed Relay Control shown that the restrictions on the dynamics are delay dependent and a consequence of the structure of the models. A ubiquitous phenomenon in control is the so-called criti- Further research directions will be discussed. cal delay. That is, if the delay in the control loop is larger than a critical value, linear stabilization by feedback be- Jacques Belair comes impossible. In contrast, a relay switch can stabilize University of Montreal an unstable system toward stable periodic motion even if Department of Mathematics subject to a large delay. The balanced inverted pendulum [email protected] serves as illustrative example, also in a discussion of the dynamics of piecewise smooth systems with delay. Pietro-Luciano Buono University of Ontario Institute of Technology Jan Sieber Faculty of Science University of Bristol [email protected] Dept. of Eng. Math. [email protected]

MS21 Stability and Bifurcation Analysis of a Nonlinear MS21 Delay Equation Model for Drilling Wheel Shimmy Caused by Distributed Delays

We study a delay differential equation model for chatter in A non-holonomic model of the shimmying wheel is stud- twist drills in which a linear vibration mode interacts with ied that uses the nonstationary values of the distributed nonlinear cutting forces. The model describes an oscilla- contact force system between the elastic tire and the rigid tor with nonlinear damping and cross-terms in the damp- ground. The governing equations are coupled partial and ing and the delay. Linear stability analysis and nonlinear integral differential equations. The existence of a travelling analysis of the primary Hopf bifurcation, by constructing wave-like solution of the PDE leads to the RFDE a centre manifold using symbolic algebra, shows that the stability of the Hopf bifurcation depends on the vibration 0 type and the cutting speed. 2 L − 1 V ψ¨(t)+ψ(t)= (L−1−2θ)ψ(t+θ)dθ+h.o.t. L2 +1/3 Emily F. Stone −1 The University of Montana Department of Mathematical Sciences [email protected] where ψ is the caster angle, and the parameters V and L are the dimensionless speed of towing and caster length, Sue Ann Campbell respectively. University of Waterloo Denes Takacs, Gabor Stepan Dept of Applied Mathematics Budapest University of Technology and Economics [email protected] Department of Applied Mechanics [email protected], [email protected] MS21 Bifurcations of Mutually Delay-Coupled Lasers MS22 Delay-coupled semiconductor lasers are an attractive Synchronizability of Complex Networks choice to study effects of time delay in coupled nonlin- We explore the interplay of network topology and synchro- ear oscillators. We present a bifurcation study of a delay nizability in the classic Kuramoto model of coupled non- differential equation model of two identical, delay-coupled linear oscillators. We go beyond the existing results for semiconductor lasers. We find a comprehensive geometri- all-to-all networks of identical oscillators by allowing for cal picture that explains the bifurcations of solutions called networks of arbitrary interconnection topology and oscilla- compound laser modes in dependence on two physically tors with uncertain natural frequencies. We use tools from relevant parameters, namely the coupling phase and the spectral graph theory and control theory to relate graph detuning. properties to the critical coupling above which all the os- Daan Lenstra cillators synchronize. We further explain the behavior of Theoretical Physics the system as a function of the number of oscillators as it Vrije Universiteit Amsterdam grows to infinity. [email protected] Ali Jadbabaie Department of Electrical and Systems Engineering Bernd Krauskopf University of Pennsylvania University of Bristol [email protected] Dept of Eng Mathematics [email protected] Mauricio Barahona Imperial College London Hartmut Erzgraber UK Vrije Universiteit Amsterdam [email protected] DS05 Abstracts 119

MS22 [email protected] Synchronization of Bursting Neurons: What Mat- ters in the Network Topology Jeff Moehlis Dept. of Mechanical and Environmental Engineering We study the influence of coupling strength and network University of California – Santa Barbara topology on synchronization behavior in pulse-coupled net- [email protected] works of bursting Hindmarsh-Rose neurons. Surprisingly, we find that the stability of the completely synchronous Philip Holmes state in such networks only depends on the number of sig- Prog. in Applied and Comp. Mathematics nals each neuron receives, independent from all other de- Princeton University tails of the network topology. This is in contrast with lin- [email protected] early coupled bursting neurons where complete synchrony strongly depends on the network structure and number of cells. Through analysis and numerics, we show that the MS22 onset of synchrony in a network with any coupling topol- Foring Synchrony: Spatial Correlations in Noisy ogy admitting complete synchronization is ensured by one Coupled Neurons single condition. Ensembles of neurons often receive spatially correlated in- Igor Belykh puts from both their environment and distant brain re- Swiss Federal Institute of Technology Lausanne gions. I will show how the ensemble statistics of coupled igor.belykh@epfl.ch nonlinear oscillators/neurons are sensitive to a correlated stochastic forcing. This interplay between self-organized MS22 and driven synchronization allows for an interesting coding strategy where spatially distributed information can be Synchronization is Enhanced in Weighted Net- coded with a temporal ensemble response. works Brent Doiron We show that synchronizability is enhanced in weighted New York University networks where the coupling retains information on the Courant Institute structure of the shortest pathlengths connecting the nodes, [email protected] or on the age of the nodes. This issues are formally treated in relation with variations in the eigenvalues of the corresponding asymmetric connecting matrices, and examples MS22 are provided for phase synchronization and complete syn- Synchronization Mechanisms of Minimal Networks chronization in a class of scale-free networks of Roessler of the Parahippocampal Region oscillators. Experimental and theoretical studies have shown that os- Dong-Uk Hwang, Andreas Amann cillatory activity at theta frequencies (8-12 Hz) can be gen- Istituto Nazionale di Ottica erated by networks having fast-firing (I) interneurons (in- Florence, Italy hibitory) and either oriens lacunosum-moleculare (O-LM) [email protected], [email protected] interneurons (inhibitory) or stellate (S) cells (excitatory). O-LM and S have an hyperpolarization-activated (h-) cur- Stefano Boccaletti rent in addition to the standard Hodgkin-Huxely ones. We Istituto Nazionale di Ottica Applicata use a map approach to explain the generation of the rhyth- [email protected] mic activity at theta frequencies and the special role played by the h-current. Mario Chavez Istituto Nazionale di Ottica Appplicata Dmitri Pervouchine Florence, Italy Boston University [email protected] Boston, USA [email protected]

MS22 Nancy J. Kopell On the Phase Reduction and Response Dynamics Boston University of Neural Oscillator Populations Department of Mathematics [email protected] We undertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulse- Horacio Rotstein like stimuli. Recalling and extending results from the Boston University literature, we compute phase response curves (PRCs) Boston, USA valid near bifurcations to periodic firing for Hindmarsh- [email protected] Rose, Hodgkin-Huxley, FitzHugh-Nagumo, and Morris- Lecar models. Phase density equations are then used to analyze the role of the bifurcation, and the resulting PRC, MS23 in responses to stimuli. In particular, we explore the in- The Dynamics of Modulated Wave Trains terplay among stimulus duration, baseline firing frequency, and population level response patterns. In this talk, we consider the dynamics of nonlinear wave trains in reaction-diffusion equations. We prove that slowly Eric T. Brown varying modulations of wave trains are well approximated Courant Institute for the Mathematical Sciences by solutions of the Burgers equation over the natural time New York University scale. In addition to the validity of Burgers equation over 120 DS05 Abstracts

long but finite time intervals, we show that the viscous School of Mathematics shock profiles in Burgers equation can be found as genuine [email protected] modulated waves, or defects, in the underlying reaction- diffusion system. Moreover, we establish the stability of these defects. This talk is based on joint work with Bjorn MS24 Sandstede, Arnd Scheel and Guido Schneider. Self-Averaging Scaling Limits of Waves in Turbu- lent Media Arjen Doelman CWI Amsterdam, the Netherlands We show under fairly general assumptions on the turbulent [email protected] refractive index field that sufficient amount of medium diversity leads to statistical stability or self-averaging of wave propagation in the sense that its limiting law is determinis- MS23 tic and is governed by one of the 6 different types of trans- Asymptotic Validity for Discrete Equations Arising port (Boltzmann or Fokker-Planck) equations depending in Nonlinear Optics on the specific scaling involved. We discuss the connection to the statistical stability of time reversal procedure. Solitons in discrete optical systems have been a topic of keen interest over the past decade. In a diffraction man- Albert Fannjiang aged waveguide array, where the waveguide’s diffraction Department of Mathematics profile is varied periodically in the direction of propagation, University of California at Davis the slow propagation of the electric field envelope is mod- [email protected] eled by a nonlocal discrete equation that exhibits soliton solutions numerically. We verify the validity of the asymptotic approximation, demonstrating that solutions of the MS24 averaged equation are close in the l2 sense to those of the Non-Standard Homogenization for an Advection- original model for long time scales, and moreover, prove Diffusion Problem that the averaged equation has stable stationary solutions, complementing previous experimental and numerical ob- We describe some interesting asymptotic features which servation of these solutions. emerge in a simple model for turbulent diffusion consisting of a large scale mean flow and small scale periodic fluctu- Jamison T. Moeser ations. While standard homogenization theory describes University of Colorado one distinguished limit, the dynamics are more complex [email protected] when the mean flow is strong. We investigate the effective transport through multiple scale analysis (involving three time scales) and numerical simulations. MS23 Justification of the Nonlinear Schrödinger Equa- Peter R. Kramer, Adnan Khan tion in Spatially Periodic Media Rensselaer Polytechnic Institute Department of Mathematical Sciences The dynamics of the envelopes of spatially and temporar- [email protected], [email protected] ily oscillating wave packets advancing in spatially periodic media can approximately be described by solutions of a Nonlinear Schrödinger equation. Here we prove es- MS24 timates for the error made by this formal approximation Eddy Viscosity of Cellular Flows using Bloch wave analysis, normal form transformations, and Gronwall’s inequality. Joint work with Kurt Busch, In two dimensions in the presence of small-scale eddies the Guido Schneider, Lasha Tkeshelashvili transport of large-scale vector quantities may be accompanied with depleted, and even “negative” diffusion. This Hannes Uecker phenomenon can be investigated by the stability analysis University of Karlsruhe of the effective (averaged) equations. The eddy viscosity Department of Mathematics is a tensor in this equations. The main questions are the [email protected] structure of this tensor, how it depends on the underlying geometry of the small-scale eddies and how accurate the predictions of the effective equations. The goal of this work MS23 is to answer these questions for a simple model where the Approximations for the Collision of Two Solitary eddies are given by a fixed periodic time-independent flow. Water Waves A particular example is a cellular flow with the streamfunction sin x/a sin y/a, a ¡¡ 1. The KdV approximation for surface water waves in a shallow canal predicts both a head on collision of solitary waves Alexei Novikov as well as an overtaking collision. While such collisions are Penn State University in fact observed experimentally, there are notable depar- Mathematics tures from the predictions of the KdV model. We discuss [email protected] these departures and propose a set of equations which govern corrections to the KdV approximation. These consist of linearized and inhomogeneous KdV equations plus an in- MS24 homogeneous wave equation. Some comparisons between Periodic Homogenization for Inertial Particles this higher order model and experimental data taken by J. Hammack and D. Henderson will be shown. We study the problem of homogenization for inertial particles moving in a periodic velocity field, and subject to Doug Wright molecular diffusion. We show that, under appropriate as- University of Minnesota sumptions on the velocity field, the large scale, long time DS05 Abstracts 121

behavior of the inertial particles is governed by an effective MS25 diffusion equation for the position variable alone. An ex- The Topology of Interfaces in Systems Under pression for the diffusivity tensor is found and various of its Coarsening properties studied. Incompressible and potential fields are studied, as well as fields which are neither, and theoretical We examine the evolution of the morphology of a two-phase findings are supported by numerical simulations. mixture during coarsening under Cahn-Hilliard (CH) and Allen-Cahn (AC) dynamics. The morphology is quanti- Andrew Stuart fied using the interfacial shape distribution, which gives University of Warwick the probability of finding a small patch of interface with a [email protected] certain mean and Gaussian curvature. The scaling properties of interfacial shape distributions and the differences in Grigorios Pavliotis these distributions for systems coarsening under CH and Department of Mathematics AC dynamics will be discussed. Imperial College [email protected] YongWoo Kwon Dept. of Materials Science and Engineering Martin Hairer Northwestern University Warwick University, United Kingdom [email protected] [email protected] Katsuyo Thornton Dept. of Material Science and Engineering MS25 University of Michigan Homological Characterization of Spiral Defect [email protected] Chaos Peter Voorhees Relating the global structure of patterns exhibited to un- Northwestern University derlying dynamics is an important aspect of the study of Dept. of Material Science and Engineering complex systems. We use homological characterizations to [email protected] build insights into the dynamics of spiral defect chaos, a weakly turbulent state of Rayleigh-Benard convection. We note observations implying asymmetries between hot and MS25 cold flows, novel measures of boundary influence and in- Residual Stress Networks in Polycrystalline Mate- dicators of system control parameters. We also find the rials: Their Origin and Character evolution of the global structure of the flow to be primar- ily stochastic unlike the locally chaotic signatures reported Crystallographic texture in conjunction with anisotropic previously. crystalline properties profoundly influences ensemble physical properties and macroscopic behavior of polycrystalline Kapil Krishan materials. Recently, polycrystalline materials with crys- Georgia Institute of Technology talline thermal expansion anisotropy were observed to de- School of Physics velop residual-stress networks with a length-scale that sur- [email protected] prisingly could encompass many grains. Microstructure- based finite-element simulations were used to elucidate the Michael Schatz origin of these stress networks in the orientation and mis- Center for Nonlinear Science & School of Physics orientation distribution functions. Residual stress isosur- Georgia Institute of Technology faces for these networks were characterized by topological [email protected] and fractal metrics. Thomas Weiss MS25 Universitaet Goettingen Pattern Classification in Controlled Drug Delivery Germany Systems [email protected]

In an effort to maintain optimal levels of drug in the body Peter S. Bullen over time, drug delivery systems have been developed that NIST facilitate a sustained or periodic release of drug. The per- [email protected] formance of these systems is significantly influenced by the microstructure. Thus, a diffuse interface model is used Edwin R. Fuller, Jr. to predict microstructure formation and release behavior. NIST Classification of the complex patterns that comprise the Ceramics Division microstructure in these systems enables us to establish rig- [email protected] orous, quantitative structure-property relationships. David M. Saylor David M. Saylor Food and Drug Administration Food and Drug Administration CDRH-OSEL-DCMS CDRH-OSEL-DCMS [email protected] [email protected]

James Warren Thomas Wanner National Institute of Standards and Technology George Mason University Materials Science and Engineering Laboratory Department of Mathematical Sciences [email protected] [email protected] 122 DS05 Abstracts

MS26 [email protected] Nontwist Systems with Many Degrees of Freedom: A Mean Field Approach MS26 Nontwist systems have received considerable attention over Nontwist Hamiltonian Systems: An Introduction the last years. However, little is known about nontwist Hamiltonians with many degrees of freedom. To address In this talk, I will give an introduction to the study of this problem we consider mean-ﬁeld coupled, nontwist nontwist Hamiltonian systems with particular emphasis on symplectic maps. Mean ﬁeld coupled Hamiltonians have applications in physics. A simple example of such a sys- proved to be a useful laboratory to study large degrees tem is the so-called standard nontwist map which has been of freedom systems. Here we focus on the role of self- used in several physical models to study the transition to consistent dynamics in the formation of coherent struc- global chaos. I will discuss some recent work on invariant tures, separatrix reconnection, and the robustness of the manifold reconnection in this map. shearless curve. Alexander Wurm Diego Del-Castillo-Negrete Department of Physics, Fusion Studies Oak Ridge National Laboratory The University of Texas at Austin [email protected] [email protected]

MS26 MS27 Vanishing Twist Near Focus-Focus Points A Coupled-Oscillator Model with a Conservation Law for the Rhythmic Amoeboid Movements We show that near a focus-focus value in a Liouville integrable Hamiltonian system with two degrees of freedom Experiments on the fusion and partial separation of plas- lines of locally constant rotation number in the image of modia of the true slime mold Physarum polycephalum are the energy-momentum map are spirals determined by the described, concentrating on the spatio-temporal phase pat- eigenvalue of the equilibrium. From this representation of terns of rhythmic amoeboid movement. On the basis of the rotation number we derive that the twist condition for these experimental results we introduce a new model of the isoenergetic KAM condition vanishes near the focus- coupled oscillators with one conserved quantity. Simula- focus point. This implies, e.g., that near the Hamiltonian tions using the model equations reproduce the experimen- Hopf-bifurcation the twist always vanishes. tal results well. Holger R. Dullin Ryo Kobayashi Loughborough University Department of Mathematical and Life Sciences Mathematical Sciences Hiroshima University [email protected] [email protected]

San Vu Ngoc MS27 Universite Grenoble Institut Fourier Introduction: Past Key Findings and Present Di- [email protected] rections I mention two key findings on collective motion of cellu- MS26 lar rhythms: 1) modes of phase difference in coupled some oscillators (Takamatsu), 2) control of taxis by using phase Normal Forms for Fold and Cusp Singularities for wave propagation. Phase equations are basically success- Symplectic Maps ful to analyze these phenomena. A recent challenge is An integrable symplectic map is characterized by its fre- done to include significant roles of visco-elasticity of cell quency mapping J → Ω(J). If Ω(J) is not one-to-one, (Kobayashi). Behavioral intelligence of cell is also tested the singularities give rise to “twistless bifurcations” upon by posing some geometrical puzzles (Takagi) and analyzed perturbation. We study the normal forms in 4D for fold by the mathematical model. and cusp singularities in the neighborhood of a rational Toshiyuki Nakagaki rotation vector. A critical parameter is the slope of the J Research Institute for Electronic Science critical set in . When the perturbation is small enough Hokkaido University the map can be converted into a Hamiltonian flow that has [email protected] an approximate second invariant. James D. Meiss Ryo Kobayashi University of Colorado Department of Mathematical and Life Sciences Dept of Applied Mathematics Hiroshima University [email protected] [email protected]

Holger R. Dullin Atsushi Tero Loughborough University Hiroshima University Mathematical Sciences [email protected] [email protected]

A.V. Ivanov St. Petersburg University Russia DS05 Abstracts 123

MS27 Method for Studying the Control of Gene Expres- Decision Making By True Slime Mold sion in Eukaryotic Cells

The plasmodium of Physarum polycephalum is a naked ag- A method is developed for incorporating diffusion and gregate of protoplasm, and this living mass migrates to seek active transport of chemicals in complex geometries into a proper environment under various stimuli. The mecha- stochastic chemical kinetics simulations. Systems are mod- nism of decision making to a conflicting stimulation in the eled using a master equation, with jump rates for diffusive plasmodium was studied by analyzing the behavioral re- motion and active transport between mesh cells calculated sponses to a mixture of attractants and repellents. The from the discretization weights of an embedded boundary avoidance to repellents shifted to the higher concentrations method. Since jumps between cells are treated as first or- as the attractant concentration became the higher. On der reactions, individual realizations of the stochastic pro- the other hand, the membrane potential deflection varied cess can be created by the Gillespie Method. The method similarly both in the presence and absence of attractants. is used to study a 3D model of transcription, nuclear ex- Thus, the interference took place not at the receptor mem- port, translation, nuclear import, and gene regulation in a brane, but at the signal integration. The mode of inter- eukaryotic cell. ference was explained quantitatively by assuming that two kinds of signaling substances, corresponding to attraction Charles S. Peskin and avoidance, bind competitively to a protein. Courant Institute of Mathematical Sciences New York University Tetsuo Ueda [email protected] Hokkaido University [email protected] Samuel A. Isaacson Courant Institute of Mathematical Sciences Seiji Takagi [email protected] Research institute for Electronic Science Hokkaido University [email protected] MS28 Reconstructing Subpopulation Connectivity Within Neuronal Networks MS27 Well-Controlled Biological Oscillator Systems for I present a mathematical framework for analyzing connec- True Slime Mold Constructed with Microfabrica- tions among a subset of measured neurons embedded in a tion Techniques larger neural network. Through analyzing a simple probabilistic model of neural response, I demonstrate how to Oscillating cells of plasmodium of true slime mold can be account for the presence of unmeasured neurons. One can considered as a coupled oscillator system. For synthetic determine connectivity patterns in terms of certain neu- and systematic analysis, we constructed coupled oscillator ral subpopulations, which are groups of neurons with a systems with living cells of plasmodium by cell pattering similar response to a stimulus. Although the results are method using microfabrication technique, where configura- presented in terms of neuronal networks, the mathemati- tion of oscillator parts and couplings can be systematically cal framework is applicable to other networks, such as gene controlled. Rich oscillation patterns and switching behav- regulatory networks. ior among the patterns observed in systems consist of more than three oscillators will be reported. Duane Nykamp University of Minnesota Atsuko Takamatsu School of Mathematics Department of Electrical Engineering and Bioscience [email protected] Waseda University atsuko [email protected] MS28 The Influence of Chromosome Flexibility on MS28 Anaphase A Chromosome Movement as Driven by Stochastic Stimulation of the Mammalian Circa- an Imperfect Brownian Ratchet Molecular Motor dian Clock The segregation of sister chromatids to daughter cells was Based on an experimental estimate of the number of one of the earliest observations in cell biology. However, molecules of key proteins within the mammalian circadian the mechanism by which they move has proven elusive, clock, we simulate our model of the mammalian circadian as no conventional motor protein seems wholly responsi- clock with stochastic molecular interactions. Interactions ble for the motion. We propose a Brownian ratchet model with promoters on the time scale of seconds are required as the driving mechanism behind chromosome segregation. for accurate 24-hour timekeeping. The stochasticity of our We find that one determinant of the mean velocity in this model scales in an expected way. We also find noise in- system is the flexibility of the chromosome itself. More- duced oscillations. This work was conducted with Charles over, as the ratchet becomes more ”perfect”, this effect is Peskin. enhanced. Also, we find that a system with multiple Brow- nian ratchets allows for some manner of load independence, Daniel B. Forger perhaps pointing to a resolution of the long standing ques- New York University tion of why long and short chromosomes move at similar [email protected] velocities. Arjun Raj MS28 New York University A Stochastic Reaction-Diffusion-Active Transport [email protected] 124 DS05 Abstracts

MS29 to the Diffusion Equation Competition Between Mixing and Segregation in Granular Tumblers We discuss the Lagrangian chaos in a particular Stokes flow that is a superposition of a regular flow and a weak per- Flowing granular materials tend to segregate, ordering con- turbation. We demonstrate that the adiabatic invariant of stituents by size, density or even surface properties. Like the flow undergoes quasi-random jumps when a streamline fluids, chaotic advection in granular flows can enhance mix- crosses the resonance of the unpertubed flow. We show ing. We investigate the competition between segregation that for multiple resonance crossings the accumulation of and mixing of glass beads in unique tumbler geometries the jumps leads to the chaotic advection and mixing, the that produce a variety of patterns. A simple continuum rate of which is governed by a standard diffusion equation. model the underlying flow gives insight into the structure of these segregation patterns. Anatoly Neishtadt Space Research Institute, James Gilchrist Moscow Lehigh University [email protected] [email protected] Igor Mezic, Dmitri L. Vainchtein University of California, Santa Barbara MS29 [email protected], [email protected] Reactions in Open 3D Flows

We study the dynamics of reactive particles advected by 3-dimensional open incompressible flows, both analytically MS30 and numerically, using a 3D generalization of the baker Propagation in Neural Fields: an Introduction map. We find that 3D reactive flows have fundamentally novel dynamical features, not found in 2D systems, due to In this introduction I will review the way in which synapti- the richer structure of the unstable manifold in 3 dimen- cally coupled neural networks may generate and maintain sions. In particular, we show that the reaction output in travelling waves of activity, whether they be in so-called 3D displays features not found in 2-dimensional systems. spiking or rate based models. For simplicity I will focus on waves in integrate-and-fire spiking networks and their Alessandro Moura analogues in firing rate models. This will set the scene Institute of Physics for the content of the two subsequent sessions, which will Universidade de Sao Paulo cover descriptions of real neural waves from the entorhinal [email protected] cortex, the effect of plastic synaptic connectivity and axonal delays on wave stability, wave generation via external Celso Grebogi stimuli, and instabilities in heterogeneous networks. Instituto de Fisica - IF Universidade de Sao Paulo/ USP Stephen Coombes [email protected] University of Nottingham [email protected]

MS29 On Chaotic Advection In 3-D Open Flows MS30 Activity Propagation in Neural Fields for General Chaotic advection is examined in two 3-D open flows which Synaptic Connections differ by their cross-section, the eccentric helical annular mixer (EHAM) and the 3-D flow between two confocal The spatiotemporal dynamics of neural populations has at- gliding ellipses. In both cases the outer boundary turns at tracted much attention in recent years. By virtue of the constant speed while the inner boundary turns at a time spatial extention of the population, transmission delay and periodic angular velocity. Conditions leading to the better nonlocal constant feedback delay are present and may af- mixing of an injected dye can be determined by analysis. fect the stability of the population. The presented work Unexpectedly, results show that better mixing is obtained examines the propagation of neural activity for general in EHAM. homogeneous connectivities involving transmission delay. In particular, conditions for plane waves are derived and Antonio Rodrigo the effect of corticothalamic feedback on traveling fronts is Universidade Nova de Lisboa studied. [email protected] Axel Hutt Esteban Saatdjian Applied Stochastic Processes, Humboldt University Berlin ENSIC - INPL LEMTA [email protected] France [email protected] MS30 Rhythms and Propagating Activity in the Entorhi- Paulo Mota nal Cortex Universidade Nova de Lisboa [email protected] The entorhinal cortex plays an instrumental role in memory formation while acting as an interface between the hip- pocampus and the neocortex. Theta and gamma frequency MS29 rhythmic oscillatory activity of pyramidal cells, interneu- Mixing in Stokes Flows: From a Dynamical System rons, and stellate cells in layers II, III, and V are thought to play an important role. Using biophysically based mod- DS05 Abstracts 125

els of these cell types and dynamical systems techniques, tations. we analyze the basic mechanisms for the formation, suppression, and propagation of these rhythms. Oliver Junge Institute for Mathematics Horacio Rotstein [email protected] Boston University Boston, USA Jerrold E. Marsden [email protected] Control and Dynamical Systems California Institute of Technology Nancy J. Kopell [email protected] Boston University Department of Mathematics Sina Ober-Bl¨obaum [email protected] Institute for Mathematics University of Paderborn Jozsi Z. Jalics [email protected] Center for BioDynamics Boston University [email protected] MS31 Discrete Mechanics, Variational Integrators, and Optimal Control MS30 Spiral Waves in Neural Field Equations The theory of discrete mechanics and associated variational integrators is applied to optimal control of mechanical sys- Neural ﬁeld equations are non-local PDEs thought to de- tems in which one attempts to optimize a cost function scribe large-scale dynamics of the cortex in various situa- subject to equations of motion in mechanics. The novel tions. We discuss the existence of rotating spiral waves in feature of our approach is that we use discrete mechan- such equations on a two-dimensional domain, and follow ics to represent the equations of motion as algebraic con- them via numerical continuation as various parameters are straints. We show that this is especially useful in problems changed. This gives insight into which physiological pa- in which one has long time problems (as in low thrust) or rameters should be manipulated to either destroy or en- in which one needs to take large time steps. The method is hance the stability of these waves. illustrated with satellite control, control of a group of hov- ercraft and coordinated control of a group of underwater Carlo R. Laing vehicles. Massey University IIMS Sina Ober-Bloebaum [email protected] University of Paderborn [email protected]

MS31 Jerrold E. Marsden Dynamics in Aerospace Applications: from Control California Inst of Technology to Design of Dynamics Dept of Control/Dynamical Syst [email protected] We will show how certain aspects of design of unsteady flow devices for military aerospace applications contribute to origin of detrimental oscillations. We will also ways of mit- Oliver Junge igation of oscillations by active and passive control as well Institute for Mathematics as by modification of design, including symmetry breaking. [email protected] The analysis of the oscillations will be done using nonlinear distributed models involving multiple oscillatory modes MS31 coupled through nonlinear terms and distributed transport delay, that are driven by broad-band disturbances. Multicomponent Dynamical Systems and Graph Theory Andrzej Banaszuk United Technologies Research Center We discuss asymptotic behavior and uncertainty propaga- [email protected] tion in nonlinear dynamical systems on graphs. We utilize digraph decomposition theory of the system to prove several results on asymptotic dynamics of large systems that MS31 depend only on their structural properties. We discuss dis- Optimal Control of Formation Flying Satellites turbance propagation in such systems and show that it depends on the node distance in the associated graph. We Future space missions like Terrestrial Planet Finder connect the theory to issues of recent interest in industrial (NASA) and Darwin (ESA) will make use of a network context. of formation flying spacecraft. In these missions, the re- quirements on the accuracy of the relative positioning of Igor Mezic the craft are extremely high. In addition, reconfigurations University of California, Santa Barbara of the formation have to be performed at regular intervals [email protected] with minimal energetical effort. In this talk, we show how a recently developed variational method for the numerical MS32 computation of optimal open-loop controls for mechanical control systems can be applied to this problem. We out- A Unified Lyapunov Function Approach for Nonau- line the method, its benefits and present example compu- 126 DS05 Abstracts

tonomous Attractors namical Systems

In this talk we present a framework for Lyapunov functions The Hausdorff dimension is a parameter which describes for nonautonomous attractors. We show that one and the the behavior of dynamical systems. In particular, there are same Lyapunov function construction yields a characteri- several method to estimate that dimension for the attractor zation of (i) pullback attractors (ii) forward attractors and of such a system. In the case of a non-autonomous system (iii) uniform attractors. Our construction is based on ideas the so called pullback attractor gives a description of the going back to Yoshizawa combined with a careful analysis dynamics. We will discuss several methods to estimate the of the attraction rates expressed via comparison functions Hausdorff dimension of those attractors. which generalize the concept of KL functions widely used in nonlinear control theory. Björn Schmalfuss University of Paderborn Stefan Siegmund [email protected] University of Frankfurt Germany [email protected] MS32 Vortex Merging Fabian Wirth University of Bremen The merging of two vortices is described using tools from [email protected] nonautonomous dynamical systems theory. It is crucial to capture the hyperbolic transient behavior by finite-time invariant manifolds. Lars Gruene Mathematical Institute, University of Bayreuth Stefan Siegmund [email protected] University of Frankfurt Germany Peter Kloeden [email protected] Department of Mathematics J.W. Goethe University Frankfurt [email protected] MS33 Distributed Coordination of Mobile Agents: From Bird Flocking to Synchronization of Coupled Os- MS32 cillators Global Behaviour in Adaptive Control Systems In this talk we provide a unified view of several distributed Adaptive controllers are used in systems where one or more coordination and flocking algorithms which have appeared parameters are unknown. Such controllers are designed in various disciplines such as statistical physics, biology, to stabilize the system using estimates for the unknown computer graphics over the past 2 decades. These algo- parameters that are adapted automatically as part of the rithms have been proposed as a mechanism for demon- stabilization. One drawback in adaptive control design is strating emergence of collective behavior (such as social the possibility that the closed-loop limit system is not sta- aggregation, schooling, flocking and synchronization) us- ble. The worst situation is the existence of a destabilized ing purely local interactions. We will show that these limit system attracting a large open subset of initial con- coordination problems, such as Vicsek’s model of coor- ditions. These situations lie behind bad behavior of the dination of self propelled particles in statistical physics, closed-loop adaptive control system. In this talk we iden- The Reynolds’ model of flocking in computer graphics, and tify and characterize the occurrence of such bad behavior the synchronization of coupled nonlinear oscillators (a well in the adaptive stabilization of first- and second-order sys- studied problem in dynamical systems and physics), can be tems with one unknown parameter. In this context we dis- all studied and analyzed rigorously under a unfied frame- cuss a number of bifurcation-like phenomena and develop work of graph theory and dynamics. Utilizing these results, corresponding normal forms. we provide a biologically plausible, vision-based coordination scheme for flocking and velocity alignment, which does Hinke M. Osinga not require velocity measurements and/or nearest neigh- University of Bristol bor communication, but instead relies on nearest neighbor Department of Engineering Mathematics sensing. We will show that by sensing the optical flow and [email protected] time-to-collision between each agent and its neighbors we can achieve coordination, even if the topology of the prox- Reza Rokni Lamooki imity graph changes with time. Finally, we use the same University of Mazandaran framework to analyze a recently proposed scheme for geo- Babol, Iran graphic routing in wireless adhoc networks which does not [email protected] rely on location information.

