78 DS09 Abstracts

78 DS09 Abstracts IP0 IP3 Jrgen Moser Lecture: Catastrophes, Symmetry- Network Topology: Sensors and Systems Breaking, Synchrony-Breaking Networks arise in innumerable contexts from mobile com- The ideas that surrounded catastrophe theory (codimen- munications devices, to environmental sensors nets, to bi- sion, unfoldings, organizing centers, ...) have shaped much ological systems at all scales. This talk explores the re- progress in bifurcation theory during the past forty years lationships between what happens on the network nodes and, indeed, in many of its applications. I will try to trace (signals, sensing, dynamics) and the underlying spatial dis- some of these developments and to indicate why this same tribution of the nodes — an especially delicate interplay in way of thinking might lead to interesting discoveries in net- the context of coordinate-free, non-localized systems. The work dynamics. key tools are an adaptation of homology theory. Algebraic topology yields an enrichment of network topology that in- Martin Golubitsky tegrates cleanly with statistics and dynamics of networks, Ohio State University and which allows for solutions to problems of coverage, Mathematical Biosciences Institute communication, and control. [email protected] Robert W. Ghrist University of Pennsylvania IP1 [email protected] Collapse of the Atlantic Ocean Circulation The Atlantic Ocean Circulation is sensitive to the patterns IP4 of atmospheric forcing. Relatively small changes in atmo- Mechanisms of Instability in Nearly Integrable spheric conditions may lead to a spectacular collapse of Hamiltonian Systems Atlantic Ocean currents, with a large impact on European climate. In ocean-climate models, a collapse is associated There are many systems that appear in applications that with the existence of saddle-node bifurcations. The chal- have negligible friction, like in celestial mechanics and lenge is to efficiently determine the position of these bifur- Astrodynamics, motion of charged particles in magnetic cations in the multi-parameter, large-dimensional dynam- fields, chemical reactions, etc. A general model for this ical systems derived from these models. In this presenta- kind of systems is to consider time periodic perturbations tion, I will give a brief overview of numerical techniques of integrable Hamiltonian systems with 2 or more degrees currently used and then focus on the issue of determining of freedom. One problem that has attracted attention for the probability that a collapse will occur before the year a long time is whether the effect of perturbations accu- 2100. mulate over time and lead to large effects (instability) or whether these effects average out (stability). The first re- Henk Dijkstra sult in proving instability for this model was presented by Institute for Marine and Atmospheric Research Utrecht Arnold in 1964, but the perturbation considered was not (IMAU) general enough (there do not appear ”large gaps” in the Utrecht, the Netherlands resonant tori). In this talk we present some mechanisms [email protected] that cause instabilities for general perturbations if the un- perturbed systems considered are a priori-unstable, that is, they can be written as a product of d rotators and n IP2 penduli, d ≥ 1, n ≥ 1. The main technique is to develop a Dynamics, Instability, and Bifurcation in the Me- toolkit to study, in a unified way, tori of different topologies chanics of Biological Growth and their invariant manifolds, their intersections as well as shadowing properties of these bi-asymptotic orbits. Part of Biological growth is at the heart of many developmental, this toolkit is to unify standard techniques (normally hy- physiological, and pathological processes. Growth relies perbolic manifolds, KAM theory, averaging theory) so that on the fine tuning of different genetic and biochemical pro- they can work together. A fundamental tool used here is cesses but, ultimately, expresses itself through a change of the scattering map of normally hyperbolic invariant man- geometry mediated by physical forces. These forces gen- ifolds. The conditions needed are explicit. This makes it erated by growth also influence growth itself, creating a possible to show that this phenomenon occurs in concrete complex feedback mechanism. The coupling between ge- examples, as the restricted three body problem in celestial ometry, stresses, and growth in elastic tissues can be mod- mechanics. eled within the framework of nonlinear anelasticity, a theory which includes both reversible and irreversible defor- Tere M. Seara mations. The consequences of growth can then be studied Univ. Politecnica de Catalunya on simple geometries or reduced models. I will explain the [email protected] basic foundational principles of nonlinear anelasticity