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118 DS17 Abstracts 118 DS17 Abstracts IP1 ations from this limit can only be due to very rare events Some Geometrical and Physical Aspects of Mor- (Large Deviations). The net result is that the deviations phogenesis from the limit behavior can be observed only on time scales that are huge (for example, the exponential of number of How is living matter organized in space and time during particles/units in the system). However, in several natural morphogenesis ? A comparative view across animal and instances deviations happen on a much shorter time scale. plant species suggests that the answer may lie in reusing After discussing this important issue and its practical rel- just a few geometric cellular principles and organ-sculpting evance in a general framework, I will focus on mean field motifs. Using examples, I will discuss the quantitative ba- synchronization models for which 1. we can sharply trace sis for three of these motifs: elongation, lumenization and the time scale of validity of the deterministic large scale folding. PDE; 2. we can describe in detail what happens after this time scale. The talk is based on papers written by sub- L Mahadevan sets of the following authors: Lorenzo Bertini, Eric Lu¸con, Harvard University Christophe Poquet, & myself. [email protected] Giambattista Giacomin Universit´eParisDiderot IP2 [email protected] Pattern Formation in the Drylands: Vegetation Patterns in Mathematical Models and in Satellite Images of the Horn of Africa IP4 Computational Approach Elucidating the Mecha- An awe-inspiring example of spontaneous pattern forma- nisms of Cardiovascular Diseases tion appears in the distribution of vegetation in some dry- land environments. Examples from Africa, Australia and Cardiovascular diseases such as aortic aneurysms and aor- the Americas reveal vegetation congregated in stripe-like tic dissections persist as life-threatening hazards. Patient- bands, alternating with striking regularity with bands of specific simulations are now common in biomedical engi- bare soil. A typical length scale for such patterns is 100 m; neering. Although they are extremely useful for grasping they may be readily surveyed in Google Maps. The typical the flow/stress distributions and for patient-specific treat- time scale for pattern evolution, however, is 100 years, so ment planning, they remain insufficient to elucidate the investigations of dynamics are a bit thwarted, with only a general mechanisms of a targeted disease. Several mathe- few early data points provided by aerial photographs from matical viewpoints should play important roles in this con- the 1950s. These ecosystems represent some of Earths most text. For instance, we introduce a geometrical characteri- vulnerable under threats to desertification, and ecologists zation of blood vessels, which vary widely among individ- have suggested that the patterns, easily monitored by satel- uals. Differences in the vessel morphology can produce lites, may tell us something about a regions possible stage different flow characteristics, stress distributions, and ulti- of collapse. This is an attractive idea, especially to us mately different outcomes. Therefore, the characterization pattern formation researchers who want to work on prob- of the morphologies of these vessels poses an important lems that have potential impact. My talk consists of two clinical question. Through close collaboration with medi- parts. In the first, I describe how one proposed early warn- cal doctors, these analyses can yield greater understanding ing sign, related to vegetation pattern morphology, can be leading to better risk assessments. framed mathematically in terms of an equivariant bifur- cation problem, and then probed by a systematic analysis Hiroshi Suito of models. In the second, I turn to investigations of image Okayama University and data, focusing on our preliminary studies of vegetation pat- Tohoku University, Japan terns in the Horn of Africa. The goal of this presentation [email protected] is to highlight the potential for models to be confronted by data, and vice versa. IP5 Mary Silber Dynamics, Mixing, and Coherence University of Chicago Dept. of Engineering Sciences and Applied Mathematics Coherent regions in geophysical flows play fundamental [email protected] roles by organising fluid flow and obstructing transport. For example, in the ocean, coherence impacts dynamics from global scales down to scales of at least tens of kilome- IP3 tres, and strongly influences the transportation of heat, Random Long Time Dynamics for Large Systems salt, nutrients, phytoplankton, pollution, and garbage. of Interacting Oscillators I will describe some recent mathematical constructions, ranging across dynamical systems, probability, and geome- A considerable amount of effort has been put into under- try, which enable the accurate identification and tracking of standing large scale dynamics of interacting particles or, such structures, and the quantification of associated mix- more generally, interacting “units” (like cells or individ- ing and transport properties. I will present case studies uals). This large scale limit, leading often to a partial from a variety of geophysical settings. differential equation, is in most cases taken under the as- sumption of finite time horizon. This of course raises the Gary Froyland crucial issue of the relevance of long time dynamics for the UNSW Australia limit PDE when dealing with the long time dynamics of [email protected] the original system, which is made of a finite number of particles or units. This issue has been successfully tack- led for stochastic systems (the large scale limit is in this IP6 case a law of large numbers) for the cases in which devi- Stochastic Arnold Diffusion of Deterministic Sys- DS17 Abstracts 119 tems and study all the macroscopic attractors and bifurcations of these systems. In 1964, V. Arnold constructed an example of a nearly integrable deterministicsystem exhibiting instabilities. In Edward Ott the 1970s, physicist B. Chirikov coined the term for this University of Maryland phenomenon Arnold diffusion, where diffusion refers to Inst. for Research in Electronics and Applied Physics stochastic nature of instability. One of the most famous [email protected] examples of stochastic instabilities for nearly integrable systems is dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. They were discovered numerically by as- CP0 tronomer J. Wisdom. During the talk we describe a class Mentoring Program of nearly integrable deterministic systems, where we prove stochastic diffusive behavior. Namely, we show that distri- See session abstract. butions given by deterministic evolution of certain random initial conditions weakly converge to a diffusion process. Conference Participants This result is conceptually different from known mathe- n/a matical results, where existence of diffusing orbits is shown. n/a This work is based on joint papers with O. Castejon, M. Guardia, J. Zhang, and K. Zhang. CP0 Vadim Kaloshin Student Icebreaker University of Maryland at College Park, USA and ETH Z¨urich On Saturday immediately before the opening reception, [email protected] there will be an informal Student Icebreaker session. Through structured activities, the primary goal is for at- tendees to leave feeling comfortable about the meeting, IP7 and having made new connections to depend on as friendly Interactions, Deformations and Bifurcations of Sin- faces during the conference. There is no additional cost, gular Patterns but in order to get a count on how many people to expect, we request that you register for this session when you reg- Singular patterns appear in multiscale systems. These sys- ister for the conference. tems exhibit the rich behavior of general systems, their singular nature provides a structure by which this may Chad M. Topaz be understood. Moreover, many natural phenomena are Dept. of Mathematics, Statistics, and Computer Science modeled by such systems. I will discuss the strong cross- Macalester College fertilization between applications and the development of [email protected] mathematical theory. Unravelling the nature of patterns exhibited by specific chemical or ecological models goes hand in hand with uncovering novel generic destabiliza- CP1 tion mechanisms as the ‘Hopf dance’. Understanding real- Control of Complex Network Dynamics and Syn- istic patterns requires analytical descriptions of deforma- chronization by Symbolic Regression tions, bifurcations and annihilation of interacting localized structures – from an ecological point of view preferably un- The control of dynamical systems is a relevant and im- der varying (climatological) circumstances. Through this, portant issue in engineering applications, like turbulence, mathematics may explain why desertification sometimes is but also for experimental designs in natural sciences and in a sudden catastrophic event, while it is a gradual process medicine, where drugs and technical devices are more and in other situations. more fine-tuned to enhance the quality of life. Large-scale systems are typically hard to control: due to the many de- Arjen Doelman grees of freedom and related possible instability of modes, Mathematisch Instituut linear control is often not effective to control a complex [email protected] system such that it remains on a well-defined orbit. We use symbolic regression to find analytical control laws for the described situations. Here, we use coupled oscillator SP1 networks as a general model for a complex
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