Stuart Townley Ali Jadbabaie University of Exeter Department of Electrical and Systems Engineering Exeter, UK University of Pennsylvania [email protected] [email protected]

MS32 MS33 Dimension for Attractors for Non-Autonomous Dy- Formations from Gyroscopic Interactions We consider gyroscopic interactions among particles moving in three-dimensional space, and demonstrate their ef- DS05 Abstracts 127

fectiveness in producing desired stable spatial patterns of and analyze the collective behavior of swarms in presence motion. Applications include formation control for teams of obstacles. We provide simulation results that success- of agile vehicles such as UAVs. We discuss how best to fully demonstrate migration, split/rejoin, squeezing, and frame the curves representing the particle trajectories. A rendezvous maneuvers for swarms of hundreds of agents. natural Lie group formulation emerges, with G = SE(3) as a symmetry group. For two-particle formations with Reza Olfati-Saber specific G-invariant interaction laws, we are able to prove University of California, Los Angeles & global convergence to specific formations (as well as non- California Institute of Technology collision). We also describe how a single particle might [email protected] interact with a fixed structure in space by exploiting gyroscopic feedback to achieve obstacle avoidance or boundary- following behavior. MS34 DNA in Flows with Controlled Mixing Eric W. Justh University of Maryland We model the recombination of very-short-strand DNA [email protected] with the assumption that two complementary strands will combine only if they are close together in position as well Fumin Zhang as alignment. We describe this as a reaction-advection- Princeton University diffusion system in position-orientation space with avenues [email protected] for control in the form of velocity fields and external potentials. We analyze the dynamics in a flow caused by an active (shear superposition) micro-mixer and compare with P.S. Krishnaprasad experiments. University of Maryland, College Park Institute for Systems Research Igor Mezic, Thomas John [email protected] University of California, Santa Barbara [email protected], [email protected] MS33 Self-Organizing Robotic Systems MS34 The fundamental problem of designing self-organizing sys- Geometry and Addressability Actuation in Cat- tems is to understand how local interactions between com- alytic Pattern Formation ponents give rise to global properties. This problem must be solved if we are to engineer predicatable and reliable We investigate the effect of two-dimensional composite ge- large-scale systems consisting of vast numbers of parts (e.g. ometry on the dynamic behavior of pulses through com- micro-robots, cells or molecules). In this talk I will de- bined experimentation and modeling. With a ”Y”-shape scribe a formal approach to modeling and designing self- structure, we show how to direct the propagation of pulses organizing systems based on graph rewriting. The ap- by varying the details of the geometry. Then we extend our proach allows us to describe massively parallel algorithms work to numerical bifurcation analysis on pulse propaga- for self-assembly, self-replication, distributed locomotion tion in a two-dimensional ring structure using matrix-free and other decentralized processes and to rigorously prove techniques. The combination of confining geometry and that they work. I will illustrate the approach by show- laser addressability on CO oxidation pulses is also explored. ing how it can be used with our self-assembling robotics platform and in MEMs self-assembly. Harm-Hinrich Rotermund Fritz-Haber-Institute of the Max Planck Society Eric Klavins Berlin, Germany Department of Electrical Engineering [email protected] University of Washington [email protected] Christian Punckt Fritz-Haber-Institute of the Max Plack Society MS33 Berlin, Germany [email protected] Self-Assembly and Collective Control of Agent- Based Swarms Liang Qiao Swarms of sensors and mobile agents can be used for mas- Department of Chemical Engineering sive distributed sensing, monitoring, modeling, and map- Princeton University ping of an environment. In this talk, we discuss a theory [email protected] for modeling and control of collective motion of swarms of particles with local interactions. We present distributed Yannis Kevrekidis swarming algorithms that allow a leaderless swarm of par- Princeton ticles to migrate from point A to B without inter-agent or [email protected] agent-to-obstacle collisions. We view a ”swarm” as a complex network of dynamical systems. A class of semiregu- lar lattice structures called ”alpha-lattices” are introduced MS34 to represent the dynamic topology of inter-agent interac- Addressable Excitable Media for Modeling Collec- tions. It turns out that self-assembly and landscape prop- tive Behavior erties of alpha-lattices are instrumental in analysis of our swarming algorithms. Formal results are established on We discuss two topics of collective behavior in the con- self-alignment and spatial order of particle-based swarms. text of excitable media models, swarming behavior and Moreover, a multi-species particle system is used to design spatiotemporal networks. Studies of controlling reaction- 128 DS05 Abstracts

diffusion waves with realistic excitability potentials are de- MS35 scribed. We also describe a study of dynamical networks Self-Similarity in Burgers Turbulence in the photosensitive Belousov-Zhabotinsky reaction. We model local nearest-neighbor interactions by the spread of I will describe simple and optimal conditions for universal- reaction-diffusion waves, while nonlocal excitations are de- ity and non-universality in Burgers turbulence (the statis- scribed by nondiffusive jumps along shortcuts defined in tics of the Cole-Hopf solution to Burgers equation with ran- the medium. dom initial data). The results depend on a striking connection between Burgers turbulence and Smoluchowski’s Kenneth Showalter coagulation equation discovered by Bertoin and our classi- West Virginia University fication of dynamic scaling in Smoluchowski’s coagulation Department of Chemistry equation. [email protected] Robert L. Pego Carnegie Mellon University MS34 Department of Mathematical Sciences Controlling Faraday Wave Interactions via Multi- [email protected] frequency Forcing Govind Menon Faraday waves form on the surface of a fluid when it is Applied Mathematics, Brown University subjected to a sufficiently strong vibration. Experiments [email protected] performed over the last decade have shown that the spatiotemporal form of the standing wave pattern depends on the frequency content of the periodic forcing function. We use MS35 a framework of equivariant bifurcation theory to show how Ultimate Dynamics on the Scaling Attractor for to design parametric forcing functions in order to enhance Smoluchowski’s Coagulation Equations or inhibit weakly nonlinear three-wave interactions in the Faraday system. We describe a basic framework for studying dynamic scaling that has roots in dynamical systems and probability Mary C. Silber theory, and consider coagulation equations with rate ker- Northwestern University nels K =2,x + y, xy. We previously classified all self- Dept. of Engineering Sciences and Applied Mathematics similar solutions (fixed points under scaling) and their do- [email protected] mains of attraction. Now we show the dynamics on the set of all limit points modulo scaling (the scaling attractor) Chad Topaz is linearized by Bertoin’s Lévy-Khintchine representation University of California, Los Angeles for eternal solutions, and this dynamics is chaotic. The [email protected] scaling attractor contains a dense set of scaling-periodic solutions, and solutions with dense trajectories, analogous Jeff Porter to Doeblin’s universal laws in probability theory. Universidad Complutense Madrid jport@fluidos.pluri.ucm.es Robert L. Pego Carnegie Mellon University Department of Mathematical Sciences MS35 [email protected] Self-Similar Conformations in Glassy Wrinkled Membranes: From Casacade-Like Wrinkling to a Govind Menon Sub-Critical Compaction Applied Mathematics, Brown University [email protected] Partially polymerized membranes display a striking mechanical transition at low temperature known as the wrinkling transition. Fluorescence and scanning electron micro- MS35 scope as well as profile measurements using an atomic force Coarsening in Thin-Film Equations: Upper Bound microscope (AFM) revealed the existence of three degrees on Coarsening Rate of wrinkling depending on the degree of the membrane polymerization. At low polymerization the membrane un- Thin, nearly uniform layers of some fluids can destabilize dergoes a cascade of wrinkling to form a folded phase with under the effects of intermolecular forces. After the ini- a roughness exponent η equal to 3, at intermediate poly- tial phase, the fluid breaks into droplets connected by an merization, the membrane is in an intermediate-wrinkled ultra-thin layer of fluid. This structure coarsens on slow- phase (similar to the crumpling of an elastic sheet) with time scale. The characteristic distance between droplets η ∼ 2.5, while at high polymerization the membrane un- and their size grow, while their number is decreasing. This dergoes an abrupt and sub-critical “compaction” to the physical process can be modeled in the lubrication approx- wrinkled-rough phase and η ∼ 2. These phases, into which imation by a so called thin-film equation for the height of the membranes lock, are glassy roughened conformations the fluid. I will discuss coarsening in thin-film equations with memory depending on the degree of polymerization. with mobility equal to the height of the fluid. These equations are gradient flows in the Wasserstein metric. Using Sahraoui Chaieb the gradient flow structure within the Kohn-Otto frame- Dept. of Theor. & Appl. Mech. work we obtain rigorous upper bound on the coarsening Univ. of Ill., Urbana-Champ. rate. The upper bound we obtain coincides with the coars- [email protected] ening rate that Glasner and Witelski conjectured for the 1-dimensional problem. Dejan Slepcev DS05 Abstracts 129

UCLA Equations Mathematics Department [email protected] It is shown that canards, which are periodic orbits for which the trajectory follows both the attracting and re- Felix Otto, Tobias Rump pelling part of a slow manifold, can exist for a two- University of Bonn dimensional reduction of the Hodgkin-Huxley equations, [email protected], [email protected] which model the generation of action potentials for a squid bonn.de giant axon. By smoothly connecting stable and unstable manifolds in an asymptotic limit, the parameter value at which canards exist for this system are predicted with great MS35 accuracy. Coarsening Dynamics of the Convective Cahn- Jeff Moehlis Hilliard Equation Dept. of Mechanical and Environmental Engineering University of California – Santa Barbara Stephen J. Watson [email protected] Engineering Sciences and Applied Mathematics Northwestern University [email protected] MS36 Resonance in Stellate Cells of the Medial Entorhi- nal Cortex: A Canard Mechanism? MS35 Viscous Entrainment: Singular and Nearly Singu- Stellate cells in layer II medial entorhinal cortex exhibit lar Spouts subthreshold resonance at theta frequencies (8-12 Hz). We study this phenomenon using a biophysical model of A small air bubble rising in syrup remains spherical. A Hodgkin-Huxley type. In the subthreshold regime our larger air bubbles deforms, developing an increasingly ta- model reduces to a three-dimensional system with one fast pered trailing end. For an even larger air bubble, the ris- and two slow variables. We show that the resonance effect ing movement so severely deforms the bubble that a thin is is the result of an underlying canard structure that also tendril of air is deposited behind the rising bubble. We an- governs intrinsic subthreshold oscillations. We generalize alyze such entrainment dynamics in a simple model prob- our studies to noisy and spiking systems. lem where a long-wavelength model describes the essential dynamics. We show that both continuous and weakly dis- Nancy J. Kopell continuous entrainment transitions are possible when the Boston University interface shape on the largest length-scales is constrained Department of Mathematics so that the base of the entrained tendril approaches a coni- [email protected] cal shape. Finally, we show that two kinds of critical transition exist in the full problem because the scale-invariant Andreas V.M. Herz dynamics supports a saddle-node bifurcation. After the Humboldt Universitaet zu Berlin bifurcation, scale-invariant solutions which can link onto [email protected] physical large-scale conditions do not exist. Tim Oppermann Wendy W. Zhang Humboldt Universitaet zu Berlin Physics Department & James Franck Institute Berlin, Germany University of Chicago [email protected] [email protected] Horacio G. Rotstein MS36 Center for Biodynamics Boston University A Heuristic Explanation of a Trapping Mechanism [email protected] That Results in Large Interspike Intervals in the HH Equations with Excitatory Coupling MS36 Excitatory self coupling in a reduced Hodgkin Huxley system is explored. The excitatory coupling causes an unex- Subthreshold Oscillations and Spiking in a Medial pected effect on a system where the neurons rapidly ’syn- Entorhinal Cortex Stellate Cell cronize’ and maintain very long ISIs (inter spike intervals). Medial entorhinal cortex stellate cells (SC) develop low- The trapping mechanism is a vortex induced by the rela- amplitude rhythmic subthreshold oscillations at theta fre- tive movement of the slow synaptic variable and the slow h quencies (8-12 Hz) when depolarized to a value below spik- inactivation variable, . Unlike many canards, this occu- ing threshold. As the membrane potential approaches spik- rance is not extremely sensitive to choice of parameters and ing threshold, SCs fire action potentials at the peak of the persists over an interval of parameter sets. oscillations but not necessarily at every cycle. In this work Jonathan Drover we show that the observed mixed-mode oscillations are gen- University of Pittsburgh erated via a canard mechanism in a SC model having a two- Pittsburgh, PA, USA component hyperpolarization-activated and a persistent- [email protected] sodium currents Nancy J. Kopell MS36 Boston University Department of Mathematics Canards for a Reduction of the Hodgkin-Huxley [email protected] 130 DS05 Abstracts

Horacio Rotstein Caltech Boston University [email protected] Boston, USA [email protected] Shane Ross USC [email protected] MS36 Panel Discussion Wang Sang Koon Control and Dynamical Systems Horacio Rotstein California Institute of Technology Boston University [email protected] Boston, USA [email protected] Jerrold E. Marsden California Inst of Technology Martin Wechselberger Dept of Control/Dynamical Syst The Ohio State University [email protected] [email protected] MS37 MS36 Controlling Mixing in Three-Dimensional, Volume- Significant Slowing of Firing Rates in Neurons Preserving Mappings Through the Canard Phenomenon The motion of a passive scalar in time-periodic, incom- Recent work on Hodgkin Huxley type neurons respectively pressible fluid flows, such as those in certain Rayleigh- neural networks showed a significant slowing of the firing Bernard instabilities, can be modeled by volume-preserving rate under certain cirumstances. This slowing of the firing mappings. Here we investigate a class of flows we call rate is accompanied by subthreshold oscillations near the ”blinking-rolls” and the optimization of mixing behavior action potential threshold. I show that canards are respon- in these maps under changes in their parameters. sible for that delay and line out how to identify this canard phenomenon in biophysical problems. Keith Julien University of Colorado Martin Wechselberger [email protected] Ohio State University Mathematical Biosciences Institute James D. Meiss [email protected] Applied Mathematics [email protected]

MS37 Paul Mullowney Channelling Chaos by Building Barriers University of Colorado [email protected] Chaos often represents a severe obstacle for the set-up of many-body experiments, e.g., in fusion plasmas or turbulent flows. We propose a strategy to control chaotic dif- MS37 fusion in conservative systems. The core of our approach Dynamics and Control of Macromolecules is a small apt modification of the system which channels chaos by building barriers to diffusion. It leads to practical We study a network of nearest-neighbor coupled oscillators prescriptions for an experimental apparatus to operate in starting with a simple coarse-grained model of a molecule a regular regime (drastic enhancement of confinement). with a backbone and side-chains. We show that this system is particularly good at reacting responsively to local- Cristel Chandre ized disturbances by amplifying them. We present analysis Centre de Physique Theorique - CNRS of an interesting transition phenomenon between global en- Marseille, France ergy minima of the system and relate this to recent results [email protected] on controllability of Hamiltonian systems. More generally, oscillator networks that exhibit such phenomena are char- MS37 acterized by nearest neighbor interactions of a node that are, in most of the phase space, much stronger than the Reaction Rates of Chemical Systems with Three or nonlinear oscillations of the local dynamics at the node. More Degrees of Freedom Igor Mezic We merge the ideas of tube dynamics and reaction is- University of California, Santa Barbara land theory into a comprehensive theory of chemical re- [email protected] action rates that overcomes some of the problems plagu- ing the Transition State Theory. By computing normally hyperbolic invariant manifolds and their stable and unsta- MS37 ble manifolds, and merging them with Monte Carlo meth- Control of Resonances in Hamiltonian Systems ods to compute the volumes of tube intersections within Poincare sections, we provide the theoretical and computa- To achieve large changes in adiabatic invariants using tional tools for computing accurate chemical reaction rates small control inputs, a conservative dynamical system must in realistic systems. posses an internal resonance. We propose a control method to use capture into resonance to transport particles. When Frederic Gabern DS05 Abstracts 131

the nominal dynamics brings particles close to a resonance block of cardiac tissue with straight fibers immersed in a surface, a control pulse forces the capture of particles into conductive bath. Intracellular conductivities were varied resonance. Captured particles is transported across the en- stochastically around nominal values with variations of up ergy levels and then released when the desired energy level to 50%. A single rotor reentry was initiated and, by ad- is achieved. justing the spatial variation of action electrical parameters the level of organization could be controlled. The single Dmitri L. Vainchtein, Igor Mezic rotor could be stabilized or spiral wave breakup could be University of California, Santa Barbara provoked leading to fibrillatory-like activity. For each level [email protected], [email protected] of organization, multiple shock timings and strengths were applied to compute the probability of shock success as a function of shock strength. Our results suggest that the level of the small-scale conductivity fluctuations is a very MS38 important factor in defibrillation. Spatiotemporal Dynamics as a Function of Connec- tivity in Cardiac Cell Cultures L. Joshua Leon University of Calgary, Canada Monolayer cultures of cardiac cells form a biological ex- [email protected] citable medium that display different spatiotemporal patterns, ranging from target patterns to multiple spiral Edward Vigmond waves, depending on experimental conditions. Monolay- Department of Electrical and Computer Engineering ers are optically mapped under different concentrations University of Calgary of pharmacological agents that alter cell-cell connectivity. [email protected] Spatiotemporal dynamics are then characterized by an entropy measurement, which facilitates comparison to a simple theoretical model. Gernot Plank Karl Franzens University at Graz, Graz, Austria Gil Bub [email protected] SUNY Health Science Center at Brooklyn [email protected] MS38 The Dynamics of Multiple Spiral Waves As a MS38 Stochastic Predator-Prey System Parallel Implicit Methods for the Bidomain Equa- tions We have found that a system containing many unstable spiral waves can be modeled as a stochastic predator-prey In the bidomain model of cardiac electrophysiology, cur- system. If cells in the excited state are considered predators rent conservation takes the form of an elliptic constraint and cells in the recovered state prey, circular orbits in phase on the potentials defined for the intracellular and extracel- space consistent with predator-prey behavior are observed. lular compartments. To satisfy this constraint, a system of The mean time to spiral wave extinction is much larger equations must be solved at each timestep, making implicit closer to the inferred fixed point, suggesting a predictive time discretizations a natural approach. We describe both role for the predator and prey quantities. linearly and nonlinearly implicit schemes for the bidomain equations, discuss their comparative accuracy and efficient Sandeep Mannava parallel implementation, and present fully resolved three- Department of Biology dimensional computed dynamics. Cornell University [email protected] Charles S. Peskin Courant Institute of Mathematical Sciences Robert F. Gilmour Jr. New York University Department of Biomedical Sciences [email protected] Cornell University [email protected] Boyce E. Griffith New York University Niels F. Otani griffi[email protected] Department of Biomedical Science Cornell University [email protected] MS38 Mathematical/computer Modeling of the Effects of Inhomogeneities on Cardiac Defibrillation MS38 Subcellular Turing Instability Mediated by Voltage Cardiac fibrillation is the deterioration of the heart’s nor- and Calcium Diffusion in Cardiac Cells mally well organized activity into one or more meandering spiral waves which subsequently break up into many me- We investigate the spatiotemporal dynamics of subcellular andering wavefronts. If it is not treated immediately it calcium alternans within cardiac cells. We show the exis- leads to death. The only way to stop fibrillation is by tence of a pattern forming instability that leads to spatially the application of an electric shock (defibrillation). This discordant calcium alternans. This instability is mediated study focuses on examining whether higher degrees of dis- by the diffusion of membrane voltage and intracellular cal- organization of the fibrillation has an effect on the shock cium, and is found to rely on the bidirectional coupling strength required to defibrillate, and whether microscopic between these species. We describe the conditions for this conductivity fluctuations favor shock success. We devel- instability to occur, and present a mathematical descrip- oped a three dimensional computer bidomain model of a 132 DS05 Abstracts

tion of the pattern formation process. flows: shear instability and breaking waves. In the context of hydrostatically balanced flows, these correspond to Yohannes Shiferaw transitions to elliptic behavior of an otherwise hyperbolic UCLA School of Medicine system, and nonlinear wave steepening. We ask whether [email protected] shear instability can occur at all in unforced flows. Along the way to answering this basic question, surprising features of the hierarchy of models describing hydrostatically MS38 balanced flows, from single-layered to continuously strati- Field Stimulation of Heart Tissue: Surface Polar- fied, are revealed. ization Versus Intramural Virtual Electrodes Esteban G. Tabak The mechanism by which electric fields terminate arrhyth- Courant Institute mias continues to puzzle investigators. Existing experi- New York Universityb mental methods provide information about epicardial man- [email protected] ifestations of electrical cardioversion, yet little is known about field effects inside the myocardium. We combined specially designed optical mapping experiments and com- MS40 puter modeling to separate the intra-myocardial and sur- Travelling Solitons in the Discrete Nonlinear face field effects. We find only minor surface polarization Schroedinger Equation during field stimulation. Intramural virtual electrodes produced even by weak fields are sufficiently strong to initiate Asymptotic methods are used to study moving discrete intra-myocardial excitation. solitons on the line, with particular attention to the existence of radiation tails. We also study discrete periodic Christian Zemlin, Arkady Pertsov waves on finite intervals. SUNY Upstate Medical University [email protected], [email protected] Oliver Oxtoby, Igor Barashenkov University of Cape Town [email protected], [email protected] MS39 Energy Spectra of the Ocean’s Internal Wave Field: Theory and Observation MS40 Embedded 2pi-Kinks in the Generalised Sine- Gordon Lattice: A New Barrier Yuri V. Lvov Rensselaer Polytechnic Institute An explanation is offered for an observed lower bound on [email protected] the wave speed of travelling kinks in Frenkel Kontorova lattices. Kinks exist at discrete wavespeeds within a parameter regime where there is resonance with linear waves MS39 (they are embedded solitons). However, they fail to exist Nonlinear Stability of Stratified Shallow Water Dy- whenever there is more than one branch in the dispersion namics relation. Novel numerical methods are used to continue curves of 2π- and 4π-kinks in propagation speed, lattice Paul A. Milewski discreteness and a tunable amount of onsite anharmonicity. Dept. of Mathematics This is joint work with Andre Aigner and Vasillis Rothos. Univ. of Wisconsin Alan Champneys [email protected] University of Bristol Dept. of Engineering Maths MS39 [email protected] Effects of Rotation on Breaking Waves MS40 Ruben Rosales Stationary and Travelling Kinks in the Discrete MIT Phi-Four Model [email protected] We introduce several new discretisations of the cubic Klein- Gordon equation (the so-called phi-four theory) which ex- MS39 hibit continuous families of (stationary) kink solutions. We Near Resonances and Symmetry Breaking in Ro- study the existence of travelling kinks in these models, as tating Turbulence well as in discretisations proposed previously. Our method is based on the asymptotic analysis beyond all orders of the perturbation theory. Leslie Smith Mathematics Dmitry Pelinovsky University of Wisconsin McMaster University, Canada [email protected] [email protected]

Oliver Oxtoby, Igor Barashenkov MS39 University of Cape Town Stability of Stratiﬁed Sheared Flows [email protected], [email protected] We contrast two distinct scenarios for mixing of stratiﬁed DS05 Abstracts 133

MS40 in a Globally Coupled System Classical and Quantum Peierls-Nabarro Barriers We evidence numerically and experimentally that advec- Usually, discrete solitons experience a potential barrier as tion can induce spectrotemporal defects in a system with they travel through the lattice, which can induce fast ra- a localized solution. We start with the study of a free- diative deceleration and trapping. We will review various electron laser, which is described by a one-dimensional discrete systems where this barrier is totally absent, de- convection-diﬀusion model with global saturation coupling. scribing their possible physical applications, then demon- Then, we show that the main features of the instability are strate that, even in the absence of a classical barrier, the kept in simple Ginzburg-Landau equations with advection. phonon Casimir energy induces a quantum energy barrier The situations of local and global coupling are compared. of generically similar type. Thus a purely quantum kink trapping mechanism arises in these systems. Christelle Bruni, David Garzella, Gian-Luca Orlandi, M.-E. Couprie James M. Speight CEA/LURE School of Mathematics France University of Leeds [email protected], [email protected] [email protected], [email protected], [email protected]

MS40 Christophe Szwaj Stokes Constant for Discrete Maps Universite de Lille (France) Laboratoire PHLAM [email protected] Alex Tobvis Department of Mathematics University of Central Florida Serge Bielawski [email protected] PhLAM/Université Lille I, France [email protected] MS40 Moving Breathers in One-Dimensional FPU and Two-Dimensional FPU-Klein-Gordon Lattices MS41 Bumps, Breathers, and Waves in a Neural Network We seek moving breather solutions in one-dimensional with Threshold Accommodation Fermi-Pasta-Ulam (FPU) chains with cubic-quartic potential energy functions. We find a family of solutions which I will discuss a continuum model of neural tissue that in- can be continuously varied from traditional breathers to cludes the effects of accommodation. The basic model is an breathing-kink travelling waves. The breathing-kink solu- integral equation for synaptic activity that depends upon tions have a kink amplitude which can be arbitrarily small, the non-local network connectivity, synaptic response, and (the quartic FPU lattice has a threshold below which no firing rate of a single neuron. A phenomenological model classical kinks exist). Our approximations use multiple of accommodation is examined whereby the firing rate is scales asymptotic techniques, which we then use to inves- taken to be a simple state-dependent threshold function. tigate breathers in two-dimensional systems. As in the case without accommodation classical Mexican- Hat connectivity is shown to allow for the existence of spa- Jonathan Wattis, Imran Butt tially localised states (bumps). Importantly an analysis School of Mathematical Sciences of bump stability using recent Evans function techniques University of Nottingham shows that bumps may undergo instabilities leading to the [email protected], emergence of both breathers and travelling waves (and in- [email protected] deed travelling breathers). Numerical simulations show that bifurcations in this model have the same generic properties as those seen in many other dissipative systems that MS41 support localised structures, and in particular those of cou- Bifurcations in Neural Fields with Delays pled cubic complex Ginzburg-Landau equations and three component reaction diffusion systems. A class of integro-differential equations modelling neural activity is considered which take into account the finite Markus Owen, Stephen Coombes speed of transmission along axons and cortico-thalamic University of Nottingham feedback delays. The stability of spatially homogeneous [email protected], equilibria and the bifurcations leading to spatial patterns, [email protected] oscillations, and traveling waves are investigated. A perturbative method is presented which allows the determination of the bifurcations for arbitrary interaction kernels. MS41 Stability of Nonlocal Neural Fields Subject to Ad- Fatihcan M. Atay ditive Stochastic Forces Max Planck Institute for Mathematics in the Sciences, German The spatiotemporal dynamics of neural populations has at- [email protected] tracted much attention in recent years. By virtue of the network of axonal connections between neurons, there are nonlocal interactions yielding various spatial patterns. The MS41 presented work discusses the stability of the neural field Spectro-Temporal Defects Induced By Advection subjected to general external stimulus. Further the emergence of the Turing instability is discussed subjected to 134 DS05 Abstracts

stochastic forces. We derive the order parameter equation Jonathan N. Blakely and find stability threshold shifts dependant on the noise US Army RDECOM level. [email protected] Axel Hutt Applied Stochastic Processes, Humboldt University Berlin MS42 [email protected] Beam Steering by Lag Synchronization in Wide- band Chaotic Arrays Lutz Schimansky-Geier Institute for Physics Chaotic oscillators are an intriguing source of waveforms Humboldt University at Berlin for ultra wideband-radar applications. The broadband and [email protected] nonrepeating nature of chaos provides an ideal combination of high range resolution and zero range ambiguity. Array synchronization via local coupling is an efficient alternative MS41 to the use of a master oscillator in a phased array; however, Q-Switching Instability in Passively Mode-Locked a difficulty in any wideband array is achieving a practical, Lasers cost-effective mechanism for beam steering. Here, we explore the use of lag and anticipating synchronization in a A new asymptotic analysis of the equations for a mode- chaotic array to achieve steering. The natural frequency of locked laser exhibiting a Q-switching instability is pro- individual oscillators is adjusted to control a uniform lag posed. The averaging of the fast time in the derivation across the array. The direction and extent of the steer- of the model introduces a non-local term in the equation ing is directly controlled by the orientation and magnitude for the population inversion. We show that this non-local of the induced lag or anticipation. Using chaotic circuits term leads to a Hopf bifurcation and we derive analytical operating at 20 MHz, we show that lag synchronization thresholds for it. Our analysis is valid for all class B lasers can be practically controlled in arrays of rf devices. Open exhibiting slowly damped relaxation oscillations which in- questions include determining stability and dynamical lim- clude solid state and semiconductor lasers. Our analytical itations for steering the larger arrays that practical appli- results are shown to compare favorably with numerical sim- cation will require. ulations. Ned J. Corron, Shawn Pethel Theodore Kolokolnikov U.S. Army RDECOM Free University of Brussels [email protected], [email protected] [email protected] Jonathan N. Blakely US Army RDECOM MS41 [email protected] Localized Vegetation Structures in Arid Land- scapes MS42 Localized vegetation structures are large-scale vegetation Hopf Bifurcations in Time-Delay Systems with patterns observed worldwide in arid and semiarid territo- Band-Limited Feedback ries. Single localized vegetation structure consists of either an isolated patch surrounded by a bare state or a bare spot We investigate the steady-state solution and it’s bifurca- in sparsely vegetated area. Their spatial distribution can tions in time-delay systems with band-limited feedback. be either ordered or random. A model and a non-linear This is a first step in a rigorous study concerning the ef- analysis are presented to account for their formation and fects of AC-coupled components in nonlinear devices with study their interactions. time-delayed feedback. We show that the steady state loses stability, generically, via a Hopf-bifurcation and we deter- Mustapha Tlidi mine whether the Hopf bifurcation is supercritical or sub- université Libre de Bruxelles critical. Furthermore, the presence of double-Hopf bifurca- Optique nonliéaire Théorique tions is shown, which indicates possible quasiperiodic and [email protected] chaotic dynamics. Daniel J. Gauthier, Lucas Illing MS42 Duke University Multimode Dynamics of a Transmission Line Os- [email protected], [email protected] cillator

A transmission line oscillator consists of a length of trans- MS42 mission line terminated with a negative resistor and a New Directions in Nonlinear Delay Optoelectronic diode. The complexity of the dynamics of this oscillator Oscillators for Chaos Generation and Application depends on the number of active modes in the line. With to Wideband Communication Systems a fixed bandwidth and a very short transmission line, only a single mode is active. A longer line allows multiple modes Ikeda-like dynamics have been already intensively studied to oscillate and interact. We present an experimental in- from a fundamental point of view, both for their extremely vestigation of a line with two active modes. simple mathematical modeling (scalar nonlinear difference differential equation) together with their high complexity Ned J. Corron, Shawn Pethel dynamical behavior. However, its experimental implemen- U.S. Army RDECOM tation was often performed by optoelectronics equivalent [email protected], [email protected] setups due to practical limitations. More recently, many such optoelectronics nonlinear delay dynamics have been DS05 Abstracts 135

developed for wideband chaos communication applications. smooth submanifolds are carefully chosen. Furthermore, New dynamical models appeared with these optoelectronic the Jacobian of the holonomy map also depends diﬀeren- oscillator. tiably on the diﬀeomorphism. Laurent Larger Miaohua Jiang Universite de Franche-Comte Wake Forest University UMR FEMTO-ST 6174 Department of Mahematics [email protected] [email protected]

Pierre-Ambroise Lacourt, Stéphane Poinsot, Aurélien Pallavisini MS43 GTL-CNRS Telecom Billiards with Moving Walls UMR FEMTO-ST 6174 [email protected], spoinsot@georgiatech- We consider a particle inside a billiard table with periodi- metz.fr, [email protected] cally moving walls and show that under certain conditions the energy of some orbits approaches infinity at an affine rate. This behavior coexists with KAM tori of lower di- MS42 mension. Dynamics of Chaotic Blocking Oscillators Mark Levi Simple modification of a well-known circuit of blocking os- Department of Mathematics cillator can lead to the onset of chaotic oscillations. The Pennsylvania State University output signal of such chaotic oscillator is a series of short- [email protected] term pulses characterized by chaotic fluctuations of time intervals between pulses and has an Ultra-Wide Band con- MS43 tinuous power spectrum. We discuss the results of theoretical and experimental studies of nonlinear dynamics of Fixed Points for Commuting Diffeomorphisms on the oscillator. The bifurcation scenarios responsible for the the Sphere onset and development of chaos are considered. Let G be a finitely-generated abelian group acting smoothly on the two-dimensional sphere. Then there exists Nikolai Rulkov G Univ of California / San Diego an index-two subgroup of that acts with a global fixed Inst for Nonlinear Science point. [email protected] Kamlesh Parwani University of Houston Alexander Volkovskii [email protected] Inst for Nonlinear Science Univ of California/San Diego [email protected] MS43 Nonuniformly Hyperbolic Dynamical Systems and Periodic Orbits MS42 High-Frequency Implementation Approaches and Consider a smooth diffeomorphism on a compact Rieman- Circuit Issues for PWL Chaotic Oscillators nian manifold preserving an ergodic hyperbolic measure of positive metric entropy. Assume that the metric entropy is The robust implementation of high-frequency chaotic oscil- not locally maximal in the class of invariant ergodic hyper- lators is essential to the development and successful tech- bolic measures. We show that there exist multiplicatively nology insertion of many proposed chaos-based communi- many periodic orbits equidistributed with respect to this cations and signal processing applications. This talk will measure. tell the story of one journey towards arriving at such a robust design, covering such topics as motivation for choos- Ilie Ugarcovici ing PWL oscillators, the two basic dynamical system ap- Rice University, Houston proaches taken (autonomous and nonautonomous) for the [email protected] oscillator implementation, and the various circuit issues encountered and overcome along the way. MS43 Christopher P. Silva, Albert Young An Example of the CLT in the Absence of Mixing The Aerospace Corporation [email protected], [email protected] The Central Limit Theorem is often obtained as a consequence of rapid mixing. However mixing is not required for the CLT. We consider the CLT for regular observations on MS43 non-mixing toral extensions of hyperbolic basic sets. We Smooth Dependence of Holonomy Maps of Hyper- consider constant extensions and show that under appro- bolic Systems priate arithmetic assumptions on the extension and under mild mixing assumptions on the base we get a CLT. For a diffeomorphism possessing a hyperbolic attractor, the stable foliation induces a holonomy map between two Alistair Windsor smooth submanifolds transversal to the foliation. Even University of Texas, Austin though this holonomy map is only a Hölder continuous [email protected] map, in general, the dependence of the holonomy map on the diffeomorphism is differentiable when the transversal Ian Melbourne 136 DS05 Abstracts

University of Surrey, UK orem and its corollaries restrict the possible spatial and [email protected] spatiotemporal symmetries of periodic solutions in equivariant phase oscillator networks and also guarantee that periodic solutions with certain symmetries will exist for MS43 open sets of such systems. The network architecture again Recent Progress in Regularity of Cohomology plays an essential role in determining which spatiotemporal Equations symmetries of the different solutions are permitted. While the results are quite general, we illustrate them in networks We will present some new results on the problem of prov- of phase oscillators coupled through biophysically realistic ing regularity of solutions of cohomology equations over a phase response curves, and H functions. dynamical system. Most of the results discussed will be for systems that have some partial or non-uniform hyperbol- Martin Golubitsky, Kresimir Josic icity, but we will discuss some parabolic systems. University of Houston Department of Mathematics Rafael de La Llave [email protected], [email protected] University of Texas Department of Mathematics Eric Shea-Brown [email protected] Courant Institute, New York University tba MS44 Persistent Gamma Oscillations and Their Possible MS44 Role in Attention The Dynamics of Cognition : Insights and Ques- We model persistent gamma oscillations, characterized in tions from Human Direct Cortical Recordings vitro by infrequent, irregular, phase-locked spiking of ex- For the last few years, we have been recording the intracra- citatory cells. We hypothesize that similar oscillations in nial electroencephalogram of epileptic patients while they vivo are a correlate of sustained attention. Subpopulation- performed a variety of cognitive tasks, involving the per- specific excitation to E-cells drives them to spike at gamma ception of complex objects, language, verbal short-term frequency. Competition between excited subpopulations memory or the generation of occular saccades. The ex- relies on acceleration (not strengthening) of the inhibitory ceptional spatial and temporal resolution of such record- population rhythm. Model M-currents abolish rhythmicity ings gives a view of the brain dynamics in relation to cog- and weaken response to specific excitation. Our findings fit nition that is out of reach by conventional techniques of well with known effects of acetylcholine in cortex. human brain mapping. We will review some attempts to Nancy Kopell understand the organization of this dynamics along spa- Boston University tial,temporal and frequency axis and expose the needs for Department of Mathematics further tools of investigation adapted to this level of obser- [email protected] vation. Jean Philippe Lachaux Christoph Borgers Inserm, Lyon . France Tufts University [email protected] US [email protected] MS44 Steve Epstein Synchrony In Clinical Neurology Boston University Center for BioDynamics Using a new measure (Synchronization likelihood or SL) [email protected] I studied normal and disturbed cognition states. During various types of cognitive tasks, SL reveals changes in specific frequency bands, related to different aspects (working MS44 memory, attention) of information processing. In neuro- Networks of Phase Oscillators: Beyond Symme- logical disorders with cognitive decline (Alzheimers and tries Parkinsons disease) SL shows a decrease L in upper alpha, beta and gamma bands. Magnetoencephalography is Reduced models of oscillators are frequently used in mod- probably more sensitive in detecting these changes than eling neuronal networks. Particularly popular are models EEG. in which the internal dynamics of the oscillators (cells) is one dimensional, such as integrate and fire, and phase os- Cornelius Stam cillators. We show that in networks of such oscillators the VU University Medical Center, Netherlands architecture (connectivity) can force the frequency of os- [email protected] cillations of subsets of cells to be equal. A corollary of this result is that cells in an all-to-all coupled network of phase oscillators all have the same frequency. These results are MS44 generalized to clusters of synchronous oscillators in net- Where Noise and Precision Join Forces: Coding of works with arbitrary architecture using the ”groupoid” de- Neural Information Via Limit Cycles scription of a network’s structure. The relations between the dynamics of the different oscillators and clusters are In biological neural networks, the noise component often described using the notions of rotation number, oscillation is of the same order as the signal strength. This, and the number, and average frequency. We also prove a special (as yet unexplained) computational efficacy of biological case of the H/K theorem for phase oscillators. This the- systems, is of particular interest for technical exploitation, DS05 Abstracts 137

since miniaturization drives hardware chips naturally to- MS45 wards these conditions. We discuss a framework in which Bifurcations From Heteroclinic Networks with Pe- noise and precision are complementarily used to encode in- riodic Orbits formation. Our approach is based on weakly coupled neu- rodynamical limit-cycle solutions, which are investigated We consider bifurcations and bifurcation failure of hetero- under natural conditions of transient temporal behavior. clinic orbits from heteroclinic networks whose nodes involve We have developed tools to show that locking is preserved periodic orbits. A dual point of view is the termination of a under a large variety of such conditions, although in these parameter path at such a network. In some cases, these bi- cases the locked states are difficult to assess. We find that furcations significantly differ from networks where all nodes the range of conditions under which coding by locking is op- are equilibria. We present results showing such differences erational, is large enough for a realization in nature, hinting for simple networks. A major application is the existence, at a large potential also in technical applications. The de- and partially the stability, of travelling waves in spatially scribed coding mechanism may be fundamental for achiev- onedimensional parabolic PDEs. ing the highly efficient computations observed in biological systems. Jens Rademacher University of British Columbia Ruedi Stoop Department of Mathematics Institute of Neuroinformatics ETHZ/UNIZH [email protected] Switzerland [email protected] MS45 Modeling Intracellular Calcium: Diffusion, Do- MS44 mains, and Dynamics Transient Stimulus-Locked Desynchronization and Response Clustering in the Brain Mathematical models of calcium dynamics often represent the interaction of intracellular calcium with endogenous Studies of transient responses of coupled oscillators to brief and exogenous calcium buffers via a system of nonlinear stimuli have led to new insights into the nature of evoked reaction-diffusion equations. For example, emispherically brain responses: (i) Apart from generating a stereotypical symmetric steady-state solutions to the full equations for response, interacting oscillators may also switch between the buffered diffusion of intracellular calcium provide esti- different responses across trials. This can typically not mates of the concentration of free calcium near open cal- be detected with standard averaging. (ii) The standard cium channels. Perturbation methods have provided ap- method for estimating transmission times fails when approximations for steady-state calcium and buffer profiles plied to oscillators. Both predictions have been verified in in two well-understood asymptotic limits. 1) An ”excess a magnetoencephalograhy study with visual pattern rever- buffer approximation” (EBA) where the mobility of buffer sal stimulation. exceeds that of calcium and the fast diffusion of buffer toward the calcium channel prevents buffer saturation (Ne- Peter A. Tass her, 1986; Naraghi and Neher, 1997). 2) A ”rapid buffer Institute of Medicine (MEG) approximation” (RBA), where the diffusive time scale for Research Centre Juelich calcium and buffer are comparable, but slow compared to [email protected] reaction, resulting in saturation of buffer near the calcium channel (cf. Wagner and Keizer, 1994; Smith, 1996). These two limits can also be applied to investigations of the re- MS45 lationship between single-channel kinetics and the collec- Stabilizing Unstable Waves tive phenomena of stochastic Ca2+ excitability, i.e., local calcium elevations known as calcium ”puffs” and ”sparks.” The interaction of calcium and buffers also plays an impor- Chris Jones tant role in the dynamics of both continuous and saltatory Department of Mathematics propagating calcium waves. University of North Carolina [email protected] Gregory D. Smith College of William and Mary [email protected] MS45 Homoclinic and Hopf Bifurcations in Calcium and FitzHugh-Nagumo Models MS46 Multiscale Simulation of Sedimentation A numerical bifurcation analysis is performed of the interaction between a homoclinic and a Hopf bifurcation Sedimentation at small Reynolds numbers is a simple-to- in models of Calcium wave propagation and a FitzHugh- define process at the microscale level of hydrodynamic in- Nagumo model. At first sight, the homoclinic orbit shows teractions between solid particles, but full simulations are some unexpected behaviour when the corresponding equi- computationally intensive. Meanwhile, there is no clear librium undergoes a Hopf bifurcation, but much can be ex- agreement on the precise form of the correct macroscopic plained by the underlying slow manifold and the stiffness continuum models, even in the dilute limit of monodisperse of these slow-fast systems. spheres. In this talk, we will demonstrate attempts to use equation-free methodologies to efficiently capture the rich Bart E. Oldeman behavior of the statistics of particle velocities and the evo- Department of Mathematics lution of the spreading front between particle-laden and University of Auckland clarified flow. [email protected] Peter J. Mucha 138 DS05 Abstracts