and, motivated by the behavior of biological systems, I will de- scribe generic instabilities and bifurcations occurring in the IP5 growth dynamics. Analysis of Large-Scale Interconnected Dynamical Systems Alain Goriely University of Arizona The problem of analysis of large-scale networked systems Program in Applied Mathematics is one of the most interesting current challenges in Dy- [email protected] namical Systems. In this talk, I will present a perspec- tive on this problem that combines an operator-theoretic and graph theoretic approach, but also incorporates, in a strong way, the geometric point of view that is so fruitful in low dimensions. This approach leads to a new proposal DS09 Abstracts 79 for model reduction that is rooted in the dynamics of the Physics: Lorentz gases as deterministic models for stochas- system rather than in energy-minimization arguments. It tic behaviour; to (2) pattern forming excitable media: hy- also enables analysis of uncertain and stochastic systems permeander of spiral waves where the spiral tip appears to - where initial conditions and/or parameter values are not undergo a random walk in the plane. For (1), Nicol and known exactly - within the same framework. Most of the the speaker proved recently that the positions of almost tools apply equally to discontinuous systems. I will moti- all gas molecules behave asymptotically like sample paths vate the approach by a number of examples, ranging from of Brownian motion. For (2), a mechanism (and experi- biomolecular models to power grid systems, that exhibit ment) is proposed for mathematically verifiable Brownian- large scale (”emergent”) transitions between states. like motion for the spiral tip. Igor Mezic Ian Melbourne University of California, Santa Barbara University of Surrey [email protected] Department of Mathematics and Statistics [email protected] IP6 The Multiscale Dynamics of Lightning and a Mov- IP9 ing Boundary Problem Living on the Edge of Noise-Driven Order Streamers pave the way of lightning strokes and of many Transient or unstable behavior is often ignored in consid- other discharges in nature and technology. They are ering long time dynamics in the deterministic world. How- plasma fingers that penetrate non-ionized matter by a ever, stochastic effects can change the picture dramatically, strong field enhancement at their tips. Each streamer so that the transients can dominate the long range be- has an intricate inner structure, and a discharge can form havior. This talk will compare a few seemingly unrelated fractal-like trees with ten thousands of streamer branches. canonical models with noise-driven regular behaviors, such First I will review how far their very fast dynamics and as coherent oscillations, synchronization, mixed mode os- multiscale structure are resolved by recent experiments and cillations and even quiescence, that are transient in the simulations. Then I will focus on derivation, tests and so- absence of noise. Perspectives combining ideas from coher- lutions of a moving boundary approximation for streamer ence resonance, meta-stability, and multiples scales show fingers. The boundary approximation with kinetic under- that the order in these models can be attributed to tran- cooling regularization revives the question of the selection sients ”stabilized” by stochastic effects. These perspectives of Saffman-Taylor fingers and similar solutions. Quantita- suggest analysis on reduced models to better understand tive predictions of the field enhancement at the streamer and predict these phenomena. tips are important practical results. Rachel Kuske Ute Ebert University of British Columbia CWI Amsterdam [email protected] [email protected] IP10 IP7 The Fluid Trampoline: Droplets Bouncing on a Systems Biology: How Dynamical Systems Theory Soap Film Can Help to Understand the Basis of Life We present the results of a combined experimental and the- The 21st century has been noted to be the century of biol- oretical investigation of droplets falling onto a horizontal ogy and new technologies and methods are vastly accelerat- soap film. Both static and vertically vibrated soap films ing the pace of discovery in biology and medicine. Systems are considered. A quasi-static description of the soap film biology, the systems level understanding and investigation shape yields a force-displacement relation that allows us to of biological phenomena, is one of those key methods that model the film as a nonlinear spring, and yields an accu- act as a driving force for progress in biological research. In rate criterion for the transition between droplet bouncing this talk the role of dynamical systems theory for the de- and crossing. On the vibrating film, a variety of bounc- velopment

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