Georgia Institute of Technology MS46 School of Mathematics Patch Dynamics With Buﬀers For Multiscale Prob- [email protected] lems

For an important class of multiscale problems, a separa- MS46 tion of scales exists between the available (microscopic) Application of Equation-Free Methods to Models model and the (macroscopic) level at which one would like of Biological Systems to observe the system. For this problenm, Kevrekidis et al. developed a so-called “equation-free” framework, of The movement of many organisms can be described as a which patch dynamics is an essential component. Patch random walk at either or both the individual and pop- dynamics is an algorithm designed to approximate the time ulation level. The rules for this random walk are based evolution of the macroscopic unknowns; it only performs on complex biological processes and it may be difficult to appropriately initialized simulations with the available mi- develop a tractable quantitatively-accurate individual-level croscopic model in small portions of the space-time domain model. However, important problems in areas ranging from (the patches). To obtain convergence, we introduce buffer ecology to medicine involve large collections of individuals, regions around the patches to avoid the (artificially intro- and a further intellectual challenge is to model population- duced) patch boundaries to affect the solution in the in- level behavior based on a detailed individual-level model. ternal region. We show our recent convergence results and Because of the large number of interacting individuals and illustrate the approach by on a set of model problems. because the individual-level model is complex, classical direct Monte Carlo simulations can be very slow, and often of Giovanni Samaey little practical use. In this case, newly developed equation- Department of Computer Science, K. U. Leuven free methods may prove a promising direction for analysis [email protected] and simulation of individual-based models. We will discuss applications of equation-free methods to biological prob- Dirk Roose lems. The biological example here is chemotaxis, but it K.U.Leuven could be any random walker which biases its movement in Dept. of Computer Science response to environmental cues. [email protected] Hans Othmer University of Minnesota Yannis Kevrekidis Department of Mathematics Princeton [email protected] [email protected]

Radek Erban MS47 School of Mathematics Curve Shortening and the Topology of Closed University of Minnesota Geodesics on Surfaces [email protected]

Ioannis Kevrekidis Sigurd B. Angenent Princeton University University of Wisconsin [email protected] Department of Mathematics [email protected]

MS46 Higher Order Accuracy in The Gap-Tooth Scheme MS47 For Large-Scale Solutions Using Microscopic Sim- Constructing Models of Global Attractors for ulators Swift-Hohenberg

We find general boundary conditions for patches of micro- In this talk, we will discuss a rigorous numerical method for scopic simulators that appropriately connect widely sep- the study and verification of global dynamics for gradient arated “teeth” to achieve high order accuracy over the systems. The procedure involved relies on first verifying macroscale. Here we explore the simplest case when the mi- the structure of the set of stationary solutions, as often croscopic simulator is the quintessential example of a par- depicted in a bifurcation diagram produced via continu- tial differential equation. We argue that classic high-order ation methods. This includes proving the existence and interpolation provides patch boundary conditions which uniqueness of computed branches of the diagram as well achieve arbitrarily high-order consistency in the gap-tooth as showing the nonexistence of additional stationary solu- scheme, and with care are numerically stable. This consis- tions. Topological information in the form of the Conley tency is demonstrated: firstly using the dynamical systems index, also computed during this verification procedure, is approach of holistic discretisation; and secondly through then used to build a model for the attractor. As illustra- the eigenvalues of selected numerical problems. tion, we apply this method to the Swift-Hohenberg PDE to produce a conjugacy between the global attractor a con- Tony Roberts structed model system. University of Southern Queensland [email protected] Sarah Day Department of Mathematics Yannis Kevrekidis Cornell University Princeton [email protected] [email protected] DS05 Abstracts 139

MS47 Alex Wurm Structure of the Attractor of the Cahn-Hilliard Dept. of Physics, Fusion Studies Equation University of Texas at Austin [email protected] Abstract: We describe the structure of the attractor of the Cahn-Hilliard equation on two-dimensional square domains in certain parameter ranges. This is accomplished by com- MS48 bining numerical results on the set of equilibrium solutions Destruction of Invariant Circles and Transport with algebraic Conley index techniques such as connection Barrier Location in the Dynamics of Quadratic matrices and transition matrices. In particular, we discuss Nontwist Maps the possibility of saddle-saddle connections. We study the scenario of destruction of invariant circles of Stanislaus Maier-Paape quadratic nontwist maps. It is proved that the main cause RWTH Aachen of their breakup is the folding property of the map. For the Institut f¨ur Mathematik maps corresponding to a point in a distinguished subregion [email protected] of the parametric plane we locate a closed invariant annu- lus and show that outside it the drifting of action occurs. Theoretical results are illustrated for a global model of a MS47 Poincar´e map associated to a poloidal section of a reversed Braids and Parabolic Equations shear tokamak.

The comparison principle for scalar second order parabolic Emilia Petrisor PDEs admits a topological interpretation: after lifting the Department of Mathematics graphs to Legendrian braids, the curves evolve as to de- Politechnic University of Timisoara crease the algebraic length of the braid. Via discretization [email protected] we define a suitable Conley index, which gives a toolbox of purely topological methods for finding invariant sets of Dana Constantinescu scalar parabolic PDEs. There is a close connection with EURATOM MEC Romania twist maps and in this context it applies to (variational) [email protected] fourth order ODEs. Robert Vandervorst Jacques Misguich VU Amsterdam EURATOM CEA Cadarache Department of Mathematics [email protected] [email protected] MS48 Robert W. Ghrist Regularity of Critical Invariant Circles of Non- Department of Mathematics Twist Maps University of Illinois, Urbana-Champaign [email protected] We study critical invariant circles of several noble rotation numbers at the edge of breakdown for area preserving Jan Bouwe Van Den Berg maps of the cylinder which violate the twist conditions. We VU Amsterdam present a high accuracy computation of about 10 million Department of Mathematics Fourier coefficients. This allows us to compute the regular- [email protected] ity of the conjugating maps and show that, to the extent of the precision, it only depends on the tail of the continued fraction expansion. MS48 Renormalization for Critical Shearless Circles Amit Apte Dept. of Mathematics We discuss the numerical evidence for universal rescaling University of North Carolina invariance of shearless critical circles of nontwist maps with [email protected]@email.unc.edu noble rotation number. It is also seen that the nearby periodic orbits map onto each under simultaneous rescaling Rafael de La Llave and shift of rotation numbers. We present a renormaliza- University of Texas tion group (RG) picture to interpret these observations in Department of Mathematics terms of critical fixed points (or higher period cycles) of the [email protected] RG operators acting on the space of maps. We also construct changes of coordinate to relate different fixed points Nikola Petrov to each other. University of Michigan Amit Apte [email protected] Dept. of Mathematics University of North Carolina MS48 [email protected]@email.unc.edu The Diagram for Shearless Torus Breakup and Sep- aratrix Reconnection in the Quadratic Nontwist Philip Morrison Map Physics Department University of Texas at Austin The appearance of a shearless torus and the reconnection of [email protected] separatrices are phenomena peculiar to nontwist systems. For the quadratic nontwist map, these can be systemati- 140 DS05 Abstracts

cally studied by investigating the trajectories of four par- University of Oldenburg ticular points in phase space that are the fixed points of ICBM, Theoretical Physics/Complex Systems transformations related to the map’s symmetry. By us- [email protected] ing these points, a diagram is obtained which reveals the breakup threshold of shearless tori and the reconnection Thilo Gross threshold of separatrices. Fachbreich Physik Universität Potsdam Susumu Shinohara [email protected] ATR Wave Engineering Laboratories [email protected] Dirk Stiefs ICBM, Theoretical Physics/Complex Systems MS49 Carl von Ossietzky University Oldenburg, Germany The Dynamics of Behavioral Choice and its Effects [email protected] on Population Dynamics

I will extend previous work on the dynamics of behavioral MS49 change to consider systems in which several interacting Evolutionary Dynamics of Mixotrophs species adaptively adjust behavioral traits that affect their interaction. A range of different behavioral models is an- We study the evolution of a population of mixotrophs living alyzed. All of the models have the characteristic that the in a water column, where light absorption triggers the build average value of the behavioral trait in each species changes up of concentration gradients. The mathematical model more rapidly when the gain in fitness per unit change in consists of an ecological part that describes the dynamics the behavior is larger. The work will examine the popula- of the population and its nutrients with partial differential tion dynamics of the entire system, and will explore how equations (PDEs) and an evolutionary part that describes these dynamics are affected by: (1) the type of interaction the change of strategies (autotroph and heterotroph abil- between species; (2) the functional relationship between ities) of these populations, parameters in the PDEs, on traits in different species and population growth parame- the slower time-scale.In this approach competition between ters; and (3) the precise form of the relationship between the resident population and a mutant population deter- the fitness gradient and the rate at which behavior changes. mines the evolutionary dynamics.Under certain conditions mutants and residents can coexist leading to evolutionary Peter Abrams branching, i.e. speciation. Dept. of Zoology, University of Toronto Canada Bob Kooi [email protected] Dept. of Theoretical Biology, Vrije Universiteit, Amsterdam The Netherlands MS49 [email protected] Adaptation Model of Phytoplankton Growth Far From Equilibrium Tineke Troost, Sebastiaan Kooijman Dept. Theoretical Biology, Vrije Universiteit Amsterdam We investigate the growth of a phytoplankton population The Netherlands in a chemostat, where the initial state is perturbed far from [email protected], [email protected] equilibrium. Experimental results are presented showing a large range of dynamics regimes, including strong over- shooting, sustained oscillations and non-intuitive response MS50 to nutrient jumps. To describe the experimental findings On the Turbulent Diffusion by Waves we develop a simple chaotic adaptation model taking into account an internal phytoplankton state. Our results give Turbulent diffusion of a passive tracer by a random wave insights into the dynamics of nutrient limited growth far field is believed to be quadratic with respect to the en- from equilibrium. ergy spectrum εk of the velocity field (i.e. proportional to 4, where is the order of the wave amplitudes). So, the Bernd Blasius wave turbulence diffusion (say, on the ocean surface or in Dept. of Physics the air) is often believed to be dominated by the turbulent University of Potsdam diffusion of the incompressible flow. In this presentation, [email protected] we consider a different mechanism of the wave turbulent diffusion and find that the wave turbulent diffusion can be more significant than previously thought. This mecha- MS49 nism works if the velocity field is compressible and statisti- Adaptive Dynamics in General Models cally anisotropic, so that the wave system has a significant Stokes drift. The contribution of this mechanism has a In general models the functions that specify the interaction lower order in . We confirm our results with numerical of model variables are not restricted to specific functional simulations. To derive these results, we develop the Sta- forms. In this way a single general model describes a large tistical near-identity transformation. We also show that a class of different systems. Despite this generality, general small deviation of the energy spectrum εk can lead to a models can be analysed efficiently in the framework of lo- drastic difference in the coefficient of turbulent diffusion. cal bifurcation theory. In this talk we apply the general modelling approach to adaptive dynamical processes. We Aleksander Balk illustrate the formulation and investigation of general mod- The University of Utah els of adaptive dynamics and discuss their advantages. [email protected] Ulrike Feudel DS05 Abstracts 141

MS50 sible origins of the unexpected oscillations. Statistical Physics of Waves Zoltan Neufeld University College Dublin David Cai [email protected] Courant institute New York Unvirsity Jerry Gollub, Paulo Arratia [email protected] Haverford College [email protected], [email protected] MS50 Anomalous Probability of Large Amplitudes in MS51 Wave Turbulence Recent Mathematical Advances on Strange Eigen- modes Brief overview of wave turbulence formalism will be presented. Generalized Random-Phase-and-Amplitude ap- Diffusive tracers tend to develop intriguing patterns when proach will be introduced. This approach will be used mixed by unsteady fluid flows. These complex patterns, or to derive evolution equations for spectrum and Probability strange eigenmodes, appear in a broad class of flows, rang- Distributions Function of waves in a self consistent manner. ing from simple laboratory mixing experiments to simu- It will be shown that stationary state of wave-turbulence lations of atmospheric dynamics. Despite their ubiquity, systems may correspond to intermittency (high probability the patterns have not been understood mathematically. of waves with large amplitudes). In this talk, I discuss recent analytic results on the existence and evolution of strange eigenmodes in two and Yuri V. Lvov three-dimensional fluid flows. I also describe similar re- Rensselaer Polytechnic Institute sults for the wilde eigenmodes observed in dynamo theory. [email protected] George Haller Massachusetts Institute of Technology MS50 Department of Mechanical Engineering Numerical Modeling of Surface Gravity Waves: [email protected] Discrete Resonances

Gravity surface gravity waves has been traditionally in the MS51 focal point of wave turbulence theory. Recenlty it became Chaotic Advection-Diffusion Problems For a Tracer possible to model numerically underlying evolution equa- Subject to Phase Transitions tions of surface waves. Spectral energy density of gravity waves will be measured and compared to measurements of We consider the planar advection-diffusion problem for a other groups and predictions of WT. Significant additional scalar tracer subject to a phase change. An important information will be measured as well, including spectral example of this class of problem is condensation of wa- fluctuations, phase statistics and probability disctribution ter vapor in the Earth’s atmosphere. The target quantity functions of waves. Difficulties of numerical modelling, for understanding in that case is the probability distribu- associated with discreteness of spectral space will be dis- tion of subsaturation, a quantity that has profound im- cussed. It will be demonstrated that statistical steady state plications for the climate of the Earth. The condensation of gravity surface waves correspond to strong intermittency problem is a prototype for a broad class of problems in- for low wave-numbers, due to the flux of probability in an volving nonlinear chemical reactions. We point out that, amplitude space. because condensation is a nonlinear process, the operations of coarse-graining concentrations on a cloud of advected Boris Pokorni particles does not commute with the operation of conden- Rensselaer Polytechnic Institute sation. This means that diffusion cannot be gotten into [email protected] the problem using the familiar trick of adding a random- walk component to deterministic trajectories. Some con- MS51 sequences of this are pointed out. A number of numerical explorations mapping out the behavior of the system are The Interplay of Mixing, Reaction, and Diffusion: presented. Experiments and Simulations Raymond T. Pierrehumbert We investigate a fast acid-base reaction in the presence The University of Chicago of chaotic advection and diffusion, in a thin layer. We Dept. of the Geophysical Sciences compare experimental results to theory and simulations, [email protected] which indicate that after a short transient, the product concentration should increase exponentially at first due to stretching, and then more slowly due to diffusion and the MS51 collision of reaction interfaces. We demonstrate experimen- Topological Kinematics of Mixing tally the close connection between reaction and stretching. However, the reaction product grows more slowly than ex- The orbits of fluid particles in two dimensions effectively pected, possibly as a result of highly non-uniform stretch- act as topological obstacles to material lines. A spacetime ing. Surprising oscillations occur on a time-scale much plot of the orbits of such particles is a braid whose prop- slower than the basic flow period, a phenomenon that is erties reflect the underlying dynamics. For a chaotic flow, not reproduced by the usual model for fast reactions, which the braid generated by the motion of three or more fluid is isomorphic to a passive scalar problem. We discuss pos- particles is computed. A “braiding exponent” is defined to characterize the complexity of the braid. This exponent is 142 DS05 Abstracts

proportional to the usual Lyapunov exponent of the flow. [email protected] Measuring chaos and mixing properties in this manner has several advantages, since neither nearby trajectories nor derivatives of the velocity field are needed. MS52 Synaptic Saturation-Induced Effects on Dynamics Matthew Finn, Jean-Luc Thiffeault of Traveling Waves in Integrate-And-fire Neural Imperial College London Networks matthew.fi[email protected], [email protected] We extend upon the previous theoretical and computational studies of traveling waves of activity in neural tis- MS52 sue by analyzing how saturated synapses change the dy- Stimulus-Induced Traveling Waves and Breathers namics of interacting spiking neurons. The addition of in Neural Networks biologically-motivated pre and post-saturated synapses to the traditional models has the desired property of prevent- We analyze the existence and stability of stimulus-locked ing potential explosion of self-generated network activity. stationary (in 1-D and 2-D) and traveling (in 1-D) pulse Asymptotic analytical solutions are compared to dynam- solutions in synaptically coupled neural networks. The netics generated by numerical simulations for one and two- works are modeled in terms of nonlocal integrodifferential dimensional integrate-and-fire neural networks. equations, in which the integral kernal represents the spatial distribution of synaptic weights and the output firing Remus Osan rate of a neuron is taken to be a Heaviside function of activ- Boston University ity. We explore the emergence of time-periodic pulse-like [email protected] solutions, or breathers, which arise as the system undergoes a Hopf bifurcation, rendering the pulse unstable. MS53 Stefanos Folias Dynamical Systems Approaches to Light Propaga- University of Utah tion in Optical Fibers [email protected] John Abbott MS52 Corning Inc. Heterogeneous Connections Cause Stability [email protected] Changes in Neural Field Propagation

Spatiotemporal pattern formation in physical and chemical MS53 systems is typically based on a dynamics with a homoge- An Epidemiological Model of Alcohol Problem neous, i.e. translationally invariant, connection topology. Treatment However, biological systems like the human cortex show homogeneous connectivity with additional strongly hetero- A model of alcohol-related problem treatment on a social geneous projections from one area to another. Here we network is developed. A discrete model using a cubic non- report how such a dynamic system performs macroscopi- linearity is used on both random and rewired-connected- cally coherent pattern formation and provide a formalism caveman network. The model modiﬁes the likelihood to for its analytic treatment. The connection topology is used develop an alcohol based on a comparison with the aver- systematically as a control parameter to guide the system age of the values at the neighboring vertices. A treatment through a series of phase transitions. We discuss the exam- model is introduced that resets the value at the vertex and ple of a two-point connection and its destabilization mech- holds it constant over a number of time steps. Signiﬁcant anism. dependence on the treatement parameters is found. Viktor Jirsa Richard Braun Florida Atlantic University University of Delaware, Newark, DE Physics Department Department of Mathematical Sciences [email protected] [email protected]

MS52 MS53 Periodic Travelling Waves In the Theta Model For Modeling and Simulation of High-Speed Machining Synaptically Connected Neurons Operations

We present mathematical results obtained about periodic The most basic manufacturing processes involve the me- travelling waves in the theta model for a linear continuum chanical working of a material, resulting in a permanent of synaptically-interacting neurons. In the case of excitable alteration of its shape to produce a ﬁnished component. neurons, we prove that periodic travelling waves exist when Many industrial organizations still use trial-and-error pro- the synaptic coupling is suﬃciently strong, and in that case totyping to select process parameters. This method is at least two periodic travelling waves of each wave-number, expensive, and it often leads to sub-optimal parameter a ‘fast’ and a ‘slow’ one, exist. We also study the limits choices. A necessary step towards improved process control of large wave-number and of small wave-number, for which is the development of better models of these operations. more information can be obtained. Some numerical results, While considerable progress has been made in the devel- and open questions, will also be presented. opment of predictive models for low-strain-rate processes, there is currently a need for improved predictive capabili- Haggai Katriel ties for high-rate processes. In this talk, a survey will be Einstein Institute of Mathematics given of some work in progress at NIST on the the mod- The Hebrew University eling and simulation of some basic high-speed machining DS05 Abstracts 143

operations. MS54 Set Oriented Methods for Optimal Control Prob- Timothy J. Burns lems National Institute of Standards and Technology [email protected] We develop a set oriented method for approximating the optimal value function and approximate optimal trajectories of nonlinear optimal control problems. The idea of the MS53 method is to employ a set oriented approach in order to ex- Control and Connectivity of Networked Dynamical plicitly construct a weighted directed graph that is a finite Systems state model of the original continuous space control system. On this graph, standard graph theoretic algorithms In this work we consider networked dynamical systems con- for computing (all source, single destination) shortest paths trol dependency on available communication links. Using can be applied in order to compute an approximate optimal Graph-theoretic tools we focus on designing networked dy- value function. The associated feedback law gives rise to namical systems decentralized control law. We model over- approximate optimal trajectories and we derive statements all system in a way that allows for both fixed and varying about the performance of this feedback. We also show how communication topology. In the case of variable communi- to generalize this approach to perturbed systems and in cations topology, communication links between individual particular, how robust feedback laws can be constructed. systems are defined based on a predefined proximity rule. Dynamical systems considered are homogeneous, with dis- Hinke M. Osinga crete time dynamics. University of Bristol Department of Engineering Mathematics Anca Williams [email protected] Portland State University [email protected] Oliver Junge Institute for Mathematics Sonja Glavaski [email protected] Honeywell [email protected] Lars Grüne Mathematical Institute MS54 Universitaet Bayreuth Nonautonomous Dynamics of Geophysical Flows [email protected]

Jinqiao Duan MS54 Illinois Institute of Technology Inertial Manifolds For Nonautonomous Skew Prod- [email protected] uct Semiflows and Applications Under the dissipative conditions, the mild solutions of a MS54 nonlinear nonautonomous evolutionary equation du/dt + Some Questions In Control of Nonautonomous Sys- Au = F(u,t) can be formulated as a skew product semiflow tems in a product phase space. We show that there exists an inertial manifold for this skew product semiflow under the Using methods of the theory of nonautonomous lin- spectral gap condition. Instead of using Lyapunov-Perron ear differential systems we generalize the statement of method, we take the approach of conic invariance and in- Yakubovich’s Frequency Theorem from periodic control cremental exponential dichotomy to fulfil the proof, based systems to systems with bounded uniformly continuous on two conic differential inequalities. The construction of coefficients. Then we study nonautonomous H∞ control inertial manifold is through an exponentially tracking in- problmes with infinite horizon. We pass from a Riccati tegral manifold with the aid of Caccioppoli homotopy. An equation to a linear nonautonomous Hamiltonian system. illustration is made by nonautonomous reaction-diffusion Using the concepts of exponential dichotomy and rotation equations. number we define a minimal attenuation value and prove stability when the disturbance is zero. Yuncheng You University of South Florida Russell Johnson [email protected] Universita di Firenze Italy [email protected]fi.it MS55 Patterns of Synchrony in Lattice Dynamical Sys- Roberta Fabbri tems University of Firenze Italy Stewart et al. have shown that flow invariant subspaces for [email protected]fi.it coupled networks are equivalent to a combinatorial notion of a balanced coloring. Wang and Golubitsky have classified all balanced two colorings of planar lattices with either Carmen Nunez nearest neighbor (NN) or both nearest neighbor and next University of Valladolid nearest neighbor coupling (NNN). This classification gives [email protected] a rich set of patterns and shows the existence of many nonspatially periodic patterns in the NN case. However, all balanced two-colorings in the NNN case on the square and hexagonal lattices are spatially periodic. We present 144 DS05 Abstracts

new results showing that all balanced k-colorings in the MS55 NNN case on square, rhombic and hexagonal lattices are Bursting in Coupled Cell Systems spatially periodic. Periodic bursting in fast-slow systems can be viewed as Ana Paula S. Dias closed paths through the unfolding parameters of degen- University of Porto erate singularities. Using this approach, we show that [email protected] bursting in coupled systems can have interesting behavior. We show that coupled systems with Z2 symmetry, Yunjiao Wang particularly in Hopf/Hopf mode interaction and symmetry- University of Houston breaking Takens-Bodganov singularities, lead to interesting USA bursting phenomena. [email protected] LieJune Shiau Fernando M. Antoneli University of Houston - Clear Lake University of San Paolo Department of Mathematics Brazil [email protected] [email protected] Martin Golubitsky, Kresimir Josic Martin Golubitsky University of Houston University of Houston Department of Mathematics Department of Mathematics [email protected], [email protected] [email protected] MS56 MS55 Stochastic Epidemic Models For the Spread of Han- Synchrony-Breaking Bifurcations in Coupled Cell tavirus in Rodents Systems Hantavirus infection is an emerging disease carried by ro- Network architecture can lead to robust synchrony in cou- dents. In rodents, hantavirus infection has little impact pled systems and surprisingly to codimension one nilpo- but in humans, infection is manifested as hantavirus pul- tent normal forms in Jacobians associated to synchronous monary syndrome with a mortality rate as high as 50%. We equilibria. We analyse codimension one nilpotent Hopf present some deterministic and stochastic epidemic models bifurcations that occur generically in three different net- for the spread of hantavirus in rodents. works. Phenomena stemming from these bifurcations in- Linda Allen clude multiple periodic solutions, solutions whose growth Dept. of Mathematics and Statistics rate is faster than the standard 1/2 power, and solutions Texas Tech University whose growth rate is slower than 1/2 power. [email protected] Toby Elmhirst University of Houston MS56 [email protected] TBA Martin Golubitsky University of Houston Julien Arino Department of Mathematics Department of Mathematics [email protected] McMaster University [email protected] MS55 Some Examples of Coupled Cell System Dynamics MS56 TBA We consider examples of coupled cell networks with synchronous dynamics that are unexpected from symmetry considerations but are natural using a theory developed by Carlos Castillo-Chavez Stewart, Golubitsky, and Pivato. Our examples include Department of Mathematics and Statistics patterns of synchrony in networks with small numbers of Arizona State University cells and in lattices (and periodic arrays) of cells that can- [email protected] not readily be explained by conventional symmetry considerations. The examples we consider include a 3-cell system exhibiting equilibria, periodic, and quasiperiodic states in MS56 different cells; periodic 2n × 2n arrays of cells that gen- The Spread of Epidemics: Hantavirus and West erate 2n different random patterns of synchrony from one Nile Virus As Examples symmetry generated solution; and systems exhibiting mul- tirhythms (periodic solutions with rationally related peri- Nitant Kenkre ods in different cells). Some of these phenomena will be Department of Physics discussed in greater detail in other talks in this minisym- University of New Mexico posia. [email protected] Matthew Nicol University of Houston [email protected] DS05 Abstracts 145

MS56 University of Kansas Non-Local Dispersal and the Spatial Spread of Dis- [email protected] ease MS57 Jan Medlock The Evans Function and Fredholm determinants University of Washington Applied Mathematics For quite general nonautonomous first order systems of dif- [email protected] ferential equations, we define the Evans function using Bohl and Lyapunov exponents, and prove that the Evans function is equal to the modified Fredholm determinant of an MS57 associated integral operator with semi-separable kernel. On the Campbell-Sigeti Conjecture Yuri Latushkin, Fritz Gesztesy The Campbell-Sigeti conjecture for dissipative dynamical Department of Mathematics systems says that the sum of positive Lyapunov exponents University of Missouri-Columbia is proportional to the width of the band of analyticity (in [email protected], [email protected] complexified time) of the solutions on the global attractor. We outline a partial analytical result towards solving this Konstantin Makarov conjecture and test it computationally for the Kuramoto- Department of Mathematics Sivashinsky equation. University of Missouri-Columbia [email protected] Alexey Cheskidov Department of Mathematics University of Michigan MS57 [email protected] Nonlocal Models for Directed Self-Assembly of Nano-Particles

MS57 We derive a set of non-local evolution equations describ- Using Svd to Approximate Spectral Intervals of ing directed self-assembly of nano-particles due to mutual Continuous Dynamical Systems attraction. We show that our models exhibit stable generalized solutions as a set of moving delta-peaks (clumpons) , We explore numerical techniques based on the continuous and there is an exact finite-dimensional dynamics describ- Singular Value Decomposition to approximate the Lya- ing the peaks. We also show that our methods can be punov spectrum and the exponential dichotomy spectrum n successfully applied in some problems of crystal growth, of -th dimensional systems of ordinary differential equa- eliminating the need for surface-tension like stabilization tions. Theoretical justifications and numerical examples in models for these phenomena. are given. Darryl D. Holm Cinzia Elia Los Alamos and Imperial College Univ. of Bari, Italy Mathematics [email protected] [email protected]

Luca Dieci Vakhtang Putkaradze School of Mathematics UNM Georgia Tech [email protected] [email protected]

MS57 MS57 An Error Analysis for the Approximation of Lya- Computation of Lyapunov Exponents for Dissipa- punov Exponents tive Systems We present an error analysis for approximating Lyapunov We compute Lyapunov exponents for both the Kuramoto- exponents using QR techniques. The bounds derived are Sivashinsky and 2-D Navier-Stokes equations. A variety a function of the LOCAL error in approximating the time of continuous and discrete QR methods are compared as dependent orthogonal change of variables Q and the de- well as some which incorporate (for the K-S equation) an gree to which there is integral separation. The stability of algorithm to compute inertial manifolds. Lyapunov exponents with respect to perturbation is closely Luca Dieci related to integral separation. School of Mathematics Erik Van Vleck Georgia Tech Department of Mathematics [email protected] University of Kansas [email protected] Mike Jolly Department of Mathematics Indiana University MS58 [email protected] Mixed Mode Oscillations For a Chemical Oscillator

Erik Van Vleck One known mechanism of formation of mixed mode oscil- Department of Mathematics lations is due to the presence of a folded node or a folded 146 DS05 Abstracts

saddle node combined with a suitable return mechanism. [email protected] The mixed mode trajectories arising in this scenario are a combination of rotations about the fold line and large Martin Wechselberger excursions of relaxation type. Simple mixed mode oscil- The Ohio State University lations (many small oscillations followed by one large) are [email protected] quite robust, but solutions invloving more than one con- sequitive large oscillation are much harder to find. In this talk we present ways of finding mixed mode oscillations MS58 based on the theoretical results developed in colaboration Mixed-Mode Oscillations in a Model of Solid Com- with Krupa and Wechselberger. The context for our study bustion is given by a chemical oscillator considered by Jeff Moehlis (J. Nonl. Sci. Vol. 12 : pp. 319-345 (2002). We study a system of three differential equations modeling solid combustion. In the regime near an Andronov-Hopf Morten Brons bifurcation, the system exhibits a variety of oscillatory be- Tech University of Denmark havior. The structure and bifurcations of stable periodic Department of Mathematics solutions are investigated in this work. [email protected] Philip Holmes Maciej Krupa Prog. in Applied and Comp. Mathematics New Mexico State University Princeton University Dept. of Mathematical Sciences [email protected] [email protected] Georgi Medvedev Drexel University MS58 Philadelphia, PA Bifurcation of Neural Models: Multiple Time [email protected] Scales and Canards Yun Yoo Singular perturbation analysis of models for bursting neu- Department of Mathematics ral rhythms relies upon decomposition of models into slow Drexel University and fast time scales. Transitions between spiking and qui- [email protected] escent dynamics correspond to bifurcations of the fast subsystems. We observe that the fast subsystems of bursting models also have two time scales. Consequently, the MS58 bifurcation analysis of the fast subsystems is surprisingly Panel Discussion complex and involves canards. This talk illustrates these phenomena with several examples. Horacio Rotstein John Guckenheimer Boston University Cornell University Boston, USA [email protected] [email protected]

MS58 Martin Wechselberger Ohio State University Mechanism of Mixed Mode Oscillations Via Folded Mathematical Biosciences Institute Node Points [email protected] Folded node points are a special type of fold points occuring in singularly perturbed systems with at least two MS58 slow variables. Typical trajectories passing through the neighborhood of a folded node point make some number Reduced Systems and Canards for Neuron Models of oscillations about the fold line and then follow the re- We study the dynamics of coupled neurons modeled as sin- laxation mechanism. Two limits of folded node are folded gularly perturbed systems of differential equations. The saddle nodes of type I and type II, both characterized by fast/slow decomposition of such systems induces a reduced the number of oscillations increasing indefinitely. All the system that is often simpler to analyze than the full sys- three singularities combined with a suitable return mech- tem. In order for the reduced system to capture the possi- anism give rise to mixed mode dynamics occurring as a ble behaviors and bifurcations of the full system, the phe- combination of small rotations and large excursions of re- nomenon of canards must be incorporated in the reduced laxation type. We present some results on mixed mode model. We will illustrate this process with examples, in- oscillations occurring in the context of folded node and its cluding a model of two coupled neurons. limits. The analysis is based on geometric singular perturbation and the blow-up method. Warren Weckesser Colgate University Morten Brons Department of Mathematics Tech University of Denmark [email protected] Department of Mathematics [email protected] Kathleen A. Hoffman University of Maryland, Balt. Co. Maciej Krupa Deapartment of Math. and Stat. New Mexico State University khoff[email protected] Dept. of Mathematical Sciences DS05 Abstracts 147

John Guckenheimer fined by employing natural symplectomorphisms between Cornell University iterated tangent and cotangent spaces. Variational princi- [email protected] ples in an extended form (the Hamilton-Pontryagin principle) and how they are related to Dirac structures play an important role in the development. Employing the MS59 Lagrange-dAlembert principle, mechanical systems with Orbital Dynamics in Extended Mass Distributions external forces as well as with nonholonomic constraints are put into the context of implicit Lagrangian systems. In this talk I shall discuss aspects of orbital dynamics in Degenerate Lagrangian systems are included in the theory an extended mass distribution. The main focus will be on of implicit Lagrangian systems and will be illustrated using a particular form of the potential that arises from the uni- L-C circuits. versal form for the density profile of dark matter halos as computed via N-body simulations. I will consider stability Jerrold E. Marsden of orbits, analogs of Kepler’s laws and the nature of tidal California Inst of Technology forcing during halo mergers. This is joint work with Fred Dept of Control/Dynamical Syst Adams, Michael Busha and Gus Evrard. [email protected] Fred Adams Department of Physics MS59 University of Michigan Rigid Multi-Frequency N-Vortex Configurations [email protected] On the Rotating Sphere

Anthony M. Bloch The problem of N-point vortices moving on a rotating unit University of Michigan sphere is described. Through a sequence of linear coordi- Department of Mathematics nate transformations, we show how to reduce the problem [email protected] to that on a non-rotating sphere, where the center of vorticity vector is aligned with the z-axis. As a consequence, we prove that integrability on the rotating sphere is the MS59 same as on the non-rotating sphere. Rigid multi-frequency Shape Dynamics and Control conﬁgurations that retain their shape while rotating about two independent axes with two independent frequencies A long thread of investigations in geometric mechanics, are obtained, and necessary conditions for one-frequency chemistry, and statistics has contributed to our current un- and two-frequency motion are derived. Examples including derstanding of shape spaces. In this talk, we explore some dipoles which exhibit global ‘wobbling’ and ‘tumbling’ dy- dynamical systems on shape spaces and associated control namics, rings, and Platonic solid conﬁgurations are shown problems. This work is in part motivated by results of to undergo either periodic or quasi-periodic evolution on a collaboration with Fumin Zhang and Eric Justh on the the rotating sphere. If time permits, we will describe new subject of interacting particle systems. results on particle transport. P.S. Krishnaprasad Houman Shokraneh University of Maryland, College Park Dept. of Aerospace Eng Institute for Systems Research University of Southern California [email protected] [email protected]

Paul K. Newton MS59 Univ Southern California Break-Up of Invariant Curves By Magnetic Field Dept of Aerospace Engineering [email protected] We consider the motion of charged particles in the plane with a periodic magnetic field perpendicular to the plane. If the magnetic field has zero average, KAM theory gives MS59 quasiperiodic motions (under some additional assump- Coupled Rigid-Bodies in Potential Fields tions). Any change of the average of magnetic field from zero destroys these tori. We study the structure of the re- Multibody systems are modeled as two or more coupled maining set, as well as the physical manifestation of the rigid bodies that are connected and can move relative to structure of this set. each other. The dynamics of such coupled rigid bodies in planar motion in a potential field is analyzed. Dynamic Mark Levi coupling between the degrees of freedom gives rise to com- Department of Mathematics plex dynamical systems that are usually not integrable. If Pennsylvania State University the potential field is central, then the free dynamics has [email protected] a symmetry corresponding to a cyclic variable. The free dynamics can be reduced with respect to this symmetry. MS59 Equilibria of the reduced dynamics, which correspond to relative equilibria of the full dynamics, are obtained. The Variational Principles, Implicit Lagrangian Sys- stability of these equilibria is analyzed using the energy- tems and Dirac Structures momentum method when it is applicable; otherwise, an Implicit Lagrangian systems are defined and some of their expansion of the Hamiltonian in normal form is used. The basic properties developed in the context of variational question of integrability of such systems can also be an- principles and Dirac structures. These systems include swered in some special cases, using normal form expan- constrained systems, as well as networks of Lagrangian mechanical systems. Implicit Lagrangian systems are de- 148 DS05 Abstracts

sions. MS60 Modelling Aspects of Vascular Cancer Anthony Bloch University of Michigan The modelling of cancer provides an enormous mathemat- Department of Mathematics ical challenge because of its inherent multi-scale nature. abloch@umich,edu For example, in vascular tumours, nutrient is transported by the vascular system, which operates on a tissue level. Harris McClamroch However, it effects processes occuring on a molecular level. University of Michigan Molecular and intra-cellular events in turn effect the vas- Department of Aerospace Engineering cular network and therefore the nutrient dynamics. Our [email protected] modelling approach is to model, using partial differential equations, processes on the tissue level and couple these to Amit Sanyal the intercellular events (modelled by ordinary differential Arizona State University equations) via cells modelled as automaton units. Thusfar, Mechanical and Aerospace Engineering within this framework we have modelled structural adapta- [email protected] tion at the vessel level and we have modelled the cell cycle in order to account for the effects of p27 during hypoxia. These preliminary results will be presented. MS60 Cellular Automaton Modeling of Biological Pattern Philip Maini Formation Centre for Mathematical Biology University of Oxford [email protected] Andreas Deutsch Technische Universitaet Dresden Zentrum fuer Hochleistungsrechnen MS60 [email protected] Application of Individual-Based Models Of Cell Movement to Primitive Streak Formation in the Chick Embryo MS60 A Cell-Oriented Approach to Developmental Mod- eling Timothy J. Newman ASU The patterns of gene expression are only part of the com- Department of Physics plex set of processes that govern the formation of tissue [email protected] structures during embryonic development. Cells need to differentiate and to migrate long distances through tissues. How do they know what to become and where to MS60 go? Cells secrete and follow gradients of diffusible chem- Chick Embryo Development: Modelling the For- icals (chemotaxis) and secrete non-diffusing extracellular mation of Primitive Streak matrix. In addition, variable adhesion molecules expressed on cells’ surfaces help them to form coherent structures Bakhtier Vasiev by differential adhesion. CompuCell is a public domain University of Dundee modeling environment which implements a simple, energy Department of Mathematics minimization framework to describe these and related mor- [email protected] phogenetic processes.

James A. Glazier MS61 Indiana University, Biocomplexity Institute Depts. of Physics and Biology, School of Informatics Nearly Inviscid Faraday Waves in Nearly Symmet- [email protected] ric Containers Parametrically driven surface gravity-capillary waves in Roeland Merks different containers with weakly broken symmetry are con- Indiana University sidered. In the nearly inviscid regime, the wave amplitudes Biocomplexity Institute interact with an associated streaming flow, and the slow [email protected] dynamics must be described by a set of coupled amplitude- streaming flow equations. This interaction can destabilize pure standing oscillations and give rise to complex time- MS60 dependent dynamics at onset. The results will be explained Cell-Based Model for Chondrogenic Patterning in and related with experiments (Simonelli and Gollub, J. the Chick Limb Bud Fluid Mech. 199, 471, 1989. Feng and Sethna, J. Fluid Mech. 199, 495, 1989).

Maria Kiskowski Edgar Knobloch Vanderbilt University Physics Department Department of Mathematics UC Berkeley [email protected] [email protected]

Maria Higuera E. T. S. I. Aeronauticos Univ. Politecnica de Madrid DS05 Abstracts 149

[email protected] mean modes can alter the formation of patterns, leading to chaotic or localized patterns. In one dimension, patterns can be destabilised by a mean mode, leading to large- MS61 amplitude localized solutions, if the coupling between the Coupled Amplitude-Mean Flow Equations for mean mode and the pattern is sufficiently strong. Hexag- Faraday Waves onal patterns, however, can be unstable for any coupling strength, and form small-amplitude localized states. Amplitude equations for nearly inviscid Faraday waves must take into account the mean flow driven in the nomi- Paul Matthews nally inviscid bulk by Reynolds stresses generated in oscil- University of Nottingham latory viscous boundary layers at the rigid walls and the [email protected] free surface. This flow in turn interacts with the waves responsible for the presence of these boundary layers. The Stephen Cox mean flows enter the theory at the same order in pertur- University of Adelaide bation theory as all other nonlinear terms, even in the zero [email protected] viscosity limit. The resulting description of the system consists of amplitude equations for the waves coupled to Michael Proctor a Navier-Stokes-like equation for the streaming flow with University of Cambridge boundary conditions obtained by matching the bulk flow [email protected] to the boundary layer solutions, and represents a new class of pattern-forming systems. In this talk I will discuss some of the properties of this system, both in small and large MS61 aspect ratio domains. The following speakers will describe Mean Flow in (Rotating or Non-Rotating) particular cases of this system, focusing on drift instabili- Rayleigh-Benard Convection ties and relaxation oscillations. Mean flows are found to be important in Rayleigh-Benard Edgar Knobloch conveciton. In particular, mean flows are found in pat- Department of Physics tern formation near convection onset, and also in turbulent University of California at Berkeley regime for large supercriticality. In the first half of this talk [email protected] we present an overview on mean flow in Rayleigh- Benard convection. In the second half we present recent work on Jose Manuel Vega how mean flow affects (or leads to) the chaotic dynamics in Universidad Politecnica de Madrid rotating convection. We also discuss how mean flow may [email protected] modify the defect velocity statistics.

Maria Higuera Hermann Riecke E. T. S. I. Aeronauticos Northwestern University Univ. Politecnica de Madrid [email protected] [email protected] Yuan-Nan Young Department of Mathematical Sciences MS61 NJIT Viscous Mean Flows in Faraday Waves: Drift In- [email protected] stabilities

The weakly nonlinear dynamics of nearly inviscid Faraday MS62 waves have been shown to be coupled to the associated vis- Discrete Bogdanov-Takens Bifurcation cous mean flow, also called streaming flow, which is produced by time-averaged Reynolds stresses at the oscillatory boundary layers attached to the solid walls and the Vassili Gelfreich free surface. This coupling has a O(1) effect in the dynam- University of Warwick, UK ics beyond threshold. A good example to illustrate this is [email protected] the drift instability of spatially constant waves in annular containers, experimentally encountered by Douady, Fauve & Thual (1989), which have been recently shown to be due MS62 to a reflection symmetry breaking of the mean flow. Bose-Einstein Condensates in the Presence Of a Magnetic Trap and Optical Lattice Elena Martin Universidad de Vigo [email protected] Todd Kapitula University of New Mexico Dept of Math & Statistics Jose Manuel Vega [email protected] Universidad Politecnica de Madrid [email protected] MS62 MS61 Discrete Solitary Waves: Some Old Aspects, Some Recent Results and Some Future Perspectives Localized Patterns and Mean Modes In this short presentation, we will discuss some of the sys- Many different physical circumstances can generate mean tems where discrete nonlinear wave equations apply, such modes or mean flows. In a pattern-forming system, these 150 DS05 Abstracts

as ones in atomic physics and nonlinear optics. We ’ll fo- scale neurobiological networks based on using a system of cus on a prototypical model, namely the discrete nonlin- difference equations to describe a neuron dynamics. The ear Schrodinger equation, that is relevant to these settings existence of intrinsic resonances of reliability of firing and and present, for each of the above two areas, an example the effects of those resonance properties on collective be- of mathematical analysis/numerical computation and the havior in a cortical network models were studied. We show corresponding insights/results that it generated in recent that network interactions can enhance the frequency range experiments. Time permitting, we’ll also examine some of reliable responses and that the resonance boundaries can very recent results on the existence and linear stability of be controlled by synaptic strength. such waves and compare them to numerical findings. Fi- nally, the prototypical unit of a discrete lattice, namely a Maxim Bazhenov double-well potential will also be touched upon and analyt- Salk Institute ical, numerical and experimental results will be presented [email protected] in that context. Nikolai Rulkov Panayotis Kevrekidis Univ of California / San Diego UMass, Amherst Inst for Nonlinear Science Dept of Mathematics [email protected] [email protected]

MS63 MS62 Deriving Phase Models From Data Dissipationless Shock Waves in a Discrete Nonlin- ear Schroedinger Equation Phase models represent a simple and powerful way of reducing complex neural models to simple equations on the It is shown that the generalized discrete nonlinear circle. The key to this reduction is the so-called phase- Schrodinger equation can be reduced in a small ampli- resetting curve (PRC) which describes how the oscillator tude approximation to the KdV, mKdV, KdV(2) or the responds to inputs. We discuss two different methods for fifth-order KdV equations, depending on values of the pa- extracting the PRC from neural firing data. We use the rameter. In dispersionless limit these equations lead to PRC to create maps for instant and synaptic firing between wave breaking phenomenon for general enough initial con- neurons and analyze the resulting equations. ditions, and, after taking into account small dispersion effects, result in formation of dissipationless shock waves. Bard Ermentrout The Whitham theory of modulations of nonlinear waves is University of Pittsburgh, used for analytical description of such waves. The talk is Pittsburgh, PA based on the works done in collaboration with M. Salerno, [email protected] A. M. Kamchatnov and A. Spire, who are gratefully ac- knowledged. MS63 Vladimir Konotop Patterns of Activity in Neural Tissue With Spa- Universidade de Lisboa, tially Modulated Connectivity Portugal [email protected] In my talk I will focus on the dynamics in networks consisting of an excitatory and an inhibitory population of neurons with spatially decaying connectivity. I will show MS62 that the presence of delays give rise to a wealth of bifur- Existence and Stability of Embedded Solitons in a cations and to a rich phase diagram. I will show that the Discrete χ(2): χ(3) Model results derived in that framework allow us to understand the origin of the diversity of dynamical states observed in We report numerical findings of embedded solitons in a large networks of spiking neurons. discrete χ(2): χ(3) model. The model not only represents a discretization of a continuous optical problem but also David Hansel has a straightforward physical realization of an array of Lab. de Neurophys. et de Physiol. du Systeme Moteur optical waveguides. Embedded solitons are computed as Universite Rene Descartes symmetric homoclinic orbits to a saddle-center at the ori- [email protected] gin for a four-dimensional reversible map. Two computer softwares called AUTO and HomMap are used for the computation. Moreover, the linear stability of these embedded MS63 solitons is determined by direct numerical integration of Reproducible Sequence Generation in Random the variational ODE around them. Neural Ensembles

Kazuyuki Yagasaki Little is known about the conditions that neural circuits Gifu University have to satisfy to generate reproducible sequences. Evi- Department of Mechanical and Systems Engineering dently, the genetic code cannot control all the details of [email protected] the complex circuits in the brain. In addition, most of the analysis in networks of dynamical systems deal with the dynamics within attractors. Nevertheless, sensory systems MS63 are known to process information during transient dynam- Oscillations, Resonances and Synchrony in the Net- ics. Here, We investigate the conditions on the connectivity works of Map-Based Model Neurons degree that lead to reproducible and robust sequences in a neural population of randomly coupled excitatory and in- We developed a new approach for analysis of complex large- hibitory neurons. We found that rhythmic sequences can DS05 Abstracts 151

be generated if random networks are in the vicinity of an Department of Mechanical Engineering excitatory-inhibitory synaptic balance. Reproducible tran- University of Rhode Island sient sequences, on the other hand, are found far from a [email protected] synaptic balance. (Results to be published in Huerta, Ra- binovich PRL DEC 2004) MS64 Ramon Huerta Damage Detection of Chaotically-Excited Un- Institute for Nonlinear Science, UCSD, manned Aerial Vehicle Wing San Diego, CA [email protected] Interlaminar failure is a major failure process in carbon- carbon composites, which are commonly used in unmanned Mikhail I. Rabinovich aerial vehicles (UAVs). To automatically assess this failure University of California, San Diego mode, we excite a UAV wing with a Lorenz driver, and ap- Institute for Nonlinear Science ply a discriminator based on a feature vector consisting of [email protected] several measures, including nonlinear cross-prediction error, chaotic amplification of attractor distortion, continuity measures, as well as other time series analysis measures. MS63 Which Model to Use for Cortical Spiking Neurons? Linda J. Moniz U.S. Geological Survey We review the biological plausibility and computational [email protected] efficiency of eleven most useful and widely used models of spiking and bursting neurons. Our goal is to iden- Mary Ann F. Harrison tify a model that is most applicable to large-scale simu- Institute for Scientific Research, Inc. lations of cortical neural networks. We discuss why the [email protected] integrate-and-fire neuron, being the simplest and the most efficient spiking model, is not appropriate for simulations Lou Pecora and should be avoided by all means. Finally, we discuss Naval Research Laboratory some reasonable alternatives. [email protected] Eugene M. Izhikevich The Neurosciences Institute Steve Trickey [email protected] Naval Research Lab Washington DC, USA [email protected] MS64 Integrated Diagnostic and Prognostic Health Man- Jon Nichols agement For Mechanical and Structural Systems Naval Research Lab With Applications to Commercial and Defense Sys- Washington, DC, USA tems [email protected]

The nation’s commercial and defense manufacturers are Steve Knudsen, Leon Luxemburg searching for new ways to manage product life-cycles to in- Institute for Scientific Research, Inc. crease aftermarket profits and enable net-centric warfare. [email protected], [email protected] Diagnostics aim to identify faults and prognostics aim to predict the nonlinear evolution of those faults. The advan- Mark Seaver tages and challenges of assessing the health of mechanical Code 5673 systems online are demonstrated using physics-based and Naval Research Laboratory data-driven dynamic models in numerous applications in- [email protected] cluding an air-handling valve, laminated composite armor specimen and thermal protection system panel. MS64 Douglas Adams Purdue University Damaging Differentiability: The Connection Be- [email protected] tween Structural Damage and Loss of Synchroniza- tion

MS64 We employ chaotic interrrogation of a metal plate with in- cremental damage in order to test for changes in the struc- Slow-Time Damage Trajectory Reconstruction ture. We use two geometric tests to compare attractors Based on Smooth Orthogonal Decomposition embedded from response data from an undamaged struc- Damage is viewed as evolving slowly in a hierarchical dy- ture and a damaged structure. Both embeddings take ad- namical system causing parameter drifts in a fast-time vantage of a singular value decomposition of the data from subsystem. The fast-time measurements are used to re- multiple sensors. The geometric methods we use are really construct the slow-time phase space trajectory of damage. tests for generalized and differentiable synchronization be- Damage tracking feature vectors are developed based on tween reponses to an identical drive signal. We show that the phase space warping concept. Short-time evolution for the test for differentiable syncrhonization shows greater each point in the fast-time phase space is used to develop sensitivity than the test for generalized synchronization, in- feature vectors. The hidden damage is identified by apply- dicating that loss of differentiable synchronization between ing the smooth orthogonal decomposition to these vectors. responses is the result of damage to the structure. David Chelidze Linda J. Moniz 152 DS05 Abstracts

U.S. Geological Survey [email protected] [email protected] Michael Todd Louis M. Pecora University of California, San Diego Naval Research Lab [email protected] [email protected] Colin Olson Jonathan Nichols Dept. of Structural Engineeirng Naval Research Lab University of California, San Diego Code 5673 [email protected] [email protected] Luke Overbey Steven Trickey Dept. of Structural Engineering Coce 5673 University of California, San Diego Naval Research Laboratory [email protected] [email protected]

Mark Seaver MS65 Code 5673 Noise Induced Dimension Changing Bifurcations Naval Research Laboratory [email protected] Dramatic dynamical systems changes may occur in the presence of noise, such as bifurcations form low dimensional stochastic behavior to high dimensional chaos. I will Daniel Pecora explore this transition to illustrate some of the universal Virginia Commonwealth University characteristics on a problem from continuum mechanics. [email protected] Lora Billings Montclair State University MS64 Dept. of Mathematical Sciences Detecting Damage-Induced Nonlinearities In the [email protected] Presence of Ambient Variation: An Information- Theoretic Approach Ira B. Schwartz Two different information-theoretics, the time-delayed mu- Naval Research Laboratory tual information and time-delayed transfer entropy, are Nonlinear Dynamical Sysytems Section used to detect damage-induced nonlinearities in struc- [email protected] tures. For linear structures, both quantities admit analytical treatment. An algorithm is described for estimating MS65 both metrics and is shown to be in agreement with theory. Damage is then introduced as a structural nonlinearity. By Multistability, Noise and Attractor Hopping comparing the IT metrics for the structural response data Multistability, noise and attractor-hopping: The role of and their linear surrogates the presence of the nonlinear chaotic saddles External noise applied to systems with a damage is detected. multitude of coexisting attractors leads to a hopping dy- Jonathan Nichols namics between various metastable states. In particular, Naval Research Lab we study the role of the chaotic saddles for the accessibil- Code 5673 ity of different states and show how bifurcations of chaotic [email protected] saddles lead to a change in the hopping dynamics. Further- more we demonstrate an enhancement of the noise-induced escape from attractors due to the existence of chaotic sad- Luke Overbey, Colin Olson, Michael Todd dles embedded in the open neighborhood of an attractor. University of California, San Diego [email protected], [email protected], [email protected] Ulrike Feudel University of Oldenburg ICBM, Theoretical Physics/Complex Systems MS64 [email protected] Structural Damage Assessment Using Stochastic Probes and Pseudo-Attractor Geometry MS65 Recent research has shown the utility of using chaotic ex- Stochastic Resonance With Frequency Sensitivity citation and state space attractor features in structural health diagnostics. Attractors are reconstructed from time Usually, stochastic resonance with external periodic forc- series measured simultaneously at different structural loca- ing is not sensitive to the forcing frequency. On the other tions, predictive maps are built to correlate the attractors, hand, it has been well known that stochastoc resonance and an error metric is formed as the metric of interest. This can appear in autonomous excitable systems, called as co- work considers extending this idea to stochastically-driven herent resonance (CS). We find that systems of CS can systems and includes new features drawn from notions of show stochastic resonance with sensitive frequency depen- interdependence and coupling. dence when they are forced by periodic forcings. Moreover, the frequency-dependent stochastic resonance effect can be Jonathan Nichols greatly enhanced when many excitable system are coupled Naval Research Lab Code 5673 DS05 Abstracts 153

together. Mario Chavez Istituto Nazionale di Ottica Appplicata Gang Hu Florence, Italy Beijing Normal University, China [email protected] [email protected] Jacques Martinerie MS65 LENA-CNRS-UPR-640 Hôpital de la Salptrière, Paris. France Characterization of stochastic resonance [email protected] Traditional quantities used to characterize stochastic resonance possess the common feature of low sensitivity MS66 to noise variation in the sense that they vary smoothly about the optimal noise level. In potential applications of Stability of Synchronization in Networks of Spiking stochastic resonance such as device development, a high Neurons sensitivity to noise may be required. Here we show that, In a network of neuronal oscillators with time-delayed cou- when the resonance is regarded as a manifestation of phase pling, we uncover a phenomenon of enhancement of neural synchronization, the average synchronization time between synchrony by time delay: a stable synchronized state exists the input and output signal has an extremely high sensitiv- at low coupling strengths for significant time delays. By ity in that it exhibits a CUSP behavior about the optimal formulating a master stability equation for time-delayed noise level. Theoretical analysis and numerical evidence networks of spike-burst Hindmarsh-Rose neurons, we show are provided to establish the cusp behavior and its gener- that there is always an extended region of stable syn- ality. chronous activity corresponding to low coupling strengths. Kwangho Park, Ying-Cheng Lai Such synchrony could be achieved in the undelayed system Arizona State University only by much higher coupling strengths. A subset of Lya- Department of Mathematics punov exponents associated with transverse directions to [email protected], [email protected] the synchronized manifold can become negative from positive at different coupling strengths, indicating synchrony on different time-scales of oscillations. These results sug- MS65 gest that synchronization of spike-burst activity is a multi- Strange Nonchaotic Attractors in Random Dynam- time scale phenomenon and burst synchrony is easier to ical Systems achieve than spike synchrony.

Whether strange nonchaotic attractors (SNAs) can occur Viktor Jirsa typically in dynamical systems other than quasiperiodi- Florida Atlantic University cally driven systems has long been an open question. Here Physics Department we show, based on a physical analysis and numerical evi- [email protected] dence, that robust SNAs can be induced by small noise in autonomous discrete-time maps and in periodically driven MS66 continuous-time systems. These attractors, which are relevant to physical and biological applications, can thus be Synchronization of Complex Brain Systems expected to occur more commonly in dynamical systems I will present an analysis of the formation of synchronized than previously thought. clusters in active dynamical networks where the active Choy Heng Lai, Xingang Wang nodes are modeled by typical neuron models resp. popula- National University of Singapore tions of neurons. The connectivity patterns are taken from [email protected], [email protected] measurements of different mammal cortex areas. Com- plex synchronization phenomena are formed and destroyed through (varying) interactions between the nodes. Con- Ying-Cheng Lai ditions for optimum synchronizability are presented and Arizona State University interpreted in terms of neural tasks. Department of Mathematics [email protected] Juergen Kurths Universität Potsdam, Germany [email protected] MS66 Detection of Multiple Time Scale Synchronization MS66 Using a recent procedure for analyzing nonlinear and non- Analysis of Functional Connectivity Via Human stationary signals we decompose a time series in distinct EEG oscillation modes. When applied to coupled oscillators, we found that synchronization arises in a finite number of Functional connectivity in human neural systems is a re- common modes. For a given coupling value, different phe- search area that has attracted much interest in recent years nomena as phase slips, anti-phase or perfect phase locking due to technological advances in equipment for collecting can be simultaneously observed at specific time scales. Im- human neural data. This talks focusses on developing and plications for the study of the build-up of synchronized analyzing coupled cell models of human neural activity in states in nonstationary and noisy systems are pointed out. order to understand the mechanisms leading to connectivity between neural regions and to suggest new techniques Stefano Boccaletti for detecting such interactions in real data. Istituto Nazionale di Ottica Applicata [email protected] John R. Terry 154 DS05 Abstracts

Department of Mathematical Sciences [email protected] Loughborough University [email protected] MS67 Fronts in Media with Nonlocal Interaction MS66 Surrogates to Test for Synchronisation We consider various lattice dynamical systems with long range interaction and related integro-differential evolution The concept of synchronisation and especially phase syn- equations. Typically, these arise in the modeling of phase chronisation (PS) has been intensively studied in the re- transitions for a binary material and from activity in fam- cent years. The corresponding studies are usually based ilies of neurons. Nonlocal analogs of the wave equation, on the computation of a measure which quantifies depen- Allen-Cahn and Cahn-Hilliard equations, and the phase- dencies of the instantaneous phases of the time series (TS). field system will be discussed. However, even though these measures may be normalised, experimental TS yield values which are not at the borders Peter Bates of the interval and hence are difficult to interpret. This Michigan State University problem can be overcome if the coupling strength between Department of Mathematics the two systems can be varied systematically and a rather [email protected] large change in the measure can be observed. The PS in natural systems, e.g. of heart beats of a mother with her foetus, frequently evades such an experimental manipula- MS67 tion. In the case of mother-foetus heart beat synchronisa- Traveling Fronts in Scalar Reaction-Diffusion tion, this problem has been tackled using ECGs obtained Equations from a group of other pregnant women (which are uncou- pled from the foetus and hence not in PS) as “natural We prove existence of a family of traveling wave solutions surrogates” and then performing an hypothesis test. But for a large class of scalar reaction-diffusion equations with even this rather innovative approach has some drawbacks. degenerate, nonlinear diffusion coefficients and monostable The natural variability and also the frequency of the heart nonlinear reaction terms. We also show that, as in the beats of the surrogate mothers is usually different from the linear diffusion case, the slowest traveling wave solution in ones of the real mother. Furthermore, the data acquisition the family yields the asymptotic rate of the propagation of can be expensive and at least in some states of the preg- disturbances from the unstable rest state in these systems. nancy problematic. In these cases it is highly convenient In addition, we give conditions on the reaction term and to generate the surrogates by a mathematical algorithm. diffusion coefficient ensuring existence of interfaces. We present an approach for the generation of surrogates, Georgi Medvedev which is based on the recurrences of a system. These sur- Department of Mathematics rogates mimic the dynamical behaviour of the system and Drexel University overcome the above mentioned problems. [email protected] Marco Thiel Institut für Physik MS67 Postdam University, Germany [email protected] Large Entire Solutions of a Reaction-Diffusion Equation with Bistable Nonlinearity

Maria Carmen Romano We consider a reaction-diffusion equation of one-space di- Am Neuen Palais,10 mension with bistable nonlinearity and assume that it has 14469 Potsdam, Germany traveling front waves with constant speed. Then one can [email protected] observe the annihilation and diverging of two fronts for appropriate initial data. We show the existence of entire solu- Jürgen Kurths tions realizing these phenomena, where the entire solution Am Neuen Palais,10 is meant by a bounded solution defined for all (x, t) ∈ R2. [email protected] Namely we prove that there exist two kinds of entire solutions such that the two fronts coming from the both sides of x-axis annihilate while the other one emanating from an MS66 unstable equilibrium converges to the diverging fronts. Simulation of EEG Properties with Random Net- works of Leaky Integrator Neurons Yoshihisa Morita Ryukoku University The human electroencephalogram (EEG) is globally char- Department of Appl. Math. and Informatics acterized by a 1/f power spectum superimposed by certain [email protected] peaks where the alpha peak around8-14Hzisthemost prominent one. The spectral power in the alpha band depends on several variables: increased arousal leads to a MS67 decrease of the alpha power (“alpha block”) and the power A Variational Approach to Traveling Waves in Gra- increases during children development accompanied by a dient Systems shift from lower frequencies. I will show how some of these characterisics can be modeled by evolving random networks We study a class of systems of reaction-diffusion equa- of leaky integrator units where a critical connectivity must tions in infinite cylinders. These systems of equations arise be reached for the onset of oscillations in the network. within the context of Ginzburg-Landau theories and describe the kinetics of phase transformation in second-order Peter beim Graben or weakly first-order phase transitions with non-conserved University of Potsdam order parameter. We use a novel variational characteriza- DS05 Abstracts 155

tion to study existence of traveling wave solutions under canonical dissipative system using viscous fluid singulari- very general assumptions on the nonlinearities. These so- ties. lutions are a special class of the traveling wave solutions which are characterized by a fast exponential decay in the Sorin M. Mitran direction of propagation. Our main result is a simple ver- Dept. of Mathematics, Applied Math Prog. ifiable criterion for existence of these traveling waves. We University of North Carolina also prove boundedness, regularity, and some other prop- [email protected] erties of the obtained solutions, as well as several sufficient conditions for existence or non-existence of such traveling waves, and give rigorous upper and lower bounds for their MS68 speed. In addition, we prove that the speed of the ob- An Alternative Projective Integration Scheme tained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially Gear et al. proposed the class of projective integration sufficiently localized. We give a sample application of our methods for problems with gaps in their eigenvalue spec- results using a computer-assisted approach. trum. These methods combine consecutive results of an explicit time stepper, to obtain an extrapolated solution. Cyrill B. Muratov We develop and analyse a variant of this method, by ex- New Jersey Institute of Technology tending the results of Sommeijer (Comput. Math. Applic., Department of Mathematical Sciences 19(6):37-49, 1990). A stability analysis of the resulting [email protected] multistep method reveals that the method can also be applied to problems with no gap in their eigenvalue spectrum.

MS68 Dirk Roose Projective Integration and Multiscale Problems K.U.Leuven Dept. of Computer Science We discuss ways in which the macroscopic behavior can be [email protected] derived from a microscopic model. The microscopic model is used to estimate the forward derivative of macroscopic Kurt Lust variables for use in any process that could use a direct University of Groningen evaluation of derivatives. We determine long-term behav- Institute of Mathematics and Computing Science ior, steady states, estimates of the slow manifold, and even, [email protected] in some cases, integrate backwards in spite of instability of the underlying problem. Christophe Vandekerckhove Dept of Computer Science C. William Gear Katholieke Universiteit Leuven Princeton University [email protected] [email protected]

MS69 MS68 Dimension Reduction Techniques for Rigorous Nu- Some Computational Examples of Equation-Free merics Dynamic Renormalization Most interesting dynamical systems require a number of I will discuss a number of modeling examples whose com- dimension reductions before they can be studied numeri- putation is facilitated in an equation-free multiscale frame- cally. In order to produce rigorous results, these reductions work. I will also present some examples of equation- must be performed in such a way that information lost may free dynamic renormalization computations; these range be recovered, at least in a coarse way, by mathematical from simple glassy dynamics in compaction models to self- analysis. In this talk, we will discuss some techniques for similar interface roughness evolution dynamics. Different dimension reduction and some of the mathematical tools examples involve collaborations with different groups, who which may be used to compute rigorous information about will be mentioned in the presentation. the full system. Yannis Kevrekidis Sarah Day Princeton Department of Mathematics [email protected] Cornell University [email protected] MS68 Continuum-Microscopic Simulation of Chemotaxis MS69 Continuum models of chemotactic behavior of bacteria Rigorous Numerics for the Two Dimensional Cahn- have typically used empirical closures to furnish trans- Hilliard Equation port coefficients. A computational approach is presented Computing explicit solutions for partial differential equa- in which microscopic simulation is carried out at the be- tions is usually a difficult task. Therefore one often ginning of a continuum time step to determine local trans- uses the computer to calculate approximations, but in port coefficient values. Adaptive mesh refinement on the many cases it is not clear whether this numerical data continuum level is used to identify large gradient regions. really approximates a solution of the partial differential Differing spatial resolutions on the continuum scale lead to equation. We present a method, based on Conley in- different coarse grained transport coefficients which must dex theory, that insures the existence of an equilibrium reconciled when carrying out fine-to-coarse grid updates. of a time dependent equation in a computed neighbor- A comparison is made among different microscopic mod- hood. This technique was introduced by K. Mischaikow els including: cellular automata, lattice Boltzmann and a 156 DS05 Abstracts

and P. Zgliczyński for steady states of the one dimensional MS70 Kuramoto-Shivashinsky equation for fixed parameter val- A Variational Problem on Stiefel Manifolds ues. We improved this method in order to compute paths of equilibria for a two dimensional problem, namely the We consider here a general class of continuous time Cahn-Hilliard equation on the unit square. The Cahn- quadratic cost, optimal control problems on Stiefel man- Hilliard equation, a parabolic equation of fourth order, was ifolds, which in the extreme dimensions yield the classical introduced as a model for the process of phase separation rigid body equations and the geodesic flow on the ellipsoid. of a binary alloy at a fixed temperature. We summarize This is related to earlier work of Moser and Veselov in the some analytical results on its set of equilibria and then discrete setting. We have already shown that this optimal present the results we obtained using the rigorous method. control setting gives a new symmetric representation of the This includes secondary bifurcations from the known main rigid body flow and in this paper we extend this represen- branches of equilibria. tation to the geodesic flow on the ellipsoid and the more general Stiefel manifold case. Stanislaus Maier-Paape RWTH Aachen Peter E. Crouch Institut für Mathematik Arizona State University [email protected] [email protected]

Ulrich Miller Anthony M. Bloch Theoretical and Applied Mechanics University of Michigan Cornell University Department of Mathematics [email protected] [email protected]

Amit Sanyal MS69 Arizona State University Computing the Dynamics of Infinite Dimensional Mechanical and Aerospace Engineering Maps [email protected] I will discuss computational methods that can be used to rigorously study the dynamics of infinite dimensional maps. MS70 The Conley index provides the theoretical underpinning for Algorithmic Mechanics on Lie Groups these techniques. I will focus on the mathematical aspects of the theory that allow one to minimize the computational Many constrained optimization problems are formulated costs. naturally as optimization problems on classical Lie groups and homogeneous spaces. Considerations of faster conver- Konstantin Mischaikow gence and better dynamical behavior lead us to embedding Department of Mathematics certain classical algorithms such as gradient flow in second Georgia Tech order equations of mechanics. Specifically we consider me- [email protected] chanical systems on Lie groups wherein the objective function does double duty in determining stiffness and damp- MS69 ing. In this talk, we investigate the basic equations for certain appealing objective functions and the asymptotic Chain Recurrence From a Combinatorial Point of behavior of the associated mechanical systems. View Uwe R. Helmke In this talk we present a new definition of the chain recur- Universitat Wuerzburg rent set of a continuous map using finite spatial discretiza- Department of Mathematics tions. This approach allows for an algorithmic construc- [email protected] tion of index filtrations for Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley’s Lyapunov function. P.S. Krishnaprasad This is a natural framework in which to develop computa- Institute for System Research tional techniques for the analysis of qualitative dynamics. University of Maryland at College Park [email protected] William D. Kalies Florida Atlantic University Department of Mathematical Sciences MS70 [email protected] Computational Geometric Mechanics and its Ap- plications to Geometric Control Theory

Konstantin Mischaikow The geometric approach to mechanics serves as the theo- Department of Mathematics retical underpinning of innovative control methodologies in Georgia Tech geometric control. These include, for example, controlling [email protected] the attitude of satellites using changes in its shape, as opposed to chemical propulsion. We will introduce some of Robert Vandervorst the discrete diﬀerential geometric machinery necessary to Department of Mathematics implement control algorithms while respecting and preserv- Vrije Universiteit, Amsterdam ing the geometry of the problem. These include discrete [email protected] analogues of Lagrangian mechanics, exterior calculus, and connections on principal bundles. Melvin Leok DS05 Abstracts 157

University of Michigan, Ann Arbor vessel formation. Department of Mathematics [email protected] Daphne Manoussaki Vanderbilt University [email protected] MS70 Nonholonomic Integrators on Lie Groups MS71 Variational integrators proved to be very effective in long- A 2-Dimensional Model of Growth Factor Induced term numerical simulations of holonomic mechanical sys- Angiogenesis tems. Recently, such integrators were adapted to work in the nonholonomic setting. This talk will give an overview We propose a nonlinear coupled system of partial differ- of the theory. Conservation of momentum and energy by ential, ordinary differential and algebraic equations, for the numerical algorithm will be discussed. This is joint growth factor-induced angiogenesis. Properties of the ex- work with Yuri Fedorov. tracellular matrix are incorporated within the conductivity. An indicator function tracks the capillary network. A set of Dmitry V. Zenkov conditions captures the network characteristics. Numerical North Carolina State University results of the capillary network growth dependence on the Department of Mathematics properties of the growth factor (its diffusivity, decay, and [email protected] consumption rate) and the extracellular matrix (anisotropy and heterogeneity) will be discussed.

MS71 Charles Patrick Modeling Healthy and Pathological Neovascular- University of Texas ization of the Retina M. D. Anderson Cancer Center [email protected] Maturation of the human fetal retina is regulated by the delivery of oxygen to diﬀerentiating retinal cells; the devel- Shuyu Sun opment of vasculature to deliver this oxygen is regulated University of Texas - Austin by the production of VEGF by these cells in response to [email protected] hypoxia. We present a model for retinal growth compris- ing coupled PDEs describing the interdependence of reti- Mary Wheeler nal diﬀerentiation and capillary distribution. Our goal is Center for Subsurface Modeling to distinguish the physiological path to retinopathy of pre- The University of Texas at Austin, USA maturity from normal retinal development. [email protected] Chris Graesser Department of Mechanical & Industrial Engineering Mandri Obeyesekere University of Illinois at Urbana-Champaign University of Texas M. D. Anderson Cancer Center [email protected] [email protected]

Marialuisa Ruiz MS71 Entelos, Inc. mlr@diﬀeomorphism.com Numerical Simulation of Capillary Formation Dur- ing the Onset of Tumor Angiogenesis

Charles Simmons A model for tumor angiogenesis consisting of a coupled David Geffen School of Medicine system of ordinary and partial differential equations is dis- UCLA cussed. The modeling of chemotaxis of endothelial cells to [email protected] biochemicals is a key feature in the model, which results in convection dominated diffusion equations. A numerical Scott Kelly method that efficiently treats problems of this type is pre- Department of Mechanical & Industrial Engineering sented. It is based on the use of characteristics, is mass University of Illinois at Urbana-Champaign conserving, and provides an effective means of evaluating [email protected] modeling scenarios. Marit Nilsen-Hamilton MS71 Department of Biochemistry, Biophysics and Molecular Mechanical Forces During Vasculogenesis and An- Biology giogenesis Iowa State University [email protected] During blood vessel formation, endothelial cells follow mechanical as well as chemical cues in their environment. The Howard Levine, Michael Smiley theory that describes these mechanochemical interactions Iowa State University assumes that the extracellular matrix (ECM) is a viscoelas- [email protected], [email protected] tic material which deforms under cellular traction, and that cell movement depends on chemoattractant gradients and ECM strain. Numerical simulations predict cell traction MS72 can reorganize the ECM into a network. I discuss the po- Mathematical Model Studies of Lagrangian tential role of chemical vs. mechanical forces during blood Stochastic Methods

Lagrangian Stochastic Models (LSMs) are being actively 158 DS05 Abstracts

developed as computationally tractable schemes to pre- Langevin models, diffusion) to explore the effects of differ- dict and analyze oceanic transport of immersed substances, ent transport parameterization types on primary produc- without full resolution of the oceanic turbulence. In order tivity and residual nutrient concentration. to understand better how LSMs can, on a fundamental physical level, model certain transport features, such as Claudia Pasquero subdiffusion and superdiffusion, we analyze them in the Geological and Planetary Sciences Division context of relatively simple mathematical model flows in- California Institute of Technology cluding a mean flow, low-frequency variability, and a tur- [email protected] bulent component. Emilio Castronovo MS72 Rensselaer Polytechnic Institute Closure theory for the effective diffusivity tensor in Dept. Mathematical Sciences forced beta-plane turbulence [email protected] Theoretical and numerical results for the stirring of a tracer Peter R. Kramer both along and across turbulent jets is presented. Such a Rensselaer Polytechnic Institute system can be taken as a model for the stirring of true trac- Department of Mathematical Sciences ers along jets in either the atmosphere and ocean, or, less [email protected] obviously, for the stirring of baroclinic potential vorticity by non-zonal flow in the ocean. The model flow considered is two-dimensional turbulence with a mean vorticity gra- MS72 dient (β) and forced randomly and isotropically at small- Application of Auto-Regressive models to La- scales. The tracer transported by the flow is forced by a grangian Prediction and Stochastic BC mean tracer gradient that is arbitrarily oriented with respect to the mean vorticity gradient. Such a tracer can Auto-Regressive (AR) models are widely used in spectral be decomposed into two independent tracers, one forced analysis, but their use as stochastic models for geophysi- by a gradient that is parallel to the vorticity gradient (and cal variables is still under development. Geophysical ap- so is stirred across jets), and another that is forced by a plications of AR models include the random flight model mean gradient that is perpendicular to the mean vorticity for Lagrangian motion and applied Lagrangian prediction; gradient (and so is stirred along jets). The effective dif- the reduced-order information filter, based on an extended fusivity tensor for the full tracer can be computed from second-order AR model in space, for the assimilation of the eddy correlations of the two decomposed tracers. The altimetric data into ocean circulation models; coupled AR across-jet diffusion is well-described by mixing length the- models for modeling Lagrangian covariance functions; and ory, while the diffusivity of the tracer stirred along the jets the use of spatio-temporal AR models for parameterizing is the result of shear dispersion. At only moderate levels stochastic boundary conditions in coastal flows. Our stud- of anisotropy in the flow, the along-jet diffusivity is two ies have shown that AR models are parsimonious models orders of magnitude larger than the across-jet diffusivity. for geophysical variables and they should be used in a large The skew flux of the tracer is, notably, of the same magni- class of parameterization and prediction problems. tude as the mean flow. Annalisa Griffa Shafer Smith University of Miami Center for Atmosphere Ocean Science agriff[email protected] Courant Institute [email protected] Arthur Mariano RSMAS/MPO MS73 University of Miami [email protected] Decay of Chaotically Advected Passive Scalars in the Zero Diffusivity Limit

Tamay Ozgokmen There has been much recent interest in the time asymp- University of Miami/RSMAS totic decay of the variance of a passive scalar advected by [email protected] a chaotic flow. In particular, there is a debate as to the relative importance of long wavelength and short wavelength Toshio Mike Chin processes in determining zero diffusivity limit of the decay JPL, California Institute of Technology rate. Here we investigate the validity and regimes of appli- [email protected] cability of these two types of mechanisms. We argue that under typical conditions the short wavelength mechanism provides an upper bound on the decay rate in the zero dif- MS72 fusivity limit. We also demonstrate the applicability of the Parameterization of Turbulent Transport in Ma- short wavelength mechanism for a particular numerical ex- rine Ecosystems ample by showing that the decay rate is insensitive to the operation of filtering out the longest wavelength perturba- Statistical properties of Lagrangian trajectories in turbu- tions [1]. The decay of variance in the case in which the lent flows can be reasonably reproduced by using stochas- scale of the flow field is much less than that of the scalar is tic models of different complexity. Here, we focus on the governed by long wavelength processes. In this case there is extension of stochastic parameterization for transport in a relation between the decay rate and the exponent in the the case the advected particles contain reactive compo- power law for the wavenumber dependence of the scalar’s nents. We use a simple model of the marine ecosystem dy- spectrum [1,2]. Finally, the long-time decay rate of the namics (nutrient, phytoplankton, zooplankton) coupled to scalar variance is sensitive to the chaotic properties of the different advection models (quasi-geostrophic turbulence, DS05 Abstracts 159

ﬂow. Examples of this will be discussed. of change of the (Renyi) entropy of the distribution, producing entropy according to a generalized Lyapunov ex- Yue-Kin Tsang ponent. Their properties govern the quantum to classical IREAP transition in decoherent quantum chaos. We discuss other University of Maryland such fundamental questions that can be explored through [email protected] experiments in chaotic advection.

Edward Ott Arjendu K. Pattanayak University of Maryland Department of Physics and Astronomy Inst. for Plasma Research Carleton College [email protected] [email protected]

Thomas Antonsen MS74 Institute for Research in Electronics and Applied Physics University of Maryland Asymptotic Properties of Equations of Biological [email protected] Excitability We consider various parametric embeddings of models of MS73 biological electrical excitability, from Hodgkin-Huxley to modern cardiac models. Tikhonov-Pontryagin style small Effective Dimensions and Chemical Activity in parameters, with slow manifold having van der Pol’s fold, Closed Flows or Zeeman’s cusp catastrophe, inevitably lead to misrep- We show how reactions (in particular, autocatalytic ones) resentation of important qualitative features of the full spread in a closed container due to diffusive chaotic mixing. systems. We introduce alternative parameterization, not Contrary to open flows, in closed flows there are no fractal amenable to Tikhonov theorem. It is based on actual prop- objects forming the skeleton of activity. Nevertheless, we erties of full models and describes phenomena that evade are characterising the process via an effective dimension traditional asymptotic description. which is converging asymptotically towards the dimension Vadim N. Biktashev of the flow. Dept of Mathematical Sciences Tamas Tel University of Liverpool Institute for Theoretical Physics [email protected] Etovos University, Hungary [email protected] Irina V. Biktasheva, Radostin D. Simitev, Rebecca Suckley Gyorgy Karolyi University of Liverpool Dept. of Structural Mechanics [email protected], [email protected], Budapest University of Technology and Economics rebecca [email protected] [email protected] MS74 MS73 Milti-Time-Scale Dynamical Systems and Their Diffusive Interfaces and Boundary Layers at High Synchronization Peclet Numbers We consider dynamical systems with two time scales where I will describe the behavior of solutions of a steady both fast and slow dynamics can be nontrivial (in particu- advection-diffusion problem on a bounded two-dimensional lar, chaotic). We show how a reduced slow system can be domain with prescribed Dirichlet data when Pe, the Peclet derived. When two such multi-scale systems are coupled, number is very large. The characteristic property of ad- an effect of partial synchronization can be observed, where vection by cellular flows is that the fluid motion is sepa- the slow variables are synchronized while the fast ones are rated into flow cells by diffusive interfaces. At high Peclet independent. As an example bursting dynamics of neurons numbers advection dominates diffusion and the solutions is considered. tend to constant in each flow cell. Boundary layers of or- Arkady Pikovsky Pe−1/2 der arise near diffusive interfaces. In this talk I Department of Physics will discuss this boundary layer structure by means of an University of Potsdam, Germany asymptotic model on diffusive interfaces. [email protected] Alexei Novikov Penn State University Polina Landa Mathematics Moscow State University [email protected] Russia [email protected]

MS73 Michael Rosenblum Chaotic Advection and Persistent Patterns in Sta- Potsdam University tistical Mechanics Department of Physics [email protected] A ﬂuid undergoing chaotic advection corresponds directly to Liouville distribution dynamics. The persistent patterns studied in ﬂuids have interesting interpretation and roles MS74 in statistical mechanics. These extremize the second rate Complex Dynamics of Two Time Scale Neuron 160 DS05 Abstracts

Models MS75 Short Time Lyapunov Exponents to Anticipate The talk focuses on homoclinic bifurcations of periodic or- Vessel Instabilities bits and tori in singularly perturbed systems following the Hodgkin-Huxley formalism. Of special interests are the The chaotic behavior of finite-time phenomena such as ves- bifurcational scenarios that underlie plausible biophysical sel capsize, is quantifiable using a short, or finite-time Lya- transitions between tonic spiking and bursting regimes of punov exponent. This talk presents an investigation of fi- a neuron, as well as, explain its bi-stability. nite time Lyapunov exponent time series for vessel capsize in regular beam seas calculated from experimental data Andrey Shilnikov using a combined numerical-experimental approach. Ad- Georgia State University ditionally, the details of the Jacobian evaluation in the Dept Mathematics Statistics combined method and comparison to finite-time Lyapunov [email protected] exponents based upon numerical simulation are discussed.

Gennady S. Cymbalyuk Armin Troesch Georgia State University Naval Architecture and Marine Engineering Department of Physics and Astronomy University of Michigan [email protected] [email protected]

Leigh Mccue MS74 Department of Aerospace and Ocean Engineering Fast and Slow Dynamics in Gene Regulation Virginia Tech [email protected] Recent experiments show high variability of gene expression throughout the cell lifecycle which operates on vastly different time scales. Some of these scales are imposed MS75 by extrinsic factors and higher level control mechanisms Capsize Assessment Using Modern Methods (e.g. circadian clock), while others are intrinsic to the gene expression itself. Usually binding/unbinding and dimer- Modern methods are analysis and modeling approxima- ization of enzymes and other molecules are fast, and the tions applied to complex real systems. The modeling re- transcription and translation steps are typically slow. In ductions necessary to produce tractable equations are fre- this talk we deduce simplified deterministic and stochas- quently called into question for physical relevancy. This tic equations for the slow transcription/translation dynam- talk reviews several such examples and suggests ways in ics. These equations differ significantly from the usual rate which critical insight can be gained by viewing model cap- equations a priori assuming local equilibrium of fast re- size experiments as extensions of more idealized nonlinear actions. Using these equations, we discuss transient and systems. non-Markovian effects in gene regulation in simple genetic circuits. Armin Troesch Naval Architecture and Marine Engineering Dmitri Volfson University of Michigan University of California, San Diego [email protected] [email protected]

Jeﬀ Hasty MS75 Jacobs School of Engineering Theoretical and Experimental Studies of Passive UC San Diego Nnonlinear Targeted Energy Transfer in Systems [email protected] of Coupled Oscillators

Dmitri Bratsun, Lev S. Tsimring We study the dynamics of passive energy transfer from University of California, San Diego damped linear oscillators to essentially nonlinear end at- [email protected], [email protected] tachments. This transfer is caused by fundamental or sub- harmonic resonance captures, and in some cases is initiated by nonlinear beat phenomena. The end attachments MS75 are capable of passively absorbing broadband energy both Two Dimensional SPH Simulation of Floating and at high and low frequencies, acting, in essence, as passive Sinking Objects broadband boundary controllers. Complex transitions due to bifurcations of periodic orbits, and experimental results SPH is a Lagrangian method invented in the seventies and are discussed. it has been extended to incompresible flows in the eighties. Since, results in good accordance with experiments have Alexander Vakakis been obtained in some cases like water collapse or confined Department of Applied Mathematical and Physical problems like sloshing in tanks. Nevertheless, SPH leads to Sciences inaccuracies mainly coming from the boundary treatment National Technical University of Athens and the viscosity formulation. These points are discussed [email protected] here dealing with solid objects moving in water. Different possible formulations are compared. MS76 Louis Delorme An Overview of Normal Form and Unfolding Escuela Técnica Superior de Ingenieros Navales Styles. Universidad Politécnica de Madrid x Ax ··· [email protected] ˙ = + is in normal form if the higher order terms DS05 Abstracts 161

belong to a complement of the space of terms removable are weaker and are probably in need of new methods. through coordinate changes in each degree. A normal form style is a choice of the complementary space. There are four Ferdinand Verhulst styles in common use: Poincare-Dulac, inner product, sim- University of Utrecht plified, and sl(2). We will review the definitions, advan- Mathematics Institute tages and disadvantages, and state of current knowledge [email protected] about these styles, extend the notion of style from normal forms to (asymptotic) unfoldings, and indicate how to compute these unfoldings in any style. MS77 Euclidean Shift-Twist Symmetry in Population James A. Murdock Models of Molecular and Cellular Alignment Iowa State University Department of Mathematics We consider the symmetry properties of a general class [email protected] of nonlocal population models describing the aggregation and alignment of oriented objects in two dimensions. Such objects could be at the level of molecules, cells or whole MS76 organisms. We show that the underlying interaction kernel is invariant under the shift-twist action of the Euclidean A Normal Form Of Thin Fluid Film Equations Pro- 2 1 vides Initial Conditions group acting on the space R ×S . We then use equivariant bifurcation theory to identify the types of spatio-angular We construct a normal form of the Navier–Stokes equations patterns that are expected to occur, and compare these for the flow of a thin layer of fluid upon a solid substrate. with experiments. This illuminates the fluid dynamics by decoupling the interesting long-term ‘lubrication’ flow from the rapid viscous Paul Bressloff decay of shear modes. The normal form clearly shows the University of Utah centre manifold of the lubrication model and demonstrates bressloff@math.utah.edu that the initial condition for the fluid thickness of the lubrication model is not the initial physical fluid thickness, but instead is modified by the initial lateral shear flow. With MS77 these initial conditions, better forecasts will be made using Deriving Information About Architecture From the lubrication model. Activity Patterns In Coupled Cell Systems Anthony Roberts We take an inverse approach to the study of coupled net- University of Sothern Queensland work activity, asking how the existence of a specified ac- Toowoomba 4352, Australia tivity pattern constrains the possible network coupling ar- [email protected] chitectures. For patterns featuring multiple synchronized clusters, we derive a linear relation, the solutions of which are precisely those architectures that are robustly compat- MS76 ible with a given pattern. The analysis of this relation Towards Unique Normal Form Computation With shows how the inclusion of a second “hidden” cell popula- Quadratic Convergence tion can broaden the range of architectures that support particular activity patterns in an “observed” population. The computation of the transformation into normal form of a vectorfield at equilibrium with respect to its linear Jonathan E. Rubin part can be done with quadratic convergence, but this is University of Pittsburgh no longer true for the computation to higher order (hyper) Department of Mathematics normal form. In this talk I’ll describe the obstructions [email protected] to this in terms of spectral sequences and show how one can try and find optimal convergence somewhere between Kresimir Josic linear and quadratic. University of Houston Department of Mathematics Jan Sanders [email protected] Free University of Amsterdam Tne Netherlands [email protected] MS77 Unstable Attractors? Prevalence in Networks of Pulse-Coupled Oscilla- MS76 tors Normal Forms and Averaging For Partial Differen- tial Equations Attractors are central to studies in many fields of science, because they determine the long-term behavior of most dis- The techniques of averaging normal forms are well estab- sipative dynamical systems. Traditionally, attractors are lished for ordinary differential equations, but far from com- viewed as being asymptotically stable invariant sets which plete in the case of partial differential equations. This talk have a neighborhood that absorbs all sufficiently close ini- is a report on the current status of rigorous averaging and tial conditions. In 1985, John Milnor introduced a con- normal form theory for partial differential equations. For cept of attractor that neither presumed nor implied sta- bounded domains and operators allowing for a semigroup bility. Here we show that unstable attractors exist and formulation we can formulate normal forms and approxi- even occur robustly in models of biological synchronization mation theorems. In the case of wave operators for un- phenomena, namely in networks of oscillators with delayed bounded domains we can apply averaging over the char- pulse-coupling (Timme et al., Phys. Rev. Lett. 89:154105, acteristics but the corresponding approximation theorems 2002). From random initial conditions, groups of synchro- 162 DS05 Abstracts

nized oscillators (clusters) are formed that send pulses al- MS78 ternately, resulting in a periodic dynamics of the network. Panel Discussion The full measure of arbitrarily weak perturbations causes a splitting-up of certain clusters such that trajectories de- The minisymposium presented a variety of examples, where part from the attractor. This is explained by the surpris- geometric blowup both made a rigorous stability analysis ing geometrical fact that these unstable attractors are sur- possible and provided a ’typical’ geometric picture of the rounded by basins of attraction of other attractors whereas existence and stability problem. The discussion will fo- the full measure of their own basin is located remote from cus on three aspects: - comparison of the geometric view the attractor. Unstable attractors do not only robustly ex- to matched asymptotic expansions - numerical implemen- ist in these systems but moreover dominate the dynamics tation of the blowup algorithm - open problems and new for large networks and a wide range of parameters. directions Marc Timme Arnd Scheel Max Planck Institut for Dynamics and Self-Organization University of Minnesota Goettingen, Germany School of Mathematics [email protected] [email protected], [email protected]

Mariana Haragus MS77 Laboratoire de Mathematiques de Besancon Robust Patterns in Coupled Cell Systems Universite de Franche-Comte, France [email protected] A coupled cell system is a network of dynamical systems, or “cells”, coupled together. For such systems, we investigate patterns whose spatio-temporal symmetries are preserved MS78 under small changes in the equations describing the cells. Evans Function and Blow-Up for Degenerate Shock We conjecture that these patters are the consequence of an Waves underlying symmetry of the system, and prove this in some cases. We consider the Evans function approach to the stability of viscous shock waves in the case of characteristic shocks, i.e. Martin Golubitsky shocks with shock speed equal to a characteristic speed at University of Houston one of the end states. In comparison to the well-understood Department of Mathematics non-characteristic case two complications arise: ﬁrst, the [email protected] slow (merely algebraic) decay of the traveling wave at the characteristic end state; second, the fact that the essential Ian Stewart spectrum of the corresponding linearized equation has a University of Warwick branch point at the origin. We show how an Evans function Mathematics Institute can still be meaningfully deﬁned in that case by means of [email protected] the blow-up technique.

Andrew Torok Nikola Popovic University of Houston Boston University Department of Mathematics Center for BioDynamics and Department of Mathematics [email protected] [email protected]

MS78 MS78 Holes in Oscillatory Wave Propagation Zero Versus Nonzero Contact Angles in Thin Film Problems Holes are almost planar interfaces for which the angles of the interface at each point, relative to a fixed planar inter- In the lubrication approximation, thin films are described by a degenerate parabolic equation for the height ht = face, tend to zero at infinity. In isotropic reaction-diffusion n systems holes bifurcate from stable planar pulsating fronts. (h hxxx)x. For 0

MS78 MS79 Stability of Viscous Shock Waves Long-time Asymptotics of a Multiscale Model for Polymeric Fluid Flows Evans function methods are a powerful tool in the analysis of the stability of viscous shock waves. However, at various We investigate the long-time behaviour of some micro- stages in the analysis problems associated with the lack of macro models for polymeric fluids (Hookean model and hyperbolicity in the underlying ODEs must be overcome. FENE model), in various settings (shear flow, general In this talk we show how rescaling and blow-up methods bounded domain with homogeneous Dirichlet boundary can be used to overcome these difficulties. This is joint conditions on the velocity, general bounded domain with work with H. Freistühler (Leipzig). non-homogeneous Dirichlet boundary conditions on the velocity). We use both probabilistic approaches (coupling Peter Szmolyan methods) and analytic approaches (entropy methods). Institute for Analysis and Scientific Computing Vienna University of Technology Tony Lelievre [email protected] University of Montreal, Canada [email protected] MS78 The Generalized Korteweg-de Vries Equation: Sta- MS79 bility of Solitary Waves in the Singular Speed Limit Surprising Dissipation in Models of Inviscid Limits

Pego and Weinstein studied the gKdV equation, calculated the Evans function and gave conditions for spectral stabil- Jonathan C. Mattingly ity. This theory cannot be applied in the singular limit Department of Mathematics c = 0 of the wave speed (a standing wave) because the Duke University Evans function is not defined anymore. The main reason is [email protected] that the absolute spectrum touches the essential spectrum and the profile loses exponential dichotomy properties, i.e. the profile of the standing wave is just algebraically decay- MS79 ing. We are able to analyse the stability properties of the Nonequilibrium Stationary States in Classical Sta- standing wave by applying the blow-up technique which tistical Mechanics allows us to extend the Evans function into this singular limit. Luc Rey Bellet Bjorn Sandstede Departmant of Mathematics University of Surrey University of Massachusetts [email protected] [email protected]

Arnd Scheel MS79 University of Minnesota Invariant Measure of Stochastic PDEs and Condi- School of Mathematics tioned Diﬀusions [email protected], [email protected]

Martin Wechselberger Eric Vanden-Eijnden Ohio State University Courant Institute Mathematical Biosciences Institute [email protected] [email protected] MS80 MS79 Noise-Induced Superpersistent Chaotic Transients Navier Stokes: Invariant Measures and Cascades Superpersistent chaotic transients are characterized by an Randomly forced Navier-Stokes system in 3D will be dis- exponential-like scaling law for their lifetimes where the ex- cussed. I will give a new existence-uniqueness theorem for ponent in the exponential dependence diverges as a param- stationary solutions under some smallness conditions on eter approaches a critical value. So far this type of transient the random forcing. The stationary solution will be con- chaos has been illustrated exclusively in the phase space structed and studied with the help of a beautiful stochastic of dynamical systems. Here we report the phenomenon cascade construction due to Le Jan and Sznitman. of noise-induced superpersistent transients in the physical space and explain the associated scaling law based on the Yuri Bakhtin solutions to a class of stochastic differential equations. The Duke University context of our study is advective dynamics of inertial parti- [email protected] cles in open chaotic flows. Our finding makes direct experimental observation of superpersistent chaotic transients feasible and it also has implications to problems of current MS79 concern such as the transport and trapping of chemically Ergodicity and Stochastic PDEs or biologically active particles in large scale flows.

Younghae Do Martin Hairer Arizona State University Warwick University, United Kingdom [email protected] [email protected] 164 DS05 Abstracts

Ying-Cheng Lai MS80 Arizona State University Reactions in Chaotically Time-Dependent Flows Department of Mathematics [email protected] The dynamics of chemically or biologically active particles is studied when they are advected by smooth open flows of chaotic time dependence. Such an advection dynamics can MS80 be modeled by a random time-dependence of the parame- Finite-Scale Hamiltonian Chaotic Scattering ters on a stroboscopic map. A general theory is developed for reactions in such random flows, and a reaction equation An adequate characterization of the dynamics of Hamilto- is derived. We show that there is a singular enhancement of nian systems at physically relevant scales has been largely the reaction, which depends only on the fractal dimension lacking. Here we investigate this fundamental issue using of the filaments where particles accumulate. as a paradigm the problem of Hamiltonian chaotic scattering. We show that the finite-scale Hamiltonian dynamics Alessandro P.S. de Moura is governed by effective dynamical invariants, which are Universidade de Sao Paulo significantly different from the dynamical invariants that 05315-970, Sao Paulo, S.P. Brazil describe the asymptotic Hamiltonian dynamics. The effec- amoura@fig.if.usp.br tive invariants depend both on the scale of resolution and the region of the phase space under consideration, and they Celso Grebogi are naturally interpreted within a framework in which the Instituto de Fisica - IF nonhyperbolic dynamics of the Hamiltonian system is mod- Universidade de Sao Paulo/ USP eled as a chain of hyperbolic systems. Our results are il- [email protected] lustrated with applications to chemical reactions in chaotic fluid flows. Gyorgy Karolyi Adilson E. Motter Dept. of Structural Mechanics Los Alamos National Laboratory Budapest University of Technology and Economics [email protected] [email protected] Tamas Tel MS80 Eötvös Loránd University, Hungary Collision Dynamics of Raindrops and Transient Inst for Theoretical Physics Chaos [email protected]

The collision dynamics for an ensemble of raindrops with distributed mass is governed by the transient (regular or MS80 chaotic) dynamics of each raindrop between successive col- Long Chaotic Transients in Couette Flow lisions, when the mean collision time is relatively short. However, the dynamics of each raindrop is different due to Joe Skufca and I investigate a 9 dimensional model of Cou- the difference in mass, and the traditional approach for the ette flow for Raleigh numbers where there is a chaotic sad- problem of chaotic scattering cannot be applied directly. dle of dimension about 5. The set of points attracted to it We study this problem by extending the ideas from the is above 8.99. We describe this set and how it changes as traditional approach. Raleigh number varies. Takashi Nishikawa James Yorke Southern Methodist University University of Maryland [email protected] [email protected]

MS80 MS81 Statistical Properties of Wave Chaotic Scattering Locomotion by Flapping as a Bifurcation and Impedance Matrices Stephen Childress Edward Ott Courant Institute of Mathematical Sciences University of Maryland [email protected] [email protected] MS81 Xing Zheng, James Hart, Sameer Hemmady, Steve Anlage University of Maryland Wake Structures and Thrust Produced by Un- [email protected], [email protected], steady Finite Aspect Ratio Propulsors [email protected], [email protected] Anguilliform and rajiform locomotion are studied experimentally using unsteady mechanical models. These models Thomas Antonsen consist of pitching panels, and undulating bodies represent- Institute for Research in Electronics and Applied Physics ing manta rays and eels. Despite signiﬁcant diﬀerences in University of Maryland their geometries and kinematics, these models produce sim- [email protected] ilar wake structures which evolve in a manner that is distinct from wakes produced by nominally two-dimensional geometries. Details of these wake structures will be discussed as well as implications for thrust production and DS05 Abstracts 165

propulsive efficiency. [email protected] Alexander Smits Princeton University MS81 [email protected] The Lamprey as Elastic Rod: A Simple Model of Locomotion James H. J. Buchholz Department of Mechanical & Aerospace Engineering We will consider the locomotion of anguilliform swimmers Princeton University such as the lamprey, eels and certain aquatic snakes. Mod- [email protected] els for this type of swimming have been proposed by either specifying the shape of the swimmer or modelling the Richard Clark swimmer as a discrete chain of stiff links. We consider a Princeton University continuum model that incorporates the muscle activity as [email protected] a change in the intrinsic curvature of an elastic rod. We will see how a wave of curvature passing through the rod creates a locomoter force propelling the rod through the MS81 water. Reduced Models for Fishlike Swimmers Tyler McMillen We study the dynamics of an articulated body in a perfect Princeton University fluid. The goal of this study is to analyze the locomo- [email protected] tion of the body due to the coupling between its shape changes and the surrounding fluid dynamics. In this talk, MS81 we present two models: the first treats motion in potential flow and the second analyzes the interaction of the body Control Design and Flow Characterization For Op- with point vortices. The model of the motion in poten- timal Foil Thrust tial flow relies on geometric mechanics for systems with In this presentation, existing models for carangiform lo- symmetry combined with numerical simulation. The artic- comotion are extended to include point vortices. Optimal ulated body model achieves forward locomotion and turn- control results for a class of second order nonholonomic sys- ing maneuvers by merely changing its shape. We address tems are used to motivate studies of frequency-modulated the problem of motion planning or trajectory design as one gait selection for this class of systems. Model and control in optimal control; that is, we seek the most efficient shape predictions are validated using digital partical image ve- changes that achieve a desired net locomotion. We then locimetry with both a pitching and heaving foil with and introduce point vortices in the above flow and study their a free-swimming fish robot. interaction with the articulated body. We need to add to this framework a mechanism that emulates vortex shed- Kristi Morgansen ding in order to describe the dynamics of the body with Dept. of Aero and Astro self-generated vortices. U. of Washington [email protected] Clancy W. Rowley Princeton University [email protected] MS82 Optimal Drug Infusion Strategies for Cancer Eva Kanso Chronotherapy California Institute of Technology [email protected] We present optimisation procedures for cancer chronochemotherapy. Six Juan Melli-Huber coupled differential equations govern the evolution of both Mechanical and Aerospace Engineering the tumour cell population and the jejunal villi population, Princeton University to be shielded from unwanted drug side effects. Maximum [email protected] tumour cell kill is the objective function, tolerability as measured by jejunal mucosa preservation the constraint. Optimisation is led with respect to cytotoxic drug infusion MS81 flow and takes into account circadian variations in drug Lagrangian Mechanics and Thrust Generation sensitivity for both targets, healthy and tumoral. Through Vortex Shedding Jean Clairambault The locomotion of marine animals often involves the devel- INRIA opment of thrust through the shedding of vortex structures. [email protected] Vortex shedding is an essentially viscous phenomenon, but can be modeled in the context of Lagrangian mechanics Claude Basdevant through the introduction of constraints generalizing Kutta Université Paris-Nord conditions. Fluid vorticity and body momentum evolve ac- Villetaneuse, France cording to coupled equations with thrust forces contribut- [email protected] ing gyroscopic terms to the equations governing body motion. We present basic elements of this approach to mod- Francis Lévi eling such systems. INSERM E 0354 Hôpital Paul-Brousse, Villejuif, France Scott Kelly [email protected] Department of Mechanical & Industrial Engineering University of Illinois at Urbana-Champaign 166 DS05 Abstracts

MS82 Vered Rom-Kedar Modelling the Malignant Progression of Brain Tu- Weizmann Institute mours of Science [email protected] Tumours arising from the brain glia cells constitute the most common form of primary brain tumours in adults. Lee Segel Astrocytomas are classiﬁed by the degree of malignant ap- Weizmann Institute of Science pearance in the pathological examination into four grades. [email protected] We have developed a detailed PDE model incorporating cellular mutation to simulate the development of a tumour from low to high grade. We use our model to gain insight MS83 into the mechanistic basis of tumour progression. Mixing and Chemical Reactions in Chaotic Flows

Eliezer Shochat Theory of fast binary chemical reaction, A + B→C,ina Weizmann Institute of statistically stationary bounded chaotic flow at large Peclet Science number Pe and large Damköhler number Da is described. [email protected] For stoichiometric condition we identify subsequent stages of the chemical reaction. The first stage correspondent to Kevin Painter formation of the developed lamellar structure in the bulk Hariot Watt University, Edinburgh part of the flow is terminated by an exponential decay, ∝ Department of Mathematics exp(−λt) (where λ is the Lyapunov exponent of the flow), [email protected] of the chemicals in the bulk. The second and the third stages are due to the chemicals remaining in the boundary Nick Savill region. During√ the second stage the amounts of A and Dept of Zoology B decay ∝ 1/ t, whereas the decay law during√ the third Cambridge stage is exponential, ∝ exp(−γt), where γ ∼ λ/ Pe.We [email protected] also discuss recent experiments confirming the theory. V. Lebedev MS82 Landau Institute, Moscow Modeling Tumor Growth Under Sub-Maximal [email protected] Densities Misha Chertkov A model describing the competition between normal and T-13, Theoretical Division malignant cells on nutrients under sub-maximal densities Los Alamos National Laboratory in the presence of angiogenesis and growth factors is de- [email protected] veloped. The natural model corresponds to a semi-linear slow-fast ODE system. The properties of this model and their dependence on the form of the interaction terms are MS83 discussed. For a large class of such systems, the growth Front Dynamics in Reaction-Diffusion Systems factors may be used to locally stabilize the healthy tissue with Anomalous Diffusion solution. A growing number of studies have pointed out the presence Eliezer Shochat of anomalous diffusion due to chaotic advection, and there Weizmann Institute of is a need to understand reactive systems in the presence Science of this type of non-Gaussian diffusion. Here we consider [email protected] front dynamics in reaction-diffusion systems with anomalous diffusion due to asymmetric Levy flights. Numerical Vered Rom-Kedar and analytical studies of the fractional Fisher-Kolmogorov The Weizmann Institute equation show exponential acceleration of fronts and uni- Applied Math & Computer Sci versal power law decay of the front’s tail. [email protected] Benjamin Carreras, Vickie Lynch, Diego Del-Castillo-Negrete MS82 Oak Ridge National Laboratory G-CSF Effect on White Blood Cells Dynamics [email protected], [email protected], [email protected] Prediction and prevention of infectious complications following chemotherapy induced suppression of white blood cells (granulocytopenia) is a necessity in oncological prac- MS83 tice. G-CSF (granulocyte colony stimulating factor) is piv- Oscillatory Chemical Reactions in Chaotic Flows otal in therapy of patients with clinical episodes of granulocytopenia. We propose that important features of G-CSF We investigate the effect of chaotic fluid mixing on oscilla- effect on granulocyte dynamics may be captured by a 2D tory and excitable chemical reactions. Mixing may quench model. The low number of required parameters simplifies the oscillations or produce coherent synchronized oscilla- the prediction of granulocytopenia in individual patients. tions in nonuniform oscillatory media and the amplitude of the oscillations depends on the stirring rate. Coherent Eliezer Shochat oscillations are also possible in excitable systems subject Weizmann Institute of to stochastic perturbation. In this case the stirring rate Science controls the period of the oscillations. The relationship to [email protected] experimental observations in chemical reactors will also be DS05 Abstracts 167

discussed. sive mixing), the fronts mode-lock to the external forcing. We have mapped out Arnold tongues associated with this Zoltan Neufeld locking behavior. We also investigate the behavior of the Los Alamos National Lab. fronts in the regime in which transport is superdiﬀusive, [email protected] concentrating both on the shape of the advancing front and on its propagation speed. Changsong Zhou Potsdam University, Germany Matt Paoletti, Tom Solomon [email protected] Bucknell University [email protected], [email protected] Istvan Kiss Department of Zoology MS83 University of Oxford [email protected] Eﬀects of Particle Inertia in Active Chaos: En- hancement of Kinetics

Jurgen Kurths We investigate the reaction kinetics of small spherical parti- Institute of Physics cles with inertia, being advected by hydrodynamical flows Univesrity of Potsdam with imperfect mixing properties. In contrast to passive [email protected] tracers, the particle dynamics is governed by the strongly nonlinear Maxey-Riley equations, which typically create chaos in the spatial component of the particle dynamics, MS83 appearing s filamental structures in the distribution of the Rock-Scissors-Paper Game in a Chaotic Flow: The reactants. The emerging nonlinear effects include tracer Effect of Dispersion on the Cyclic Competition of trapping and kinetics acceleration. Microorganisms Izabella Benczik, Adilson Enio Motter Numerical simulations and experiments have shown that Max-Planck-Institut für Physik komplexer Systeme the outcome of cyclic competition is affected by the spatial Dresden, Germany distribution of the competitors. Short range interaction [email protected], and limited dispersion allows for coexistence of different [email protected] species whose competition would destroy the species diversity in a well mixed environment. We study the inter- Takashi Nishikawa mediate situations of imperfect mixing, typical in aquatic Southern Methodist University media, and the transition between the two regimes, in a [email protected] model of cyclic competition between toxin producing, sensitive and resistant phenotypes. It is found, that chaotic mixing, by changing the structure of the spatial distribu- Tamas Tel tion, induces oscillations in the populations. The coherence Institute for Theoretical Physics of the oscillations increases with the strength of mixing, Etovos University, Hungary leading to the extinction of some species beyond a critical [email protected] mixing rate. When mixing is non-uniform in space (e.g. there are obstacles or vortices in the flow), coexistence can Z. Toroczkai be sustained at much stronger mixing, by the formation of Los Alamos National Laboratory partially isolated regions, that prevent global extinction. [email protected] The heterogeneity of mixing may enable toxin producing and sensitive strains to coexist for very long time at strong Celso Grebogi mixing. Instituto de Fisica - IF Universidade de Sao Paulo/ USP Zoltan Neufeld [email protected] Los Alamos National Lab. [email protected] MS84 Gyorgy Karolyi Stability of Nonlinear Defect Modes in Bragg Grat- Dept. of Structural Mechanics ing Fibers Budapest University of Technology and Economics [email protected] We consider light confined by a defect in a Bragg grat- ing optical fiber that supports multiple bound states. Istvan Scheuring This system has no energy-minimization principle, but is Dept. of Plant Taxonomy and Ecology seen in numerical experiments to select a sort of ground Eotvos University state. We present results of numerical and and analytical [email protected] study, including eigenvalues computed numerically using Evans functions, to discuss the stability of these competing modes. MS83 Roy H. Goodman Experimental Studies of Front Propagation in a New Jersey Institute of Technology Reaction-Advection-Diffusion System Department of Mathematical Sciences We present experiments on propagation of chemical fronts [email protected] in an alternating vortex chain that can oscillate and/or drift laterally. For the oscillating vortex chain (with diffu- Michael I. Weinstein 168 DS05 Abstracts

Columbia University crystals in three spatial dimensions. Coupled-mode equa- Dept of Applied Physics & Applied Math tions describe resonantly interacting Bloch waves in stop [email protected] bands of the photonic crystal. We study the linear boundary-value problem for stationary transmission of four counter-propagating and two oblique waves on the plane. MS84 Well-posedness of the boundary-value problem is proved Vector Soliton Interactions in Birefringent Optical by using the method of separation of variables and gener- Fibers alized Fourier series. For applications in photonic optics, we compute integral invariants for transmission, reflection We derive a simplified system of Hamiltonian ordinary dif- and diffraction of resonant waves. ferential equations (ODEs) from ones derived by Ueda and Kath for a coupled pair of nonlinear Schrodinger equations Dmitry Pelinovsky also studied by Tan and Yang. Using matched asymp- McMaster University, Canada totic expansions for separatrix crossing, we determine ”res- [email protected] onance windows” of initial conditions for which the interacting solitons are reflected after two passes. Numerical simulations of these two ODE models agree quite well with MS84 our asymptotic theory of capture and resonance. Anomalous Scattering Behavior Near Guided Elec- tromagnetic Modes Roy H. Goodman New Jersey Institute of Technology Guided electromagnetic modes in periodic slabs or pillars Department of Mathematical Sciences are responsible for frequency-localized anomalous behavior [email protected] of energy transmission through the structures. Typically, these anomalies occur near the frequency and wave vector Richard Haberman of a truly bound mode that becomes leaky upon perturba- Southern Methodist University tion of these parameters or the material or geometric prop- Department of Mathematics erties of the structure. We model these phenomena using [email protected] the spectral theory of boundary integral operators and the extended Maxwell system for lossy systems. (Joint work with S. Venakides, D. Volkov, and A. Figotin) MS84 Localized Structures in Two-Dimensional Photonic Stephen P. Shipman Lattices Department of Mathematics Louisiana State University New models describing wave propagation in transversely [email protected] modulated optically induced waveguide arrays are proposed. In the weakly guided regime, a discrete nonlinear Schrodinger equation with the addition of bulk diffraction MS84 term and an external “optical trap” is derived. In the de- Semi-Analytical Methods to Treat Dynamic Pho- focusing regime the optical trap induces a stable localized tonic Crystal Systems mode. In the limit of strong transverse guidance, the dynamics is governed by a model which represents the optical Electromagnetic states associated with photonic crystals analogue of wave action. undergoing rigid time-dependent translations in position space are investigated. It is demonstrated that Bloch wave Ziad Musslimani vector remains a conserved quantity and that an analogue University of Central Florida of Blochs theorem for time-dependent states can be formu- Department of Mathematics lated. For the time-dependent translations involving har- [email protected] monic rigid vibrations of the photonic crystal it is shown that inter-band transitions can be induced between dis- Keith A. Julien tinct photonic bands of a crystal, thus enabling non-linear Department of Applied Mathematics applications with strictly linear materials. University of Colorado at Boulder Maksim Skorobogatiy [email protected] Génie physique Ecole Polytechnique de Montréal Mark Ablowitz [email protected] Dept. of Applied Mathematics University of Colorado [email protected] MS85 Weakly Interacting Waves and Bumps in Synapti- Michael I. Weinstein cally Coupled Neural Media Columbia University Dept of Applied Physics & Applied Math We analyze the existence and stability of stationary N– [email protected] pulses and traveling wave trains in a one-dimensional neuronal network. The network is modeled in terms of a nonlocal integrodifferential equation whose integral kernel rep- MS84 resents the spatial distribution of synaptic weights. We use Modeling of Wave Resonances in Low-Contrast singular perturbation theory to derive a set of N coupled Pphotonic Crystals ODEs for the dynamics of N weakly interacting pulses, establishing a direct relationship between the explicit form of Coupled-mode equations are derived from Maxwell equa- the interactions and the structure of the long-range synap- tions for modeling of low-contrast cubic-lattice photonic DS05 Abstracts 169

tic coupling. pled Cortical Interneurons Paul Bressloff Fast-spiking (FS) neurons in the cortex are connected by University of Utah both electrical coupling and inhibitory synapses. The ex- bressloff@math.utah.edu tensive electrical coupling and the high level of excitability make the FS cell network well-suited to support propagated waves of activity. However, the thin spikes and large after- MS85 hyperpolarizations of FS cells and inhibitory connections Persistent Activity and Gap Junctions between FS cells should work against wave propagation. We discuss conditions for the existence of waves and the It has been noted that gap junctions are most common properties of these waves in models of cortical FS cell net- between inhibitory interneurons in the cortex. We sug- works. gest that the reason that there are few observations of gap junctions between excitatory cells is due to their ability Tim Lewis to disrupt ersistent activity. We show that they do this University of California through two mechanisms. First, GJs provide an decreased at Davis membrane resistance so that synaptic excitation is less ef- [email protected] fective. Secondly, they can destabilize the asynchronous firing state making it more difficult for maintained persistent activity. MS85 Understanding Timing and Structure of Cortical Bard Ermentrout Waves From Data University of Pittsburgh, Pittsburgh, PA Waves, a fundamental mechanism for conveying informa- [email protected] tion, are observed in many areas of the brain but their structure is not well-understood. We have developed several visualization techniques such as latency-ordered space- MS85 time overlays, reordered parallel coordinate visualizations An Intracellular Ca2+ Subsystem As a Biologically and subspace projections. These techniques permit de- Plausible Source of Intrinsic Bistability in a Net- tailed comparisons of activation for data from different tri- work Model of Working Memory als or from different variables in the same trial. We show how application of these techniques to cortical waves pro- We explore a network model of working memory in an in- vides insight into underlying mechanisms. tegrodifferential form similar to those proposed by Amari. The model incorporates an intracellular Ca2+ subsystem Kay A. Robbins whose dynamics depend upon the second messenger [IP3]. Dept of Computer Scienc and Cajal Neuroscience This Ca2+ subsystem endows individual units with in- Research Cent trinsic bistability for a range of [IP3]. This full network University of Texas at San Antonio sustains [IP3]-dependent persistent (bump) activity in re- [email protected] sponse to a brief transient stimulus. Our results highlight the importance of second messenger activity time scales. MS86 Chris Fall PCR in in-vitro Diagnostics - Dynamics and Data NYU Center for Neural Science Analysis New York University [email protected] This talk will describe methods for data analysis using dynamical models and statistical methods in in-vitro diagnostic applications of Polymerase Chain Reaction (PCR). MS85 Models and parameter spaces of PCR and related biochem- Some Neural Examples of “Equation-Free” Mod- istry will be presented. Data will be presented to outline elling the scope of the data analysis problem. Methods and results for determining the parameters with enough accuracy “Equation-free” modeling allows one to analyse the effec- and robustness to allow for their use in diagnostic applica- tive equations governing the macroscopic dynamics of a tions will be described. system, even if they cannot be explicitly derived, providing that the microscopic dynamics can be simulated and that David Eyre, Aldo Bernasconi there is some separation of timescales. These techniques Idaho Technology can be used to do bifurcation analysis of realistic networks [email protected], aldo [email protected] of neurons. We demonstrate this for macroscopic steady states of small networks, and for spatiotemporal patterns such as bumps and waves. MS86 Computational Cell Biology in the Pharmaceutical Carlo R. Laing Industry Massey University IIMS In contrast to industries that have already embraced com- [email protected] puter simulation in guidance of product design and discovery, computational biology, in particular mechanistic simulation of cellular processes, remains a marginal (al- MS85 beit growing) activity in the pharmaceutical industry. I Wave Propagation in Networks of Electrically Cou- will present a few success stories – and pitfalls – from my personal experiences applying biological simulation and mathematics to cancer drug discovery and cardiac safety 170 DS05 Abstracts

assessment. I will also emphasize the mathematical tools mechanical systems including non-smooth behaviour like and techniques I used, or learned to use, along the way. those with Coulomb like friction. As an example serves a Finally, I will describe a few of the biological simulation rotated Froude pendulum harmonically excited. It is as- activities that some of the more innovative drug discovery sumed that in a place of pendulum fixation the Coulomb and biotechnology companies are currently pursuing. friction occurs and that the pendulum pivot rotates with a constant velocity. The classical Melnikovs technique is used Dean Bottino to predict chaotic behaviour in our one-degree-of-freedom The BioAnalytics Group mechanical system. The obtained results are verified by nu- [email protected] merical experiments. The expected chaotic thresholds are drawn in the control parameter planes. In what follows for one fixed value of the first control parameter the associated MS86 critical value of the second control parameter is computed Panel Discussion owing to its prediction given by the homoclinic bifurcation condition. Then numerical experiment is carried out us- Eric N. Cytrynbaum ing the first control parameter. A range of changes of the University of British Columbia bifurcation parameter includes the predicted value of the Department of Mathematics second parameter owing to the applied Melnikovs method. [email protected] The numerical verification relies on finding a value of the bifurcation parameter when the homoclinic bifurcation occurs, and on its comparison with the value predicted an- MS86 alytically. In addition, phase portraits and Poincar maps Virtual Patients in Clinical Trial Design are used for identification of chaos. It has been shown that owing to application of the Melnikovs technique, the criti- Virtual Patients developed by Entelos scientists provide cal values of the parameter associated with the homoclinic novel methodologies for predicting clinical outcomes and bifurcation have been analytically predicted properly since optimizing clinical trial designs. Developed within a pro- they have been also identified by the numerical compu- prietary simulation platform, Virtual Patients are explicit tations. Owing to existence in the non-perturbed system representations of observed clinical phenotypes and provide four homoclinic orbits (in the case of relatively high rotat- a biological context for predicting compound efficacy and ing velocity) four analytical chaos criterions have been also identifying biomarkers of therapeutic outcome. A case- computed. The carried out numerical analysis fully coin- study demonstrating the application of Virtual Patients to cides with the predicted bifurcations region separated by a optimize a Phase I trial design, which led to substantial periodic window. 1. J. Awrejcewicz, M. M. Holicke, Mel- time and cost savings, will be presented. nikov’s Method and Stick-Slip Chaotic Oscillations in Very Weekly Forced Mechanical systems, International Journal Cynthia J. Musante of Bifurcation and Chaos, 9(3), 1999, 505-518. 2. J. Awre- Entelos, Inc. jcewicz, D. Sendkowski, How to predict stick-slip chaos in [email protected] R4, Physics Letters A, 330, 2004, 371-376. Mariusz Holicke MS86 Technical University of Lodz Application of a Computational Fluid Dynamics [email protected] Model to Predict Uptake of Hydrogen Sulfide in Rat Nasal Passages Jan Awrejcewicz Department of Automatics and Biomechanics An anatomically accurate computational fluid dynamics Lodz Technical University model of rat nasal passages was used to simulate inspi- [email protected] ratory airflow and calculate wall mass flux of inhaled hydrogen sulfide (H2S). The reaction-diffusion equation was solved with a nonlinear boundary condition describing MS87 elimination of H2S in nasal tissue by first-order and sat- Dynamics at a Degenerate Graze for An Impact urable pathways. Rate constants were computed from an Oscillator air-tissue compartmental model that was fit to experimental measurements of nasal H2S uptake in rats. We give some rigorous results on dynamics for an impact oscillator close to a degenerate graze. These results under- Paul M. Schlosser pin earlier analysis by [Budd and Dux, Phil. Trans. R. Soc. NCEA, U.S. EPA Lond. A 347 (1994), 365–389] and others on intermittency [email protected] and chattering.

Jeﬀry Schroeter, Anna McElveen, David Dorman, Julia David Chillingworth Kimbell University of Southampton CIIT Centers for Health Research School of Mathematics [email protected], [email protected], [email protected], [email protected] [email protected] MS87 MS87 Control of Near-Grazing Dynamics in Impact Os- Prediction of Chaos in a Mechanical System with cillators Friction A method is presented for controlling the persistence of a We are aimed to illustrate how the classical Melnikovs tech- local attractor near a grazing periodic trajectory in a piece- nique can be extended to analyse even complex lumped wise smooth dynamical system in the presence of jump dis- DS05 Abstracts 171

continuities. Speciﬁcally, a discrete, linear feedback strat- between grazing and chattering in a simple impacting pen- egy is employed to sustain the existence of an attractor dulum. We will also show how grazing can organise the near the grazing trajectory, such that the deviation of the basin of attraction for systems with coexisting limit cycles. attractor from the grazing trajectory goes to zero as the system parameter values approach those corresponding to Chris Budd grazing contact. University of Bath [email protected] Harry Dankowicz Virginia Polytechnic Institute and State University Petri T. Piiroinen Engineering Science and Mechanics Department of Engineering Mathematics [email protected] University of Bristol [email protected] Jenny Jerrelind Department of Aeronautical and Vehicle Engineering Royal Institute of Technology MS88 [email protected] Odor Coding in Olfactory Networks

With computer model of olfactory system we explored in- MS87 trinsic and synaptic mechanisms involved into coding olfac- Co-dimension-two Grazing-sliding Bifurcations in tory information. Interaction between principle and local Filippov Systems neurons in the antennal lobe created odor specific temporal patterns of activity decoded by the neurons at downstream Many systems of relevance to engineering applications are levels of processing. A combination of nonlinear intrinsic modelled as sets of ordinary differential equations with dis- properties of the downstream cells and local inhibitory cir- continuous right-hand sides. These systems are termed as cuits favored coincidence detection tuned to select correla- Filippov systems and are known to exhibit so called sliding tions in the input spike trains and promoting sparse odor bifurcations. We will analyse two distinct codimension-two representation. scenarios occuring when a limit cycle exhibiting grazing- sliding is non-hyperbolic. A dry-friction oscillator where Maxim Bazhenov the aforementioned types of codimension-two events were Salk Institute found will be analysed, and its dynamics around the [email protected] codimension-two points will be subsequently explained. Piotr Kowalczyk MS88 Dept. Engineering Mathematics Computation with Spikes University of Bristol [email protected] In the neocortex, neurons project to distant regions resulting in conduction delays of tens of milliseconds. Syn- chronous spiking of such neurons may not be effective to MS87 fire a postsynaptic cell, since the spikes arrive to the cell at Chaotic Oscillations in Piecewise Linear Systems drastically different times. To excite the cell, the neurons With Hysteresis must fire with certain polychronous spiking patterns determined by the delays. Simulating a spiking network with de- The proposal of this talk is to analyze the occurrence of lays and STDP, we show that neurons spontaneously poly- chaotic oscillations when a piecewise linear system is built chronize, i.e., self-organize into groups and fire time-locked by applying a switching element with hysteresis in the feed- but not synchronous spike-timing patterns with millisecond back loop. The analysis is performed for systems of second precision. We found more groups than neurons resulting and third orders and it is shown that chaotic oscillators of in unprecedented memory capacity of the system. double-scroll type can be obtained and that the simplicity of their structure allows to establish a systematic procedure Eugene M. Izhikevich for the construction of chaotic oscillators. The Neurosciences Institute [email protected] Pedro L. D. Peres Faculty of Electrical Engineering and Computation State University of Campinas - Unicamp MS88 [email protected] A Unified Approach to Determining Embedding Dimension and Time Delay for Multivariate Time Ubirajara Moreno Series Department of Automation and Systems Federal University of Santa Caterina - UFSC Typically there are two problems in multivariate attrac- [email protected] tor reconstruction. One is finding the time dely for each time series and the other is establishing the embedding dimension. The latter means deciding which series to use to MS87 enable a complete unfolding of the attractor. Autocorre- Grazing and Chattering in Vibro-Impact Systems lation or mutual information is used in the time-delay estimation and false-nearest neighbor statistics are used for Grazing and chattering are two nonsmooth phenomena the embedding dimension. We approach these two seem- that appear frequently in vibro-impact systems. These ingly independent problems by returning to Taken’s em- events are also well known to generate additional prob- bedding theorem and examining what the mathematical lems in numerical analysis. To gain some understanding of requirement is to add another component to an attractor when to expect such events we will examine the interplay reconstruction. We show that the above two problems are 172 DS05 Abstracts

really the same problem and there is a uniﬁed approach to time-dependent regimes to broad stationary distributions. both that logically leads to ﬁnding correct time delays and simultaneously choosing which time series to use. Alexander Neiman Department of Physics and Astronomy Linda J. Moniz Ohio University U.S. Geological Survey [email protected] [email protected] Lutz Schimansky-Geier Louis M. Pecora Institute for Physics Naval Research Lab Humboldt University at Berlin [email protected] [email protected]

Thomas L. Carroll Michael Zaks Naval Reseach Laboratory Department of Stochastic Processes, Institute of Physics [email protected] Humboldt University of Berlin [email protected] Jon Nichols Naval Research Lab Xaver Sailer Washington, DC, USA Institute of Physics [email protected] Humboldt University of Berlin [email protected] MS88 Beamforming with Repulsive Phase Synchroniza- MS89 tion in an Array of Nonlinear Oscillators Propagation of Singularities in Director Fields

An array of oscillators with all-to-all repulsive coupling is We shall discuss some results on the simplest possible heat considered. When the oscillators are identical, the dynam- flow of director fields with values in the unit sphere. There ics of the array settle to a synchronized regime, character- turns out to be a fascinating interplay between the ex- ized by a zero mean field. However, for non-identical os- istence of point singularities, also called defects, and the cillators, the mean field is non-zero for arbitrary coupling nonuniqueness of the flow caused by a certain degree of strength. We develop a theoretical description of repulsive freedom in prescribing the evolution of the singularities. synchronization based on a phase approximation of the dy- The singularities can be created spontaneously, starting namics. We also study the mean field response of the array from smooth data. Part of the lecture will be devoted to to external periodic driving. The concept can be used in the construction of traveling wave solutions, and several conjunction with a phased array antenna to improve beam- open problems will be presented. forming characteristics. Michiel Bertsch Michael Gabbay Istituto per le Applicazioni del Calcolo M. Picone Information Systems Labs [email protected] [email protected]

Lev S. Tsimring MS89 University of California, San Diego Phase Locking for Flame Propagation in Periodic [email protected] Media In some situations, ﬂame propagation can be described Nikolai Rulkov qualitatively by sharp-interface models with energy depo- ISL Inc. sition at the interface. For propagation in periodic media [email protected] we examine phase-locking eﬀects (which are analogous to the ones observed for the classical van der Pol oscillator). Michael L. Larsen In some sense the free-boundary PDE system behaves like ISL a fancy nonlinear oscillator. Amazingly, many features of [email protected] PDE dynamics are reproduces in a 3-dimensional projection of the original dynamical system. We use correlation dimension maps as a tool for investigating underlying dy- MS88 namics. This is a joint work with Michael Frankel of Indi- Noise Controlled Oscillations in Globally Coupled ana University-Purdue University Indianapolis. Elements Victor Roytburd We demonstrate that noise can play a certain ordering Department of Mathematical Sciences role in ensembles of globally coupled oscillators. As ex- [email protected] amples the active rotators (the Kuramoto model) and the FithHugh-Nagumo excitable systems are taken. On approximating the instantaneous distribution of the oscilla- MS89 tors by the Gaussian one, the Langevin equations are re- Signal Transmission by Autocrine Cells in Model duced to dynamical systems which govern the evolution of Epithelial Layers the cumulants. Bifurcation analysis discloses that increase of noise leads from stationary states through complicated Autocrine signaling induced by growth factors is crucial in various stages of development and in adult multicellular organisms across species. At the present level of complexity, DS05 Abstracts 173

systematic evaluation of cell communication mechanisms University of Warwick is next to impossible without mathematical modeling of Mathematics Institute cell signaling networks. In this talk, I will discuss recent [email protected] results of our mechanistic modeling and analysis of Epider- mal Growth Factor Receptor (EGFR)-mediated cell com- Laurette S. Tuckerman munication. Our modeling and analysis allows to charac- LIMSI-CNRS terize signal transmission in epithelial layers in terms of [email protected] the biophysical and geometric parameters of the problem.

Stas Shvartsman MS90 Lewis Sigler Institute for Integrative Genomics Strange Saddles in Pipe Flow and Chemical Engineering Department, Princeton University The fact that turbulent trajectories in the transition region [email protected] can decay without prior indication and without the influence of noise suggests that the turbulent state cannot be Cyrill B. Muratov an permanently sustained attractor. The most likely phase New Jersey Institute of Technology space structure then is that of a chaotic saddle: they are Department of Mathematical Sciences characterized by an exponential distribution of life times [email protected] and a positive Lyapunov exponent for the motion close to the saddle. There is experimental evidence for such saddles in plane Couette flow and numerical evidence in plane MS89 Couette, pipe flow and several models. Surprising Aspects of Traveling Waves with Non- linear Diffusion: Bacterial Growth Vortex Diffu- Bruno Eckhardt sion Models Philipps Universität Marburg Fachbereich Physik Equations with nonlinear diffusion, like in the well known [email protected] porous medium equation, arise in variety of systems. In this talk we will discuss two examples of reaction diffusion type equations with nonlinear diffusion, which admit mov- MS90 ing fronts. The first case concerns a bacterial growth prob- Patterns of Vortices and Jet Streams in Rotating, lem, the second describes the penetration of a vortices in Stratified Shear Flows superconductors. Both equations admit moving front solutions which exhibit some surprising behavior: the bacterial Rotating, stratified flows are nearly 2D and the inverse cas- growth fronts do not properly map onto a moving bound- cade of energy often leads to large, turbulent vortices and ary problem in the thin front limit, while the nonlinear jets. In general, the flows are not unique, and there are vortex fronts show a regime of singular behavior which is several basins of attraction of the flow - each character- different from that expected naievely on the basis of their ized by its own pattern of vortices and jet streams. The similarity with the porous medium equation. transport properties of each pattern vary markedly, so in a geophysical, or climate-change context, the robustness of Wim van Saarloos each pattern and how patterns are selected due to small Instituut-Lorentz changes in the environment are important. We explore Leiden University pattern selection using numerical simulation and statisti- [email protected] cal mechanics. Philip Marcus MS90 University of California at Berkeley Patterns of Turbulence [email protected]

Plane Couette flow – the flow between two infinite parallel plates moving in opposite directions – undergoes a discon- MS90 tinuous transition from laminar flow to turbulence as the Complex Behavior of Large Scale Features of Fully Reynolds number is increased. Due to its simplicity, this Turbulent Flows flow has long served as one of the canonical examples for understanding shear turbulence and the subcritical tran- It is commonly believed that the very large level of noise sition process typical of channel and pipe flows. Only re- arising from small-scale structures of turbulent flows pre- cently was it discovered in very large aspect ratio experi- cludes their large scale features from displaying complex ments that this flow also exhibits remarkable pattern for- behaviour. At variance with this widely accepted view, mation near transition. Steady, spatially periodic patterns we will present recent experimental results obtained in two of distinct regions of turbulent and laminar flow emerges different flow configurations which show that non-trivial spontaneously from uniform turbulence as the Reynolds large-scale behaviour can persist even when considerable number is decreased. The length scale of these patterns small-scale activity is present. is more than an order of magnitude larger than the plate separation. It now appears that turbulent-laminar patterns Louis Marié are inevitable intermediate states on the route from turbu- Laboratoire de Physique des Océans lent to laminar flow in many shear flows. I will explain Université de Bretagne Occidentale how we have overcome the difficulty of simulating these [email protected] large scale patterns and show results from studies of three types of patterns: periodic, localized, and intermittent. MS91 Dwight Barkley Verification of Hyperbolicity and 174 DS05 Abstracts

Non-Hyperbolicity Via Topological Methods Konstantin Mischaikow Department of Mathematics In this talk, we propose two rigorous computational meth- Georgia Tech ods, one for detecting homoclinic tangencies, and the other [email protected] for proving hyperbolicity of invariant sets. These methods are combinations of several tools and algorithms, including the interval arithmetic, the subdivision algorithm, the MS91 Conley index theory, and the computational homology the- A Numerical Method for Proving Hyperbolicity of ory. Applying these methods, we investigate the structure Complex Henon Mappings of the parameter space of the Hénon family. We describe a computer program to establish whether for Zin Arai given a,c, the complex Henon diffeomorphism, H : C2 → Department of Mathematics C2, given by H(x, y)=(x2 + c − ay, x), is hyperbolic. A Kyoto University Henon map is hyperbolic if over each point in its chain re- [email protected] current set, there is a splitting of the tangent bundle into two directions, which are uniformly expanded/contracted by the map. Hyperbolic maps display interesting dynam- MS91 ical behavior, and are amenable to analysis. We will also Covering Invariant Manifolds with Fat Trajectories discuss several examples of program output. An invariant manifold of a flow can be defined as the image Suzanne Lynch Hruska of a smooth curve of initial points under the flow. To com- Indiana University pute the stable and unstable manifolds of a fixed point, [email protected] for example, a small sphere is used in the appropriate eigenspace of the linearize flow. These manifolds can be quite complicated, and the computational problem is es- MS92 sentially a mesh generation problem. We use an approach Nonlinear Dynamics and Instabilities of Coupled used previously for implicitly defined manifolds that repre- Microbeam Arrays for Scanning Probe Microscopy sents the manifolds as the union of a set of spherical neighborhoods, which is the geometric dual of the tiling used in mesh based methods. This avoids the many difficulties Oded Gottlieb with advancing a mesh, but require a method of finding a Faculty of Mechanical Engineering neighborhood of a point on the invariant manifold. Near Technion, Haifa, Israel the starting curve it is straightforward to find such a neigh- [email protected] borhood. The key result we use is a set of equations for the evolution of the neighborhood along a trajectory (a fat trajectory). Since trajectories may diverge, we also de- MS92 rive a result that ensures the existance of an interpolation Dynamics of Nonlinear Coupled MEMS Res- point in the cleft of the diverging trajectories. The result onators is a covering of the manifold, with neighborhoods of points which either lie on a trajectory or are interpolated from We are studying the dynamics of nonlinear coupled os- the interior of a simplex whose vertices lie on trajectories. cillators, motivated by recent experiments with arrays of The method based on these results works for any dimen- MEMS resonators at Caltech and Cornell. Our studies to sions (of the manifold and the embedding space), though date have focused on the weakly nonlinear regime where large pieces of manifolds of dimension larger than four are we have looked at (1) The response of 1-dimensional ar- probably too expensive to compute. We demonstrate the rays of coupled oscillators to parametric excitation; and algorithm on the stable manifold of the origin in the Lorenz (2) The synchronization of coupled mechanical resonators equations. with a distribution of frequencies. We have obtained exact results for the parametric excitation of small arrays using Michael E. Henderson secular perturbation theory [1], as well as an amplitude IBM Research equation to describe the slow dynamics of the parametric TJ Watson Research Center excitation of large arrays [2], with many features in com- [email protected] mon with Faraday waves. We have investigated a model of synchronization, based on reactive coupling and nonlinear frequency pulling [3] (rather than the more common linear MS91 dissipative models), obtaining a phase diagram for the on- Rigorous Numerics for Continuation Methods in set of synchronization, exhibiting interesting hysteretic be- Infinite Dimension havior. [1] PRB 67 (2003) 134302; [2] cond-mat/0411008; [3] PRL 93 (2004) 224101. In this talk, we will introduce a rigorous numerical method to continue the equilibria of dissipative parameter depen- Ron Lifshitz dent partial differential equations. This method is based School of Physics and Astronomy on a standard finite dimensional path-following algorithm Tel Aviv University adapted to the infinite dimensional case. We will present [email protected] a theorem of existence and uniqueness for the equilibria branches of the original PDE and discuss the computa- Michael C. Cross tional complexity of the method. Department of Physics California Institute of Technology Jean-Philippe Lessard [email protected] Georgia Institute of Technology [email protected] DS05 Abstracts 175

MS92 [email protected] The Stochastic Dynamics of Micron and Submicron Scale Mechanical Oscillators MS93 The stochastic dynamics of micron and submicron scale Linking Mechanistic Models with Epidemiological oscillators of arbitrary geometry immersed in a viscous Data: Parameter Estimation in the Face of Incom- fluid will be discussed. It will be shown that by using the plete Information fluctuation-dissipation theorem it is possible to calculate, using deterministic calculations, the stochastic dynamics The connection of ecological time-series data with mecha- that would be measured in experiment. This approach is nistic models poses severe statistical challenges. Typically, used to investigate the motion of single and multiple os- the dynamical models themselves are stochastic and non- cillators for a variety of experimentally realistic geometries linear, only a few state variables can be observed, and these for use in making single molecule measurements. only with error. To deal with these problems, researchers have often been forced to incorporate unpalatable assump- Mark Paul tions in their models for the sake of the statistics. We Department of Mechanical Engineering describe the method of simulated composite likelihood,a Virginia Tech general approach that requires no such assumptions. We [email protected] apply the method to measles, whooping cough, and cholera incidence data, using mechanistic continuous-time models for transmission dynamics to obtain insight into the dy- MS92 namics of epidemics. Sensors Utilizing Parametric Resonance Subhash Lele Statistics Kimberly Turner University of Alberta Dept Mech Env Eng [email protected] UC Santa Barbara [email protected] Pejman Rohani Institute of Ecology MS93 University of Georgia [email protected] Complex Intermittent Dynamics in Population Bi- ology: A Tale of Weak Transverse Stability Aaron A. King New dynamical scenarios with extreme complex behaviors University of Tennessee can help us to understand the dynamical complexity of eco- Ecology and Evolutionary Biology logical systems. One of these scenarios leads to a type of [email protected] intermittence, named on-off intermittency that is related to the presence of an invariant subspace and to its weak Mercedes Pascual transversal unstability. In the dynamics of a community Dept. of Ecology and Evolutionary Biology this subspace can be defined by the extinction of a rare University of Michigan species. The weak invading capacities of this species are [email protected] responsible of the intermittent dynamics, owing to species competitions, species interactions and/or to the fluctuations of the environmental capacities. This kind of inter- MS93 mittent dynamics could explain the extreme fluctuations Panel Discussion seen on marine communities. Bernard Cazelles James Nichols, Linda J. Moniz CNRS UMR 7625 U.S. Geological Survey Ecole Normale Superieure jim [email protected], [email protected] [email protected] MS95 MS93 Solutions of the NLS-Whitham Equations and Gen- The Dynamical Detective: Using Nonlinear Models eration of Intense Optical Pulses to Test Ecological Hypotheses

Roughly one in three animal populations exhibits some- Gino Biondini what cyclic dynamics. A plethora of explanations exists State University of New York at Buﬀalo but we rarely know which applies to a given population: Department of Mathematics empirical tests are usually infeasible, or inconclusive be- biondini@buﬀalo.edu cause multiple mechanisms are operative. I will describe case studies where statistically rigorous comparisons of MS95 process-based nonlinear models have resolved longstand- ing questions or overturned established explanations for Asymptotic Stability of Ground States in NLS long-term cycles, and a laboratory model system where My talk will focus on the stability of nonlinear boundstates this approach has been validated experimentally. that bifurcate from the linear ones. Variational techniques Steven Ellner have been developed to study their orbital stability. More Department of Ecology and Evolutionary Biology recently, center manifold methods have been used to show Cornell University their aymptotic stability in supercritical regimes, for ex- 176 DS05 Abstracts

ample cubic NLS in 3-d. I will present a reﬁnement of turbed systems that contain both slow and fast segments. the latter that includes results for the critical regimes, for Canards are trajectory segments that lie on an unstable example cubic NLS in 2-d. slow manifold. Because of their instability, it is almost impossible to compute canards by solution of initial value Eduard Kirr problems forward in time. We explain this more fully with University of Chicago illustrations and describe the use of boundary value meth- [email protected] ods to compute periodic orbits containing canards. John Guckenheimer MS95 Cornell University Thermally Induced Dynamics and Pattern Forma- [email protected] tion in Optical Parametric Oscillators

Optical parametric oscillators (OPOs) are an important MS96 source of laser-quality light in the far infrared. When op- Slow-Fast Time Scales and the Phase-Resetting Re- erated at high average powers, absorption can lead to sig- sponse of Cardiac Oscillators nificant heating of the gain medium, changing the cavity properties and leading to thermal lensing and deformation Injection of a brief stimulus pulse phase-resets the spon- of the transverse beam profile. To understand this process taneous periodic activity of cardiac pacemaker celss: an better, we consider the formation and evolution of patterns earlier stimulus generally delays the next action potential, in reduced models of OPOs coupled to a diffusion equation while a later one causes an advance. In a slow-fast system for the temperature. the transition from delay to advance can appear discontinuous, as seen in experiments on some cardiac prepara- Richard O. Moore tions. Using continuation methods in AUTO, we show that New Jersey Institute of Technology the apparent discontinuity occurs when stimuli transport [email protected] state-points to either side of the stable manifold of the weak unstable manifold. MS95 Leon Glass Persistence and Stability of Discrete Vortices in McGill University Discrete NLS Department of Physiology [email protected] Dmitry Pelinovsky McMaster University, Canada Michael R. Guevara [email protected] Department of Physiology McGill University [email protected] MS96 Computing Eusebius J. Doedel One-Dimensional Manifolds of Poincare Maps in Department of Computer Science Slow-Fast Systems Concordia University [email protected] We present an algorithm to compute the one-dimensional stable and unstable manifolds of a saddle-periodic orbit on Trine Krogh-Madsen a Poincaré section by setting up the problem as a boundary McGill University value problem. The start point of the orbit is varied along [email protected] a piece of manifold previously computed, such that the end point traces out a new section of manifold. In this way, we compute manifolds for a slow-fast chemical oscillator for MS96 the first time. A Unifying Framework for Reduction Methods

Hinke M. Osinga The identification of low-dimensional manifolds containing University of Bristol the essential dynamics of multiscale systems is essential to Department of Engineering Mathematics the construction of reduced models that can be simulated [email protected] efficiently. Here, we examine two reduction techniques, the Zero-derivative Principle of Gear and Kevrekidis and the Bernd Krauskopf Computational Singular Perturbation of Lam and Gous- University of Bristol sis. We review briefly their approximation properties and Dept of Eng Mathematics construct a unifying framework by showing that the tech- [email protected] niques’ successive iterations generate new state space coordinates similar to the Fenichel coordinates. James P. England Department of Engineering Mathematics Antonios Zagaris University of Bristol Boston University [email protected] [email protected]

MS96 MS97 Computing Canards in Relaxation Oscillations Extreme Ship Motion Prediction Capabilities of

Relaxation oscillations are periodic orbits in singularly per- DS05 Abstracts 177

the US Navy ing and for capsize in the neighborhood of wave crests.

Ships of the U.S. Navy must not only survive in large seas, Kostas Spyrou but also continue to operate effectively. Predictions of large School of Naval Architecture and Marine Engineering amplitude motions and loads are attained through a mix of National Technical University of Athens model testing and numerical simulations which must be of [email protected] sufficient fidelity to capture the dominant nonlinear forces. While model testing will never be replaced, there is increased confidence in the ability to use simulation tools to MS98 assess maximum wave loads and dynamic stability perfor- Attractors for Stochastic Lattice Dynamical Sys- mance. tems

William Belknap We consider a one-dimensional lattice with diffusive near- Seakeeping Division of Hydromechanics Department est neighbor interaction, a dissipative nonlinear reaction Naval Surface Warfare Center Carderock Division term and additive independent white noise at each node. [email protected] We prove the existence of a compact global random attractor within the set of tempered random bounded sets. An interesting feature of this is that, even though the spa- MS97 tial domain is unbounded and the solution operator is not On the Practical Ergodicity of Parametric Rolling smoothing or compact, pulled back bounded sets of initial data converge under the forward flow to a random com- Experimental and numerical results regarding the problem pact invariant set. This is joint work with Kening Lu and of ergodicity of parametric roll in longitudinal irregular Hannelore Lesei. waves indicate that temporal averages can be associated to very large coefficients of variation, even though their Peter Bates expected values are theoretically correct estimators of en- Michigan State University semble averages. It has been shown that, in some cases, the Department of Mathematics analysis of time series of typical length (30min full scale) [email protected] could be useless if carried out on a single realization. Alberto Francescutto MS98 University of Trieste, Italy Modulation Equations and Stochastic Bifurcations tba for Large Domains

Gabriele Bulian We consider a class of SPDEs (e.g. stochastic Swift- Naval Architecture, Ocean and Environmental Hohenberg equation) on large domains. Near its change of Engineering stability we rigorously verify, under the appropriate scal- University of Trieste, Italy ing, that solutions can be approximated by a periodic wave, [email protected] which is slowly modulated by the solutions to a complex stochastic Ginzburg-Landau equation. This approxima- Claudio Lugni tion also extends to invariant measures of these equations. INSEAN J.w.w. M.Hairer (Warwick) and G.Pavliotis (Imperial). Italian Ship Model Basin, Rome Dirk Blomker [email protected] RWTH Aachen [email protected] MS97 Nonlinear Ship Rolling Motion and Capsizing in MS98 Random Waves Pullback Attractors for Asymptotically Compact TBD Nonautonomous/Random Dynamical Systems Jeﬀrey Falzarano We introduce the concept of pullback asymptotically com- School of Naval Architecture and Marine Engineering pact non-autonomous/random dynamical system as an ex- University of New Orleans tension of the similar concept in the autonomous frame- [email protected] work, and prove a result ensuring the existence of a pullback attractor for a non-autonomous/random dynamical system under the general assumptions of pullback asymp- MS97 totic compactness and the existence of a pullback absorbing Ship Instabilities Due to the Asymmetric Surging family of sets. Finally, we illustrate the theory with a 2D of Ships in a Following Seaway Navier-Stokes model in an unbounded domain.

The strongly nonlinear response of a ship in a following Tomas Caraballo seaway where it could be performing large-amplitude lon- Departamento de Ecuaciones Diferenciales y An´alisis gitudinal oscillations around a mean forward speed will be Num´erico discussed. An improved mathematical model of ship surg- University of Sevilla, 41080 Sevil, S ing in steep following wave has been developed. Analytical [email protected] prediction formulae for the higher limit of asymmetric surging (threshold of global surf-riding) will be reported. Also MS98 will be discussed the role of asymmetric surging for broach- Smooth Conjugacy for Random Dynamical Sys- 178 DS05 Abstracts

tems cal measures.

The study of reducing a nonlinear deterministic dynami- Rohit Kumar cal system to the simplest possible form (called a normal Thomas Jefferson High School of Science and Technology form) goes back to Poincare and Birkhoff. In this talk, I [email protected] will report our recent work on smooth conjugacy for random dynamical systems. Our results include extensions of Steven Weinstein Poincare and Sternberg’s theorems in deterministic cases Children’s National Medical Center to random dynamical systems, which are based on the Lya- [email protected] punov exponents. Random dynamical systems arise in the modeling of many phenomena in physics, biology, climatol- Tim Sauer ogy, economics, etc. when uncertainties or random influ- Department of Mathematics ences, called noises, are taken into account. These random George Mason University effects are not only introduced to compensate for the de- [email protected] fects in some deterministic models, but also are often rather intrinsic phenomena. This is joint work with W. Li. Steven J. Schiff Kening Lu George Mason University Brigham Young University The Krasnow Institute [email protected] sschiff@gmu.edu

MS99 MS99 Synchronization Measures in Seizure Prediction Are Seizures Temporally Interdependent? and Detection The effect seizures may have on subsequent seizures has We present results of applying various measures of dy- received little attention. In an experimental (rat) model, namical system synchronization, including new measures short (long) seizures followed short (long) interictal peri- derived from the Intrinsic Timescale Decomposition, to ods with statistically significant probability. Non-random long ECoG recordings. These measures are compared with seizure recurrence in humans was investigated using pro- other nonlinear dynamical measures, results from real-time longed ECoG; the results were compared with randomized automated seizure detection algorithm outputs, and expert surrogate data. Significant nonlinear dependencies of the visual analysis. Implications for seizure prediction algo- interseizure interval were found (p<0.05) in over half of rithms using these measures will be discussed. 24 subjects. Similar outcomes were obtained for the hourly seizure frequency and measures of seizure severity. Mark G. Frei Supported by NINDS/NIH Grant NS046060-01. Flint Hills Scientific, L.L.C. [email protected] Sridhar Sunderam, Mark Frei Flint Hills Scientific Ying-Cheng Lai [email protected], [email protected] Arizona State University Department of Mathematics Ivan Osorio [email protected] University of Kansas Flint Hills Scientific, L.L.C. Thomas Peters [email protected] Flint Hills Scientific, L.L.C. [email protected] MS99 Do Changes in the EEG Dynamics Permit a Pre- Ivan Osorio diction of Epileptic Seizures? University of Kansas Flint Hills Scientific, L.L.C. Epilepsy patients suffer from so far unforeseeable seizures. [email protected] To enable short term medical intervention a reliable seizure prediction method is desired. Recently, several seizure Mary Ann Harrison prediction methods have been proposed. We present a Institute for Scientific Research, Inc. methodology to assess and compare seizure prediction [email protected] methods. It is applied to statistically evaluate several prediction methods. Furthermore, we discuss electroencephalogram related characteristics of seizure prediction MS99 methods. Based on these, extensions of existing prediction Dynamical Evolution of Seizures methods are suggested.

We developed a novel geometrical interpretation and nu- Matthias Winterhalder merical approach to canonical multivariate linear discrim- Center for Data Analysis and Modelling ination of Fisher (1936). We found signiﬁcant extraction University of Freiburg, Germany of unique initial, middle, and terminal phases from 21 of [email protected] 24 scalp and intracranial recordings using discrimination of dynamical measures. No consistent increased synchro- Bjoern Schelter, Thomas Maiwald, Ariane Schad nization was evident within the initial or terminal phases Center for Data Analysis and Modeling of these seizures. These results argue for an evolution of University of Freiburg, Germany seizure patterns that can be partitioned based on dynami- [email protected], DS05 Abstracts 179

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

Armin Brandt, Andreas Schulze-Bonhage Epilepsy Center MS100 University Hospital Freiburg, Germany Localized Structures in Two-dimensional Photonic [email protected], Lattices [email protected] Ziad Musslimani Jens Timmer University of Central Florida University of Freiburg Department of Mathematics Department of Physics [email protected] [email protected]

MS101 MS100 Extracting Low-Order Stochastic Models From The Stability of Periodic Patterns in Singular Per- Data turbed Reaction-Diffusion Equations We present a numerical technique to derive optimal Recently, the stability of localized pulse solutions to sin- stochastic models (Markov chain or SDEs) for the descrip- gular perturbed reaction-diffusion equations of Gray-Scott tion of the evolution of a few interesting collective variables and Gierer-Meinhardt type has been established with the in large-sized dynamical systems. The technique is based use of Evans function methods. Apart from these localized on constructing the stochastic model whose eigenfunctions solutions, these equations also posses large classes of often and eigenvalues are the closest (in some appropriate norm) highly non-trivial, stationary, spatially periodic patterns. to the one gathered from the observations. The technique In the present talk, the stability of these solutions will be is validated on an example arising from climate dynamics, studied with the use of a generalization of the Evans func- namely by extracting a stochastic model for the evolution tion based on Floquet theory. In the context of the gen- of the first few principal orthogonal modes observed in a eralized Gierer-Meinhardt equation, it will be shown that general circulation model. the spectra associated to the stability of the patterns can be computed explicitly. Moreover, bifurcations between Eric Vanden-Eijnden different patterns, and the associated changes in stability, Courant Institute can be studied in full detail. This talk is based on joint [email protected] work with Harmen van der Ploeg (Amsterdam). Daan Crommelin Arjen Doelman Courant Institute CWI Amsterdam, the Netherlands New York University [email protected] [email protected]

MS100 MS101 Mixing in the Kuppers-Lortz Mode Mathematical and Computational Strategies for Stochastic Multiscale Problems: Coarse-Graining, Keith Julien Loss of Information University of Colorado [email protected] Markos A. Katsoulakis UMass, Amherst MS100 Dept of Mathematics [email protected] New Results on the Stability of Pulse-like Solutions to a Coupled Nonlinear Klein-Gordon System MS101 When subject to sufficient twist, an elastic filament kept under tension typically undergoes a writhing bifurcation. Analysis of Multiscale Methods for Stochastic Dif- Near threshold, the corresponding dynamics may be mod- ferential Equations eled by two coupled nonlinear Klein-Gordon equations, We analyze a class of numerical schemes proposed in [25] which are envelope equations for the amplitudes of the lo- for stochastic di erential equations with multiple time- cal deformation and twist of the filament. I will present scales. Both advective and di usive time-scales are con- a necessary and sufficient condition for the spectral sta- sidered. Weak as well as strong convergence theorems are bility of pulse solutions to these envelope equations. This proved. Most of our results are optimal. They in turn al- result, obtained by Hamiltonian methods, completes and low us to provide a thorough discussion on the e ciency as extends the analysis of S. Lafortune and J.L. (Physica D well as optimal strategy for the method. 182, 103-124 (2003)), in which a sufficient condition for the instability of “non-rotating” pulses was found by means of Di Liu Evans function techniques. I will also discuss a different Courant Institute, New York University way of numerically evaluating the Evans function, and test [email protected] it against the above analytical results. This work is joint with Stéphane Lafortune and Silvia Madrid-Jaramillo. MS101 Joceline Lega Modulation Equations: Stochastic Bifurcation in University of Arizona, USA 180 DS05 Abstracts

Large Domains [email protected]

We consider the stochastic Swift-Hohenberg equation on a large domain near its change of stability. We show that, MS102 under the appropriate scaling, its solutions can be approxi- Quasi-Periodicity in Dissipative and Conservative mated by a periodic wave, which is modulated by the solu- Systems tions to a stochastic Ginzburg-Landau equation. We then proceed to show that this approximation also extends to Kolmogorov-Arnold-Moser (of KAM) theory was devel- the invariant measures of these equations. This is joint oped for conservative dynamical systems that are nearly work with D. Bl¨omker and M. Hairer. integrable. Integrable systems in their phase space usually contain lots of invariant tori, and KAM theory establishes Grigorios Pavliotis persistence results for such tori, which carry quasi-periodic Imperial College motions. We sketch this theory, which begins with Kol- [email protected] mogorov’s pioneering work (1954). Henk Broer MS101 University of Groningen Stochastic Modelling of Complex Dynamics Ex- Department of Mathematics hibiting Metastability [email protected]

Christof Schuette MS102 University of Berlin Hamiltonian Torus Bifurcations Related to Simple Germany Singularities [email protected]

Heinz Hanssman MS101 Technische Universiteit Aachen Reduced Dynamics for a Class of Conservative Sys- [email protected] tems

Novel approach to derivation of reduced dynamics for a MS102 class of energy-conserving systems is discussed. Asymp- Computation of Invariant Tori in Quasi-Periodic totic analysis and microcanonical averaging are utilized to Systems derive closed-form reduced equations for long-term evolution of slow subset. No ad-hoc assumptions for the sta- We explain the parameterization method to compute in- tistical properties of the fast heat bath are made; instead, variant tori and their whiskers in quasi-periodic systems. single microcanonical simulation is utilized to estimate sta- We apply the method to study two examples, that are tistical properties of the fast modes for all energy levels. quasi-periodic perturbations of the Hnon map and the Truncated Burgers-Hopf model is utilized as an example. standard map. Andy Majda Rafael de La Llave NYU University of Texas [email protected] Department of Mathematics [email protected] Eric Vanden-Eijnden Courant Institute Alex Haro [email protected] Universitat de Barcelona Universitat de Barcelona Gran Via 58508007 Barcelona Ilya Timofeyev (Spain) Dept. of Mathematics 716865 Univ. of Houston [email protected] MS102 Interactions of Torus Bifurcations in Maps with Di- MS102 mension ¿2 Torus Destruction and Chaotic Behavior in a Sys- tem of Two Coupled Oscillators In this talk I describe two remaining cases of codimension 2 bifurcations for maps. First a normal form analysis is A one mass system with two degrees of freedom is con- given, which in turn is applied to an example. These bi- sidered here. Using the normal form method of averaging furcations need maps with dimension at least 3 or 4, but we derive conditions for stability of the periodic solution mechanical systems with only a few building blocks may and spot a T2-torus by a Neimark-Sacker bifurcation. We display such bifurcations already as we will show. show how this torus breaks down, through one period doubling (Shilnikov’s scenario), by numerically following the Hil Meijer changes in the involved manifolds. The strange attractor University of Utrecht and chaotic behavior following the breakdown of the torus Mathematics Institute are studied in detail. [email protected] Taouﬁk Bakri Mathematics Institute, Utrecht University, PO Box 80.010 3508 TA Utrecht DS05 Abstracts 181

MS102 film damping appears. This phenomenon introduces a fre- Continuation ofQuasi-Periodic Tori quency (and consequently an amplitude) dependent force. In this research the contribution of the squeeze-film phe- We present a new algorithm for the continuation of quasi- nomenon in MEMS dynamics is simulated by a nonlinear periodic invariant tori of ordinary differential equations in amplitude dependent function. sufficiently many parameters. The proposed method requires neither the computation of a base of a transversal G. N. Jazar bundle, nor re-meshing during continuation. It is inde- North Dakota State University pendent of the stability type of the torus and can step over [email protected] quasi-periodic bifurcation points. We demonstrate the performance of the method with examples. MS103 Frank Schilder Touchdown Dynamics in Electrostatic MEMS Department of Engineering Mathematics University of Bristol [email protected] John A. Pelesko University of Delaware [email protected] MS103 Routes to Pull-in MS103 We study the nonlinear dynamics of electrostatic MEMS to Flow Control inside Micro-Fluidic Systems: Mod- determine the underlying mechanisms leading to the pull-in eling, Sensing, and Feedback Control Design phenomenon. Pull-in occurs due to the non-existence of an equilibrium configuration corresponding to the applied DC Micro-fluidic technology has the potential to allow hand- voltage. We found several dynamic mechanisms where pull- held devices with the functionality of existing biological in developed even though there was a stable equilibrium and chemical laboratories, it can be used to create wearable position corresponding to the applied DC voltage. Any of or implantable drug delivery platforms, and it allows direct these mechanisms can lead to pull-in, depending on the handling of biological materials such as cells, proteins, and excitation level and initial conditions of the device. DNA. In our research we have found that feedback control is often required for robust micro-fluidic performance (as Eihab Abdelrahman, Ali Nayfeh in the UCLA electro-wetting devices) and it allows new Virginia Tech capabilities in other cases (as in our multi-particle steer- Department of Engineering Science and Mechanics ing system). This talk will address our efforts to integrate [email protected], [email protected] research in system design and feedback control with the rapid progress being made in micro-fluidic systems. Re- sults will be shown for two physical systems. The first is MS103 the Electro-Wetting-On-Dielectric (EWOD) system devel- Discontinuity-Driven Design and Control of Impact oped at UCLA by CJ Kim. Here a grid of electrodes is used Microactuators to locally change surface tension forces on liquid droplets: by choosing the electrode firing sequence it is possible to Impact microactuators rely on repeated collisions to gener- move, split, join, and mix liquids in the droplets. I will ate gross displacements of machine elements. Their design describe our modeling, vision sensing, and control results relies on an understanding of the critical transition between for this system. In particular, I will show our algorithms nonimpacting and impacting dynamics. A normal-form for precision control of droplet splitting and joining, and analysis is presented that predicts the character of such I will describe how feedback can be used to correct for transitions from a set of conditions that are computable in an unknown external environment. The second system is a terms of system properties at grazing. The analysis also micro-fluidic ”no-laser tweezer” system that can be used to suggests opportunities for using passive design or active steer many particles at once. I will show how we use flow control to regulate the near-grazing response. control to create an underlying, time-varying fluid flow that carries all the particles at once along their desired trajec- Harry Dankowicz tories, and I will describe the status of our experiments at Virginia Polytechnic Institute and State University Maryland and at NIST. Because the system does not re- Engineering Science and Mechanics quire lasers and high quality optics with long optical path [email protected] lengths, it is cheap and it can be miniaturized. This system is being used to steer cells for a ”cell clinics” project at Xiaopeng Zhao the University of Maryland. I will close the talk by outlin- Department of Engineering Science and Mechanics ing some of the key motivations, benefits, bottlenecks, and Virginia Polytechnic Institute and State University remaining open challenges in integrating feedback control [email protected] and micro-fluidic systems research. Benjamin Shapiro MS103 University of Maryland Effects of Squeeze Film Damping on Stability and College Park, MD Dynamics of Microbeam-based MEMS [email protected]

Damping in microelectromechanical systems strongly affects their performance, design, and control. Typical mi- MS103 croresonators employ a parallel-plate capacitor, in which Generalized Parametric Resonance in Electrostat- one plate is actuated electrically and its motion is detected by capacitive changes. Under such conditions a squeeze- 182 DS05 Abstracts

ically Driven MEM Oscillators some application to competition reaction diﬀusion systems. Wenxian Shen Steven W. Shaw Department of Mathematics Department of Mechanical Engineering Auburn University, Auburn, AL 36830 Michigan State University [email protected] [email protected]

Jeffrey Rhoads MS104 Dept. of Mechanical Engineering The Effects of Noise on Transient Pattern Forma- Michigan State University tion [email protected] Transient phase separation processes can create compli- Jeffrey Moehlis, Barry Demartini cated evolving patterns. Typical underlying models often Dept. of Mechanical Engineering include both deterministic and stochastic partial differen- University of California-Santa Barbara tial equations, and it is therefore natural to ask in what [email protected], [email protected] way stochasticity impacts the dynamics and the geometry of the produced patterns. In this talk I will present both theoretical and computational results addressing similari- Kimberly Turner ties and differences in stochastic models which are obtained Dept Mech Env Eng from deterministic ones by adding an additive noise term UC Santa Barbara of variable intensity. [email protected] Thomas Wanner George Mason University MS104 Department of Mathematical Sciences Shanon’sLimit for a Class of Nonlinear Circuits [email protected] We will consider a class of singularly perturbed nonlinear differential equation under external forcing via noisy chan- MS105 nels modeling a modulation scheme of converting analog Periodic Orbits and Chaotic Sets in a Low- signals to digits. Shannon’s information limit is defined for Dimensional Model for Turbulent Shear Flows this class of equations and is shown to be computable by using a PDE method based on the Fokker-Planck equation. We analyse a low-dimensional model for turbulent shear Generalizations of this method will also be given. flows. The model is derived for sinusoidal shear flow, in which fluid between two free-slip walls experiences a sinu- Shui-Nee Chow soidal body force, and is based on Fourier modes represent- School of Mathematics, ing important physical structures. The model illustrates Georgia Institute of Technology, Atlanta, GA 30332 many phenomena observed and speculated to exist in the [email protected] transition to turbulence, including subcritical and intermittent transition, exponential distributions of turbulent MS104 lifetimes, and unstable equilibria and periodic orbits. Dynamics of Infinite-dimensional Stochastic Sys- Holger Faisst tems Philipps Universitat Marburg Fachbereich Physik We present a non-linear multiplicative ergodic theorem [email protected] for infinite-dimensional stochastic semiflows on a Hilbert space. The theorem yields the existence of local stable and unstable manifolds in the neighborhood of hyperbolic sta- Jeff Moehlis tionary solutions of semilinear stochastic evolution equa- Dept. of Mechanical and Environmental Engineering tions and stochastic functional differential equations. University of California – Santa Barbara [email protected] Salah Mohammed Department of Mathematics Bruno Eckhardt Southern Illinois University, Carbondale, IL 62901 Philipps Universität Marburg [email protected] Fachbereich Physik [email protected] MS104 Principal Spectrum Theory for Random Parabolic MS105 Equations POD-Galerkin Models of Open Shear Flows: What Can we Expect of Them? In the current talk, I will present some recent joint work with Janusz Mierczynski on principle spectrum for random The last two decades of theoretical fluid mechanics have parabolic equations. Among others, I will show that the seen increased interest in low-dimensional models of flows. principal Lyapunov exponent of a random parabolic equa- However, the reduced complexity of low-dimensional mod- tion is greater than or equal to the principal eigenvalue of els does comes at a cost that is not always well understood. the associated time-averaged equation, which has an im- We will look at the way low-dimensional models can com- portant biological implication, that is, heterogeneous envi- plement the toolkit of the fluid dynamicist, and what their ronment favors population’s persistence. I will also discuss limitations are. The use of low-dimensional models leads DS05 Abstracts 183

to questions that are sometimes ignored, at a peril. Model Based Design for the Cylinder Wake Dietmar Rempfer Control oriented Galerkin flow models must combine sim- Illinois Institute of Technology plicity and ample dynamic range. Enablers include ‘sub- [email protected] grid’ estimation of turbulence and pressure representations, modes from multiple operating points, and actuation models. An invariant manifold defines the models dynamic MS105 envelope. It must be respected and can be exploited in Control-Oriented Models of Channel Flow observer and control design. These ideas are benchmarked in the cylinder wake system and validated by a systematic Reduced-order models of linearized flow in a plane chan- DNS investigation of a 4-dimensional Galerkin model of nel are presented, using several different model-reduction the controlled wake. techniques, including Proper Orthogonal Decomposition (POD), balanced truncation, and a new method called bal- Bernd Noack anced POD. These models are used to obtain observers and Technical University Berlin controllers to damp disturbances, and to explore funda- [email protected] mental limits in performance. The eventual goal is to use feedback to stabilize the laminar flow and delay transition Gilead Tadmor to turbulence. Northeastern University [email protected] Clancy W. Rowley Princeton University [email protected] MS106 The Colley Matrix Ranking Method

MS105 The National Championship in college football is, to my Stability and Accuracy of Periodic Flow Solutions knowledge, unique in major sports in that it features a Obtained by a POD-Penalty Method playoff of only two teams in a single game to determine the national champion. While accurate seeding is obviously im- We develop a new penalty method to derive low- portant in any tournament, nowhere is it more paramount dimensional Galerkin models for fluid flows with time- than in a two-team, one-game playoff. As such, several dependent boundary conditions. We then outline a proce- different inputs are used, including ”human” polls, which dure based on bifurcation analysis in selecting the proper amount to surveys of sportswriters and coaches across the values of the penalty parameter(s) that yield asymptoti- country, but also six computer ranking systems, of which cally stable periodic solutions of the highest possible ac- the Colley Matrix is one. The method was developed with curacy. We illustrate this new approach by studying flow simplicity in mind, focusing only upon wins and losses, past a circular cylinder using direct numerical simulation and ignoring such factors as margin of victory, chronology data, and a wave-structure interaction problem using par- and play at home or away. The method is, as such, less a ticle image velocimetry data. predictor of future outcomes than a hindcaster of accumu- George Karniadakis lated merit, and is therefore ideal for determining which Brown University teams ”deserve” to play for the national championship. [email protected] The mathematical method is an iterative extension of a simple formula used by Laplace to determine probabilities of binary outcomes (such as football games). The iterative Sirod Sirisup method has an available linear solution, called the Colley Brown University Matrix method. Division of Applied Mathematics [email protected] Wes Colley ODU [email protected] MS105 Coarse Dynamics of Plane Couette Flow MS106 We describe a technique for the efficient computation Adaptive Scheduling of plane Couette flow when only a well-resolved and computationally-costly simulation is available. Instead of Conventional schedules have been found to be insufficient using the standard POD-Galerkin technique, we adopt a in many sports to determine the best k teams (Gibbons, framework which does not require explicit equations for the Olkin, and Sobel, 1978; Weiss, 1986). Tournaments, by evolution of these coherent structures. Rather, a computa- being adaptive, determine the best teams with greater pre- tional superstructure is designed to combine short burts of cision and efficiency. However, the methods used to seed DNS with relatively coarse integration of the large scales teams in tournaments is suboptimal. We suggest an adap- to extract the dynamics of the dominant features. tive scheduling algorithm that attempts to optimize the probability of correctly selecting the best k teams, and Troy R. Smith demonstrate the method with college football data. California Institute of Technology [email protected] Mark Johnson St. Louis Cardinals SportMetrika MS105 [email protected] Control Oriented Empirical Galerkin Models and Matthew S. Johnson 184 DS05 Abstracts

Baruch College, City University champion. SportMetrika matthew [email protected] Paul K. Newton Univ Southern California Dept of Aerospace Engineering MS106 [email protected] A Multi-Dimensional Ranking Approach to Intran- sitive Relations Kamran Aslam USC Modern information systems are challenged with extract- [email protected] ing useful information from large data sets collected from sprawling, and potentially unreliable networks. A strik- Joseph Keller ingly similar task is to accurately rate football teams, which Stanford University play short disparate schedules, and whose performance [email protected] may vary signiﬁcantly from week to week. Many models provide a one-dimensional ’rating’ of a team’s strength. We will discuss the limits of such schemes and explore general- MS106 ized approaches to multi-dimensional evaluations for teams Least Squares-Gaussian Sports Predictions: Some in competitive sports. Fundamental Conclusions

Ken Massey Since 1971, I have been using a least squares approach Hollins University to rate teams in American college and professional foot- Dept of Mathematics ball, American college basketball, Australian Rules foot- [email protected] ball, European soccer (England, Italy, Norway and Ger- many), Super 12 rugby union and the Zurich Premiership MS106 in rugby union. The ratings are based on win margin adjusted for home advantage. Predictions follow using the Random Walker Rankings for College Football ratings, a shrinking factor, and home advantage. The re- Using only win-loss information of games, we develop a sults of those 30+ years provide insight into sports and family of rankings defined by random walks on the bi- sports predictions, such as the relative accuracy of various ased graph of teams (vertices) and games played (edges). prediction schemes (no real difference, actually), home ad- These ranking rules are easily explained in terms where vantage in various sports (regular season and playoff com- the random walkers represent fickle voters. We investigate petition), ease of scoring as it affects predictability and statistical properties and examine the asymptotics of the gambling strategies. rankings at extreme values of the bias parameter. We also Ray Stefani investigate the connection between the rankings and the Cal State Long Beach underlying community structure of the network of games [email protected] played.

Mason A. Porter MS107 Georgia Institute of Technology School of Mathematics and Center for Nonlinear Science Coherent Spontaneous On-going Activity in Cortex [email protected] It has been shown experimentally that spontaneous cortical activity in the absence of sensory inputs modulates Peter J. Mucha stimulus evoked activities and is correlated with behavior. Georgia Institute of Technology In the visual cortex, there is a close relationship between School of Mathematics ongoing spontaneous activity and the spontaneous firing of [email protected] a single neuron. There are dynamical switching amongst these spontaneous cortical states, which may span sveral Thomas Callaghan hypercolumns spatilaly and are closely associated to ori- School of Mathematics entation maps. We will present our theoretical modeling Georgia Institute of Technology results to illustrate a possible mechanism underlying this [email protected] spontaneous cortical activity and discuss further experimental evidence consistent with our theoretical model. MS106 David Cai Probabilistic Models for Tennis Courant institute New York Unvirsity A model for computing the probability of winning a game, [email protected] a set, and a match in tennis is described, based on each player’s probability of winning a point on serve. Both two out of three and three out of five set matches are consid- MS107 ered, allowing a 13-point tiebreaker in each set, if necessary. Experimental Status of Electical Waves In Vivo The question of whether it is advantageous to serve first or receive first is answered. Then the probability of each player winning a 128 player tournament is calculated. Data David Kleinfeld from the 2002 US Open and Wimbledon tournaments are Physics used both to validate the theory as well as to show how pre- University of California at San Diego dictions can be made regarding the ultimate tournament [email protected] DS05 Abstracts 185

Samar Mehta Department of Mathematical Sciences Neurobiology New Jersey Institute of Technology Uviversity of California at San Diego [email protected] [email protected] MS108 MS107 Modulated Structures in a Modulationaly Stable Reverse-Correlation Techniques and Cortical Ar- Optical Interferometer chitecture Evidence of periodic dissipative structures with an intrin- Reverse-time correlation measurments give the average ori- sic wavelength in a nonlinear optical system devoid of Tur- entation dynamics of individual neurons within a highly ing instability is given. They are found in the transverse excited visual cortical neuronal network. The resulting ori- field distribution of a nonlinear interferometer formed by a entation tuning curves provide specific information about Liquid Crystal Light Valve with feedback, operating near the nature of cortico-cortical connections, in particular, the nascent bistability. Their existence is related to a ro- strength and extent of cortical inhibition. We present a set bust transition from flat tomodulated rotationally invari- of models that uncover and explain the connection between ant two-dimensional fronts. The analytical expression of the experimentally observed tuning curves and the relevant the threshold associated with that transition as well as the cortical architecture. wavelength of the emerging structure are derived. These predictions are in close agreement with numerical simula- Gregor Kovacic tions. Rensselaer Polytechnic Inst Dept of Mathematical Sciences Gregory P. Kozyreff [email protected] OCIAM, Mathematical Institute, Oxford University kozyreff@maths.ox.ac.uk MS107 Kinetic Theories of Neuronal Networks MS108 Effects of Transverse Flow on Pattern Formation in We will compare various closures in the reduction of a Nematic Liquid Crystal Layer with Optical Feed- integrate-and-fire neuronal networks to kinetic theory. We back : Experiments and Theory will discuss mathematical aspects of these kinetic theories and present results of the kinetic theories and full numeri- In a previous study we have evidenced that the presence of cal simulations of integrate-and-fire neuronal networks. a transverse flow in a one dimensional (1D) optical system leads to convective instability patterns. Here, we derive the Adi Rangan different types of convective 2D patterns and their convec- Courant Institute of Mathematical Sciences tive and absolute thresholds. We show e.g. that one type New York University of patterns is always convective, and that the presence of [email protected] transverse flow can lower the threshold of pattern formation. Experiments and theory are in very good agreement. MS107 Eric Louvergneaux Synchrony-Dependent Propagation of Signals in Laboratoire de Physique des Lasers Atomes et Molécules Networks of Neurons Université des Sciences et Technologies de Lille [email protected] To examine experimentally how neural signals are transmitted, a feedforward network of live neurons was repro- Christophe Szwaj duced in brain slices using a computer-driven iterative pro- Universite de Lille (France) cedure. When inputs were delivered to the first layer, Laboratoire PHLAM neurons in successive layers fired synchronously. Although [email protected] synchrony was robust and persisted under a wide range of network configurations, its spread can be limited by lateral inhibition. Synchrony appears to be an important means Pierre Glorieux of encoding and transmitting signals through neural net- Laboratoire PHLAM, CERLA works. Universite de Lille 1 (FRANCE) [email protected] Alex Reyes, Cristina Marchetti Center for Neural Science Gonzague Agez New York University PhLAM - Universite de Lille (France) [email protected], [email protected] [email protected]

Majid Taki MS107 Universite des sciences et technologies de Lille Panel Discussion [email protected] We have brought together neurophysiologists, modelers and theoreticians to highlight the recent work on waves MS108 and coherent structures of neural activity. We end this Nonlocality and Convective Instability in Diﬀusive two-part minisymposium on ”Waves and Coherent Struc- Systems. tures in Neural Systems” with a panel discussion. We consider a large class of diﬀusive systems with nonlocal Louis Tao nonlinearity. We show that the stability of all the uniform 186 DS05 Abstracts

states of this class of systems is determined by a single tem (LCLV with feedback). Our presentation describes dispersion relation with two parameters. By using a non- the experimental system and the conditions under which perturbative analytical approach we are able to analyse patterning occurs. this dispersion relation and show that there are cases in which the convective instability window is so large that in John Sharpe practise only noise sustained patterns can be observed. California Polytechnic State University, USA Francesco Papoff [email protected] Department of Physics, University of Strathclyde, Scotland Pier Luigi Ramazza papoff@phys.strath.ac.uk Istituto Nazionale di Ottica Applicata Florence, Italy Roberta Zambrini [email protected] Department of Physics University of Strathclyde Nilgun Sungar, Karl Saunders [email protected] Cal Poly USA [email protected], [email protected] MS108 Bistability Between Different Dissipative Solitons in Nonlinear Optics MS108 Nonlinear 2D Diffractive Feedback Systems: A Call We report the observation of different localized structures for Applications coexisting for the same parameter values in an extended system. The experimental findings are carried out in a In the presentation we consider such applications of the nonlinear optical interferometer, and are fully confirmed nonlinear 2D-feedback systems as parallel image processing by numerical simulations. The existence of each kind of and atmospheric and adaptive optics. localized structure is put in relation to a corresponding delocalized pattern observed. Quantitative evaluation of Mikhail Vorontsov the range of pump parameter allowing bistability between The Army Research Laboratory and the University of localized structures is given. The phenomenon reported Maryland, results to be robust in parameter space. College Park, USA [email protected] Gregory Kozyreff Mathematical Institute Oxford university MS109 kozyreff@maths.ox.ac.uk Bifurcation of Relaxation Ocillations

Mustapha Tlidi The talk deals with bifurcation of relaxation oscillations in université Libre de Bruxelles two dimensions, emphasizing the transient canard oscilla- Optique nonliéaire Théorique tions. It relies on joint work with Robert Roussarie. [email protected] Freddy Dumortier Limburgs Universitair Centrum, Belgium Umberto Bortolozzo [email protected] Istituto Nazionale di Ottica Applicata Firenze, Italy [email protected] MS109 Periodic Waves in a Class of Singularly Perturbed Pier Luigi Ramazza Diffusive Two-Predator-One-Prey Systems Istituto Nazionale di Ottica Applicata Florence, Italy We consider a class of singularly perturbed diffusive two- [email protected] predator-one-prey systems and establish the existence of periodic traveling waves which provide mechanism for co- Luc Pastur existence. The main tool employed is the geometric singu- University of Orsay lar perturbations for turning points and invariant manifold Paris, France theory. [email protected] Weishi Liu University of Kansas MS108 [email protected] Pattern Formation in a Liquid Crystal Light Valve Through Alternation of Dynamics MS109 It has recently been discovered that modulation of param- Asymptotic Expansions for the Lagerstrom Model: eters in spatially extended nonlinear systems can have a A Geometric Approach striking impact on pattern formation. For example, rapidly switching between two states, each of which is homoge- The present work is a continuation of our geometric sin- neous, can lead to the formation of patterns. We have gular perturbation analysis of the Lagerstrom model prob- found experimentally similar phenomena, not previously lem. We reinterpret Lagerstrom’s equation as a dynamical described, in a spatially extended nonlinear optical sys- system which we analyze by means of invariant manifold theory as well as of the blow-up technique. We derive rig- DS05 Abstracts 187

orous asymptotic expansions for the Lagerstrom problem constraining potential. We rigorously justify the formal within this framework, thereby establishing a connection to limit obtained by multi-scale expansion. the method of matched asymptotic expansions. We explain the structure of these expansions and demonstrate that the Chongchun Zeng occurrence of the well-known logarithmic switchback terms University of Virginia therein is caused by a resonance phenomenon. Department of Mathematics [email protected] Nikola Popovic Boston University Center for BioDynamics and Department of Mathematics MS110 [email protected] High-Dimensional Chaos in the Kuramoto Model A novel high-dimensional chaotic behavior is found in the MS109 Kuramoto model of coupled phase oscillators. A half of Combustion Fronts in Porous Media With Two the Lyapunov spectrum appears to be positive and the Layers Lyapunov dimension almost attains the total system dimension. We find that the phase chaos phenomenon is We investigate a simplified model of combustion in a porous typical for different distributions of the natural frequencies medium with two layers. Each layer admits a traveling and also, for other ensembles of oscillators both regular and combustion wave with a certain speed, and heat can diffuse chaotic, e.g. for networks of coupled Rssler systems. between the layers. We show: (1) If the coefficient of heat diffusion between the layers is small, and the speeds of the Yuri L. Maistrenko two traveling waves are close, then the two-layer system Institute of Mathematics, Kiev, Ukraine admits a traveling combustion wave. (2) If the coefficient of Forschugszentrum Juelich, Germany heat diffusion between the layers is large, then the two-layer [email protected] system admits a traveling combustion wave, regardless of whether the speeds of the individual traveling waves are close. This wave is approximately an average of waves in MS110 the individual layers. The proofs use geometric singular Synchronization, Desynchronization and Noise in perturbation theory. Numerical simulations indicate that Electroreceptors of paddlefish these traveling waves are the dominant feature of solutions. The electrosensory system of paddlefish is used as a model Jesus Carlos da Mota system to study general mechanisms of synchronization Instituto de Matematica e Estatistica and desynchronization. Our approach is to create neuro- Universidade Federal de Goias (Brazil) electronic models, composed of afferents from two different [email protected] electroreceptors in an in vivo preparation of the paddlefish, linked via a computer interface. Interface provides Stephen Schecter feedback to drive afferents to bursting regimes and cou- North Carolina State Univ pling between electroreceptors. Using our hybrid model, Department of Mathematics we validated experimentally a new approach for disrupt- [email protected] ing the synchronization of different oscillators, based on intermittent multisite transient phase resetting.

MS109 Alexander Neiman Department of Physics and Astronomy Blow-Up Analysis of Delayed Hopf-Bifurcations Ohio University Differential equations with slowly varying parameters can [email protected] behave quite differently from the corresponding static bifurcation problems. In the case of Hopf bifurcations the David F Russell well known phenomenon of bifurcation delay occurs in an- Center for Neurodynamics, Univ. of Missouri at St. Louis alytic systems. We give a geometric analysis of this phe- St. Louis, MO 63121, USA nomenon. The analysis is based on the choice of a suitable [email protected] integration path in complex time and the blow-up method for singularly perturbed differential equations. Peter A. Tass Institute of Medicine (MEG) Peter Szmolyan Research Centre Juelich Institute for Analysis and Scientific Computing [email protected] Vienna University of Technology [email protected] MS110 Complete Synchronization, Directed Percolation MS109 and Finite Amplitude Lyuapunov Exponents Wave Equations with Strong Potentials Transition to complete synchronization in spatially ex- In this talk, we consider vector valued wave equations tended systems is analysed showing the existence of two where there is a strong constraining potential. This poten- scenarios Besides the standard analogy with Kardar-Parisi- tial achieves its minimum on a submanifold. We study the Zhang equation a connection with directed percolation is limit of solutions of finite energy as the constraining poten- found which is then investigated by implementing finite- tial tends to infinity. This problem can also be viewed as amplitude Lyapunov exponents. realizing holonomic constraints (to submanifolds in configuration spaces) to Hamiltonian motion by using a strong Roberto Livi 188 DS05 Abstracts

Dipartimento di Fisica Institute of Medicine (MEG) Firenze, Italy Research Centre Juelich livi@ﬁ.infn.it [email protected]

Antonio Politi National institute of Optics, Florence, Italy PP0 [email protected] Lyapunov Exponent Analysis of Biological Signal- ing Network Dynamics

Francesco Ginelli We employ a ﬁnite-time Lyapunov exponent analysis ap- Department of Physics proach to identify separatrices in phase space, delineat- Wurzburg, Germany ing qualitatively diverse signaling network dynamics gov- [email protected] erning pre-steady-state cell fate decisions. Application of this approach to a model of the protein network regulating Alessandro Torcini cell death responses to cytokines yields insights concerning Istituto Sistemi Complessi CNR this regulation not obtainable using steady-state bifurca- INOA - FIrenze, Italy tion analysis. This approach should facilitate similar in- [email protected] vestigations of larger signaling networks. George Haller MS110 Massachusetts Institute of Technology Eﬀective Desynchronization of Coupled Oscillators Department of Mechanical Engineering By Nonlinear Delayed Feedback [email protected]

We propose a new method for desynchronization of ensem- Peter Sorger bles of interacting and strongly synchronized oscillators. Department of Biology, Division of Biological Engineering We show that the stimulation input in the form of delayed MIT self-feedback combined nonlinearly with instantaneous sig- [email protected] nal can have a desynchronizing effect on the oscillations. The proposed method represents a robust and demand- controlled noninvasive technique with which different de- Douglas Lauffenburger grees of desynchronization can be achieved and controlled. Division of Biological Engineering, Dept. of Chem. Eng. This makes the method particularly attractive for applica- MIT tions, in particular, for deep brain stimulation. lauff[email protected] Oleksandr Popovych Bree B. Aldridge Institute of Medicine Massachusetts Institute of Technology Forschungszentrum Juelich, Germany Biological Engineering Division [email protected] [email protected]

MS110 PP0 Rotating Spiral Waves with Phase-Randomized Noise-Induced Dynamics in a Free-Electron Laser Core in Nonlocally Coupled Oscillators We study the effect of noise on the dynamics of free- electron laser oscillators. We find that the small amount of Yoshiki Kuramoto noise present in any experiment leads to a behavior quali- Department of Mathematics tatively different from the noise-free (deterministic) equiv- Hokkaido University, Sapporo , Japan alent. This is attributed to the nonnormality of the modes [email protected] involved, induced by the convective nature of the system. The results are found very similar to the ones obtained in Shin-ichiro Shima the case of mode-locked lasers. Department of Physics, Kyoto University, Japan Serge Bielawski s [email protected] PhLAM/Université Lille I, France [email protected] MS110 Model-Based Development of Desynchronizing Christelle Bruni, Gian-Luca Orlandi, M.-E. Couprie Deep Brain Stimulation CEA/LURE France For the therapy of severe neurological diseases permanent [email protected], high-frequency (¿ 133 Hz) deep brain stimulation (DBS) [email protected], [email protected] is performed. This treatment has been developed heuris- tically. To improve treatment with DBS, with methods CHRISTOPHE Swaj from nonlinear dynamics and statistical physics novel DBS PhLAM/Université Lille I techniques were designed to restore the physiological firing France mode by mild but effective desynchronization. In a pilot [email protected] study in patients during depth electrode implantation the novel DBS technique turned out to be superior. DAVID Garzella Peter A. Tass CEA/LURE DS05 Abstracts 189

France switching at branch points of equilibria and limit cycles [email protected] and detects limit points, Hopf points, limit points of cycles, branch points of cycles ... MatCont makes the MAT- LAB odesuite for time integration interactively available PP0 and can use the MATLAB Symbolic Toolbox for comput- Dynamical Models of Finger Biomechanics and ing derivatives whenever it is installed. In the case of limit Neuromuscular Control cycles MatCont discretizes the BVP exactly as in AUTO and CONTENT, i.e. by orthogonal collocation. The sys- We analyze the periodic motor patterns of middle finger tems that arise in this way are typically sparse and their control of a trackball, during a psychophysics task in which sparsity increases with the number of test intervals used in the subject has to match a constant velocity using the the discretization. In MatCont and Cl MatCont the spar- trackball. We explore minimal hybrid (DAE) kinematic sity of the linearized systems is exploited by using the Mat- models of the system, and investigate the roles of feedback lab sparse matrix routines. Among the recent additions to control both from ‘pre-flexes’ and from central motor sys- MatCont we mention the computation of normal form co- tems. efficients for bifurcations of limit cycles. We present the use of MatCont in a few interesting neural models. John Guckenheimer Cornell University Yuri A. Kuznetsov [email protected] Mathematical Institute Utrecht University Robert Clewley [email protected] Department of Mathematics Cornell University Annick Dhooge [email protected] Ghent University Department of Applied Mathematics and Computer Francisco Valero-Cuevas Science Neuromuscular Biomechanics Laboratory [email protected] Cornell University [email protected] Willy Govaerts Dept. of Applied Mathematics and Computer Science Ghent University PP0 [email protected] Pattern Selection in a Hopf Bifurcation Forced at Multiple Resonant Frequencies PP0 In Faraday surface waves complex super-lattice patterns A Biophysical Model for Sleep-Wake Timing and quasi-patterns can be excited by multi-frequency forcing. Motivated by this observation we investigate the com- The transitions between states of sleep and wake are regu- petition between various spatially periodic patterns in a lated by a network of neurons in the brainstem and hy- complex Ginzburg-Landau equation describing a Hopf bi- pothalamus. We model this network as a collection of furcation with frequency ω forced simultaneously with fre- coupled oscillators, and we incorporate circadian, home- quencies ω,2ω, and 3ω. From the Ginzburg-Landau equa- ostatic, and other biophysical elements to simulate both tion we derive amplitude equations for various planforms normal and pathological transitions. By exploiting geomet- and study their stability analytically. We then investi- ric properties of the network, we are also able to capture gate their dynamics in simulations of the Ginzburg-Landau waking behaviors on two timescales: seconds and minutes. equation. Nancy J. Kopell Hermann Riecke Boston University Applied Mathematics Department of Mathematics Northwestern University [email protected] [email protected] Cecilia Diniz Behn Jessica Conway Boston University Northwestern University Department of Mathematics and Center for BioDynamics Applied Math [email protected] [email protected] Thomas Scammell, Takatoshi Mochizuki PP0 Department of Neurology Beth Israel Deaconess Medical Center MatCont: a Software Package for Dynamical Sys- [email protected], tems with Applications to Modelling Neural Activ- [email protected] ity.

MatCont is a graphical MATLAB package for the inter- Emery Brown active numerical study of a range of parameterized non- Department of Anesthesia and Critical Care linear problems. The current version is freely available Massachusetts General Hospital at: http://allserv.rug.ac.be/ ajdhooge where also a slightly [email protected] more general non - GUI version Cl MatCont is available. Both MatCont and Cl MatCont allow to compute curves of equilibria, limit points, Hopf points,limit cycles and ﬂip, PP0 fold, torus and branch points of limit cycles. It does branch Bifurcations of Stable Sets in Noninvertible Planar 190 DS05 Abstracts

Maps winding number.

Many applications give rise to systems that can be de- Alexander Wurm scribed by noninvertible maps. These fold the phase space, Department of Physics, Fusion Studies so that different regions have different numbers of pre- The University of Texas at Austin images. This makes computing backward orbits and stable [email protected] sets more complicated as they may cross critical curves where the number of pre-images changes. We compute the Amit Apte stable sets of a two-dimensional noninvertible map, which Dept. of Mathematics define the boundaries of basins of attraction, using a re- University of North Carolina cently developed algorithm. [email protected]@email.unc.edu Hinke M. Osinga University of Bristol Kathrin Fuchss, P. J. Morrison Department of Engineering Mathematics Department of Physics and Institute of Fusion Studies [email protected] The University of Texas at Austin [email protected], [email protected] Bernd Krauskopf University of Bristol PP0 Dept of Eng Mathematics 1,2,3 Some Examples of Local Bifurcations of [email protected] Higher Codimension

James P. England We present examples of local bifurcations of codimension Department of Engineering Mathematics 1, 2 and 3 that appear in ecological models. The bifurca- University of Bristol tions are shown in three-parameter bifurcation diagrams. [email protected] Among others we show an example of a codimension-3 1:1-resonant double Hopf bifurcation. This bifurcation is formed by codimension-1 Hopf bifurcation surface in the PP0 shape of a Whitney umbrella. The bifurcation diagrams Frequency Dynamics of Semiconductor Lasers Sub- have been produced by combining analytical calculations ject to Filtered Optical Feedback with an algorithm for numerical triangulation of implicitly defined surfaces. This process is also explained. If part of the emission of a semiconductor laser is fed back into the laser after passing through an optical filter, charac- Thilo Gross teristic oscillations of the frequency on the time scale of the Fachbreich Physik delay time are found. By performing a bifurcation analy- Universität Potsdam sis of the corresponding delay differential equation model [email protected] we show how such oscillations emerge from certain Hopf bifurcations of the basic solutions of the system, which are Dirk Stiefs, Ulrike Feudel known as external cavity modes. ICBM Theoretical Physics/Complex Systems Carl von Ossietzky Universitaet Oldenburg Bernd Krauskopf [email protected], [email protected] University of Bristol Dept of Eng Mathematics [email protected] PP0 Phase Transitions in Equilibrium and Non- Daan Lenstra Equilibrium Systems with Xy-Model-Like Interac- Afdeling Natuurkunde en Sterrenkunde tions Vrije Universiteit Amsterdam [email protected] We study noise- or temperature-driven phase transitions of various dynamical systems consisting of elements with XY- Hartmut Erzgraber model-like interactions. For systems comprised of moving, Vrije Universiteit Amsterdam self-propelled elements, a second-order phase transition is Afdeling Natuurkunde en Sterrenkunde obtained. We establish an equivalent transition for im- [email protected] mobile elements interacting through network connections. For the Kosterlitz-Thouless transition of the XY-model, we present a novel analysis based on the statistics of vortex PP0 trajectories in space-time, which illustrates the unbinding Reconnection/Collision Processes and Meanders in transition. the Standard Nontwist Map Cristian Huepe Using the standard nontwist map as a single-harmonic Department of Engineering Sciences and Applied model map for nontwist systems, meandering tori and pe- Mathematics riodic orbits between dimerized odd-periodic orbit chains Northwestern University are investigated. The presence of these ”non-KAM” orbits [email protected] gives rise to more intricate reconnection/collision sequences than previously observed. We present such non-standard sequences and numerically establish parameter ranges for PP0 their applicability. We further study the destruction of Electrokinetic Effect on Electrolytic Flow in Mi- the ”central” torus, i.e., the meandering torus of extremal DS05 Abstracts 191

crochannels Department of Mathematics University of Utah Due to the size effect of the microchannels the electrolytic [email protected] phenomenon is dominant which is negligible in bulk flow. In the present work an analytic solution to electrokinetic flow in a parallel plate microchannel has been developed. PP0 The Navier-Stokes equations have been modified to take Alternative Determinism Principle for Topological into account the electro-viscous effect. The effect of elec- Analysis of Chaos trokinetics on rate of boundary layer growth is discussed. The equations for fully developed friction factor and Nus- The topological analysis of chaos based on a knot-theoretic selt number have been developed and interpreted in the characterization of unstable periodic orbits can only be ap- physical domain. plied to three-dimensional systems. Still, its core principles, determinism and continuity, apply in any dimension. Abhishek Jain We propose an alternative framework where these prin- Department of Mechanical, Aerospace and Nuclear ciples are enforced on triangulated surfaces rather than Engineering, curves. As a first step towards a formalism applicable in Rensselaer Polytechnic Institute, Troy, New York higher dimensions, we show that our approach simplifies abhijain [email protected] significantly the computation of topological entropies of three-dimensional periodic orbits. PP0 Marc Lefranc Noise Effect on Dynamic System of a Two Dimen- PhLAM/Université Lille I, sional Stoichiometric Discrete Producer-Grazer France Model [email protected]

The grazer can become extinct while having plenty of pro- Michel Nizette ducer in a completely deterministic system. An explana- Université Libre de Bruxelles tion for this lies in the bad nutritional quality of the pro- Theoretical Nonlinear Optics Department ducer that precludes the grazer from efficiently converting [email protected] the consumed food into its own biomass. In experiments, data are collected on discrete time intervals and it is observed that many prey in nature have non-overlapping gen- Pierre-Emmanuel Morant erations. In our paper, we use a non-overlapping discrete PhLAM/Université Lille 1 model to see how producer quality can pull the systems out France of oscillations and how it halts chaotic dynamics. We dis- [email protected] cuss the behavior of our discrete dynamic system: in which condition the grazer becomes extinct but the producer still PP0 exhibits chaotic behavior and in which condition the grazer becomes extinct but the producer has stable equilibrium. Homogeneous Three-Cell Networks Species diversity in nature is accomplished by coexistence. Coupled cell systems are networks of differential equations. Utilizing a realistic model that consists of two interact- The architecture of a coupled cell network is a graph indi- ing species producer and grazer, we discovered a stochastic cating which cells are identical and which cells are coupled phenomenon where noise can enhance the coexistence and to which. We show that there are 34 homogeneous three- thereby promote species diversity. We use scaling-law and cell networks with each cell having at most two inputs as phase plane to explain this interesting phenomenon. opposed to only two such two-cell networks. We classify Yun Kang codimension-one synchrony-breaking bifurcations in these Mathematics and Statistics networks, showing some surprising features. Arizona State University Martin Golubitsky [email protected] University of Houston Department of Mathematics PP0 [email protected] Calcium Waves Mediated Calcium Alternans in Cardiac Cells Maria Leite University of Houston Contraction of heart cells occurs when calcium ions en- [email protected] ter the cell through L-type calcium channels, and trigger calcium release from the sarcoplasmic reticulum (SR) by the opening of ryanodine receptors embedded on the SR PP0 membrane surface. This process is known as calcium in- Dynamic and Steady State Analysis of the Gastric duced calcium release (CICR). Calcium alternans can be Mucus Gel the result of unstable CICR. Another example of unstable CICR is calcium waves. By using mathematical model, we Existence of gels in biological systems requires reconsider- suggest that calcium waves can cause calcium alternans. ation of standard modeling techniques. The stomach contains a gel mucus layer that is responsible for protecting James P. Keener the stomach from autodigestion. To supply this protec- University of Utah tion, the gel layer maintains a large gradient in pH. We [email protected] discuss the mechanisms responsible for this phenomena at steady state. Also, we consider the dynamic response of Young-Seon Lee the gel to a surge of acid in the stomach corresponding to 192 DS05 Abstracts

the presence of food. Japan [email protected] James P. Keener University of Utah Toshiyuki Nakagaki [email protected] Research Institute for Electronic Science Hokkaido University Frank Lynch [email protected] University of Utah Department of Mathematics Ryo Kobayashi [email protected] Department of Mathematical and Life Sciences Hiroshima University PP0 [email protected] Modeling the Circadian Rhythm of Melatonin Se- cretion and Metabolism Atsushi Tero Hiroshima University We have measured plasma melatonin concentration and to- [email protected] tal urinary output of the metabolite for three days in constant lighting conditions. The secretion and metabolism of Tetsuo Ueda the melatonin are modeled with a pharmacokinetic model. Hokkaido University The onswitch and offswitch of the secretory pulse are mod- [email protected] elled as the nonlinear transform of a ”clock”. We compare three clocks: cosine with 24-hour period; cosine with estimated period; van der Pol oscillator outside its limit cycle. PP0 A Developmental Model of Ocular Dominance Col- Matthew R. Marler umn Formation on a Growing Cortex University of California at San Diego Department of Psychiatry We derive an activity-based developmental model of ocu- [email protected] lar dominance column formation in primary visual cortex that takes into account cortical growth. The resulting evo- Dan Kripke, Jeffrey Elliott lution equation for the densities of feedforward afferents UCSD Psychiatry from the two eyes exhibits a sequence of pattern forming [email protected], [email protected] instabilities as the size of the cortex increases. We use linear stability analysis to investigate the nature of the transitions between successive patterns in the sequence. We PP0 show that these transitions involve the splitting of existing Recurrence Plot Extension for Spatial Data ocular dominance columns, such that the mean width of an OD column is approximately preserved during the course of Classically introduced recurrence plots (RPs) can only be development. This is consistent with recent experimental applied to one-dimensional data like phase space vectors observations of postnatal growth in cat. and time series. We develop an extended and generalized RP approach which enables to analyze spatial (higher- Andrew Oster dimensional) data regarding recurrent structures. Result- University of Utah ing RPs have higher dimensions (eg 4). Hence, the mea- [email protected] sures used to evaluate classic RPs are extended to assess higher-dimensional recurrence plots. Developed approach is applied to assess bone structure from 2D pQCT images PP0 of human proximal tibia. (ESA-Project contract #14592) Nonlinear Oscillations in a Microelectromechanical System Norbert Marwan Institute of Physics, The dynamics of an electret-based, capacitive micro- University of Potsdam converter is described by a nonlinear set of ODEs, where [email protected] the equation of a damped, driven oscillator is coupled, through a non linear term, to two first-order, non-linear differential equations. The system, which can admit pe- PP0 riodic, steady-state solutions, exhibits behaviors typical Functions and Formation of Circulation Network of non-linear, Duffing-like oscillators, as jump phenom- in An Amoeboid Organism ena and hysteretic frequency response curves. In fact, for particular combinations of the physical parameters of the To evaluate performance in a complex survival task, system, multiple steady-state solutions appear. The fre- we studied the morphology of transportation network quency response curves and the stability properties of the in an amoeboid organism (true slime mold Physarum solutions are analyzed with a semianalytic approach. It is polycehalum) when presented with multiple separate food also proved, through perturbative analysis, that the system sources. The organism comprises a network of tubular el- always acts as a linear oscillator under appropriate combi- ements through which intracellular signals and the viscous nations of parameters: in this case the non-linear coupling sol are transported and circulated. I report functions of term reduces to a viscouslike term, physically interpretable the network shape and mathematical model for network as electromechanical damping. formation. Fabio Peano, Gianni Coppa Yasumasa Nishiura Politecnico di Torino RIES, Hokkaido University Dipartimento di Energetica DS05 Abstracts 193

[email protected], [email protected] The limiter control acts on pitch characteristic of the wing. Its effectiveness to suppress LCO and chaos beyond nom- Cristina Serazio inal flutter speeds is demonstrated to effectively suppress Politecnico di Torino vibrations and offer gust alleviation. Dipartimento di Matematica [email protected] Erik Bollt, Pier Marzocca Clarkson University [email protected], [email protected] Antonio D’Angola Universita’ degli Studi della Basilicata Chrissy Rubillo [email protected] Clarkson University MAE Dept [email protected] Federico Peinetti Politecnico di Torino Torino, Italy PP0 [email protected] The Dynamic Range of Bursting in a Synaptically Coupled Network of Square-Wave Bursters

PP0 The synchronized bursting of neurons in a certain region of Study on Autoresonant Excitation of Non Linear the brain stem likely plays a role in the respiratory rhythm. Modes in Electron Plasmas The relevant cells engage in square-wave bursting in the absence of coupling, if and only if certain experimentally Recent experiments performed at UC Berkeley showed the manipulable parameters are in a subset of their possible possibility of creating large-amplitude synchronized BGK ranges. We explain how excitatory synaptic coupling be- modes in a pure electron plasma (W. Bertsche et al., Phys- tween cells expands this bursting range, by elucidating the ical Review Letters, 91, 2003). The present work concerns dynamical mechanisms underlying transitions from silence analytical studies and simulation results aimed at explain- to bursting (symmetric and asymmetric) to tonic spiking ing the generation mechanisms and the stability of large- (symmetric and asymmetric). This analysis also explains amplitude nonlinear modes in non translation-invariant subtle effects of parameters on burst duration and on other systems, starting from the theoretical work by L. Fried- burst characteristics. land et al. (Physics of Plasmas, 11, 2004). David H. Terman Fabio Peano, Gianni Coppa The Ohio State University Politecnico di Torino Department of Mathematics Dipartimento di Energetica [email protected] [email protected], [email protected] Jonathan E. Rubin Federico Peinetti University of Pittsburgh Politecnico di Torino Department of Mathematics Torino, Italy [email protected] [email protected] Martin Wechselberger PP0 Ohio State University Parameter Study of Nonlinear Subgridscale Models Mathematical Biosciences Institute for the Large Eddy Simulation of Incompressible [email protected] Flow Problems Janet Best, Alla Borisyuk In this poster, we examine the effects of various parameters Mathematical Biosciences Institute in the numerical simulation of incompressible viscous flow [email protected], [email protected] problems using nonlinear subgridscale models. Specifically, we examine subgridscale models in which artifical viscosity is introduced into the system via a p-Laplacian regulariza- PP0 tion, and investigate the dependence of the solution to the Mode dynamics in 2D microcavity lasers parameters p, the order of the p-Laplacian term, and µ, the coefficient of artificial viscosity. We study the lasing dynamics of a stadium-cavity laser by using a model based on the Maxwell and optical- John P. Roop Bloch equations. We numerically investigate the bifurca- Virginia Tech tion of stationary lasing states when pumping strength is Department of Mathematics increased. A detail analysis is presented for a mode-locking [email protected] phenomenon, which yields an asymmetric emission pattern in a symmetric cavity.

PP0 Satoshi Sunada, Takahisa Harayama Jet Reaction Control of Nonlinear Wings ATR Wave Engineering Laboratories [email protected], [email protected] We model jet reaction torquer control of chaotic and limit- cycle oscillations (LCO) of aircraft wings. Instead of the Kensuke Ikeda discrete, hinged control surfaces, the vehicle control sys- Ritsumeikan University tems may be implemented with a combination of propul- [email protected] sive/jet forces. A 2-D nonlinear airfoil model is considered. 194 DS05 Abstracts

Susumu Shinohara PP0 ATR Wave Engineering Laboratories Pattern formation in plamodial slime mold on [email protected] hard/soft surface

Morphology of the plasmodial slime mold, Physarum poly- PP0 cephalum, in various media was investigated. Plasmodium Development of Travelling Waves in Weakly Un- shows disk-like or dendrite shape under rich nutrient or stable Shallow Water Systems harmful environment, respectively. Interestingly, the same situation occurred on hard/soft surface of agar medium Shallow water systems have been observed to develop into and plasmodium prefer hard surface without nutrient to roll waves, which are traveling waves formed after a long soft surface with small amount of nutrient. Contact angles time or distance. To predict these traveling waves, we de- of plasmodium were smaller on harder surface. Plasmod- termine the asymptotic behaviour of small disturbances as ium moves in direction of small contact angles irrespective they evolve from boundary conditions in a flat inclined of existence of food. channel. When the Froude Number is greater than two the system is unstable when disturbed from uniform flow. Atsuko Takamatsu For the weakly unstable problem, we determine the evo- Department of Electrical Engineering and Bioscience lution of the solution over space by applying perturbation Waseda University theory. It is found that the solution is dominated by the atsuko [email protected] evolution of the solution along one characteristic. We also show the shock conditions governing the asymptotic solution and use these conditions to determine the approximate PP0 shape of the resulting travelling wave, or roll wave, from Pulsating Outbreaks in An Sir-Epidemic Model the solution. We present both asymptotic and numerical with Temporary Immunity results for periodic disturbances in transition, as well as the travelling wave solutions for the shallow water and the The SIR-epidemic model considers that recovered individ- asymptotic evolution equations. Finally, we present an ap- uals are permanently immune, while the SIS model consid- plication of these results to the problem of transition to ers recovered individuals to be immediately re-susceptible. slug flow in channels. We study the case of temporary immunity in an SIR based model with delayed coupling between the susceptible and Richard Spindler removed classes. We perform a numerical and analytical University of Vermont bifurcation analysis of the resulting DDE and describe how Dept. of Math and Statistics temporary immunity leads to recurrent outbreaks and how [email protected] model parameters affect the severity and period of the outbreaks. Jun Yu University of Vermont Thomas W. Carr [email protected] Southern Methodist University Department of Mathematics [email protected] PP0 Large Linewidth Enhancement Factor in a Mi- Michael Taylor crochip Laser Department of Mathematics Southern Methodist University We evidence experimentally that the linewidth enhance- [email protected] ment factor alpha can take a rather large value (around unity) for a non semiconductor laser like a Nd:YAG microchip laser. This raises the hope to observe a rich dy- PP0 namic when the laser is subjected to external feedback but Dendritic cable with active spines: a modelling in a much direct way than with semiconductor lasers since study in the Spike-Diffuse-Spike framework the characteristic frequencies are much lower for microchip lasers (MHz instead of Ghz). The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine head dy- Christophe Szwaj namics is modeled with a simple integrate-and-fire process, Universite de Lille (France) whilst communication between spines is mediated by the Laboratoire PHLAM cable equation. We develop a computational framework [email protected] that allows the study of multiple spiking events in a network of such spines embedded on a simple one-dimensional Eric Lacot cable. In the first instance this system is shown to sup- Lab. Spectro port saltatory waves with the same qualitative features as UJF-Grenoble (France) those observed in a model with Hodgkin-Huxley kinetics [email protected] in the spine head. Moreover, there is excellent agreement with the analytically calculated speed for a solitary saltatory pulse. Upon driving the system with time-varying Olivier Hugon external input we find that the distribution of spines can Lab. Spectro play a crucial role in determining spatio-temporal filtering UJF-Grenoble properties. In particular, the SDS model in response to [email protected] periodic pulse train shows a positive correlation between spine density and low-pass temporal filtering that is consistent with the experimental results of Rose and Fortune (J Neurosci 1997, 17; J Experiment Biol 1999, 202). Fi- DS05 Abstracts 195

nally we demonstrate one of the ways to incorporate noise PP0 that arises in the spine-head. Stabilization of Pulses with Algebraic Decay

Gabriel J. Lord The Ginzburg-Landau equation has various unstable soli- Heriot-Watt University tary pulse solutions, which have, however, been observed [email protected] in systems with two competing instability mechanisms. In such systems, the Ginzburg-Landau equation is coupled to Stephen Coombes a diffusion equation. We use an Evans function method to University of Nottingham show that the effect of the slow diffusion can indeed stabi- [email protected] lize a pulse when higher-order nonlinearities are taken into account. Current work adapts this approach to the case Yulia Timofeeva where the diffusion equation has a neutrally stable mode, Heriot-Watt University introducing pulses that decay algebraically rather than ex- [email protected] ponentially. Nienke Valkhoff PP0 KdV Institute A Collective Motion Algorithm for Tracking Time- University of Amsterdam Dependent Boundaries [email protected]

We present a numerical method that allows a formation of communicating agents to follow a concentration gradi- PP0 ent and track the boundary of this concentration surface. Structured Pseudospectra in Structural Engineer- The algorithm, allows the agents to move in space in a ing non-stationary environment. Our method is applicable to studying motions of swarms in biology, as well as to engi- The stability of a system is usually studied by analysing the neering applications where boundary detection is an issue. spectrum of an associated linear operator. More accurate information, however, can often be obtained by considering Ira B. Schwartz pseudospectra of the operator. We discuss applications of Naval Research Laboratory pseudospectra to stability and vulnerability problems in Nonlinear Dynamical Sysytems Section structural engineering. In particular, we focus on a novel [email protected] method for the computation of pseudospectra with respect to a class of structured perturbations. Ioana A. Triandaf Thomas Wagenknecht Naval Research Laboratory Bristol Laboratory for Advanced Dynamics Engineering Plasma Physics Division [email protected] [email protected]

PP0 PP0 Floer Homology for Braids on the Two-Disc A Karhunen-Loeve Method of Characterizing Spatio-Temporal Chaos We are interested in 1-periodic solutions/periodic points of Hamiltonian systems on the two-disc. Such solutions can We present a method of characterizing chaos in complex be viewed as braids. For various types of braids we can spatio-temporal patterns, that allows distinguishing be- deﬁne an invariant via Floer homology. The latter can be tween a complex pattern which is involved and could ap- computed using the standard Conley index. Applying this pear chaotic, yet consisting of a variety of coexisting peri- tool allows us to obtain various types of forcing results for odic or quasiperiodic motions and a spatio-temporal pat- periodic solutions. tern trully chaotic. We exemplify our method on a system of four globally-coupled Ginzburg-Landau equations Jan Bouwe Van Den Berg derived from the weak electrolyte model for the electro- VU Amsterdam convection of nematic liquid crystals. Department of Mathematics [email protected] Ira B. Schwartz Naval Research Laboratory Nonlinear Dynamical Sysytems Section Wojciech T. Wojcik [email protected] Department of Mathematics Vrije Universiteit Amsterdam [email protected] Ioana A. Triandaf Naval Research Laboratory Plasma Physics Division Robert Vandervorst [email protected] VU Amsterdam Department of Mathematics [email protected] Iuliana Oprea Naval Research Laboratory tba PP0 Detection of Symmetric Homoclinic Orbits to Saddle-Centers in Reversible Systems

We study the persistence of symmetric homoclinic orbits 196 DS05 Abstracts

to saddle-centers in reversible systems under small perturbations and develop a perturbation method for detecting such orbits. To illustrate the theory we give an example for a four-dimensional system arising from a model of a nonlinear-optical medium with both quadratic and cubic nonlinearities. Homoclinic orbits in the ODE system correspond to embedded solitons, i.e., isolated solitary waves which reside in the continuous spectrum of the linearized system and whose tail amplitudes exactly vanish, in the PDE optical model. Kazuyuki Yagasaki Gifu University Department of Mechanical and Systems Engineering [email protected]

PP0 Classical Invariant Theory in Normal Form Com- putations

The normal form for a dynamical systemx ˙ = F (x)inthe vicinity of a ﬁxed point x = 0 is completely characterized by the polynomial solutions to the Homological equation

AP (x) − Ax ·∇P (x)=0 where A = ∇F (0) is the linearization at the ﬁxed point. When A is nilpotent, there is an obvious connection between the Homological operator and Classical invariant theory, which can be exploited to obtain a polynomial representation of the normal form to all orders. We extend this approach to include cases where A has a nonzero semi- simple part. Using these ideas along with Gr¨obner basis techniques, we determine the normal forms for all the codimension 2 bifurcations with one reversing symmetry. Shankar C. Venkataramani, Lay May Yeap University of Arizona Department of Mathematics [email protected], [email protected]

PP0 Transient Spatiotemporal Chaos on Complex Net- works

Transient spatiotemporal chaos in a network model based upon the Gray-Scott cubic autocatalytic reaction-diffusion system is examined for different topologies. Motivated by recent studies on the ”small-world” network topology, irregular connections are added to the network’s original regular ring structure. We find that a single added connection can significantly decrease the life time of spatiotemporal chaos. A connection spanning a relatively small portion of the entire network (< 15%), however, tends to increase the system’s transient time, an effect that is explained via the added connection’s effect on the dynamics local to it. Finally, we find that the addition of two connections can transform the system’s spatiotemporal chaos from transient to asymptotic. Renate A. Wackerbauer University of Alaska Fairbanks Department of Physics ff[email protected]

Saﬁa G. Yonker University of Alaska, Fairbanks Physics Department [email protected]