Dynamical Systems Seminars
The Dynamical Systems seminar is held on Monday afternoons at
4:00 PM in MCS B37. Tea
beforehand is at 3:30 PM in MCS 233A.
- September 16 Kick-off social
We will meet this week in MCS B39 at 4:00PM. Come by for refreshments and to meet everyone!
- September 23 Ira B. Schwartz
(U.S. Naval Research Laboratory )
Title: Swarming dynamics: From Nature to Theory to Robotics
Abstract:Swarming behavior continues to be a subject of immense interest because of its centrality in many naturally occurring systems in physics and biology, as well as its importance in engineering applications such as robotics. Here we examine the effects on coherent swarm pattern formation from aspects of communication, such as latency effects, link topology and environmental uncertainty. With the availability of ever more cheap and powerful computing, interest in the use of mixed-reality and swarm experiments has grown considerably in the physical sciences. Broadly speaking, these experiments consist of a simulated, or virtual model coupled directly to a physical experiment. Within the physical experiment, it is typical to find a good deal of uncertainty and noise since it is connected to the real world, and thus subjected to random perturbations. In contrast, the virtual part of the coupled system represents a somewhat idealized version of reality in which noise can be eliminated entirely. Thus, mixed-reality systems have very skewed sources of uncertainty spread through the entire system.
In this talk, we consider the pattern formation of delay-coupled swarms theoretically and experimentally to illustrate the idea of mixed-reality. Motivated by physical experiments, we then consider a model of a mixed-reality system, and show how noise in the physical part of the system can influence the virtual dynamics through a large fluctuation, even when there is no noise in the virtual components. We quantify the effects of uncertainty by showing how characteristic times of noise induced switching between swarm patterns scale as a function of the coupling between the real and virtual parts of the experiment. This work is done in collaboration with Klimka Szwaykowska, Thomas Carr, Victoria Edwards and Jason Hindes.
- September 30 Denis Patterson
Title: Spatially extended models of savanna dynamics
Abstract: We introduce two interacting particle system models of vegetation dynamics (one macroscale and one mesoscale) based on the interaction rules from the mean-field Staver-Levin model of forest-savanna-grassland evolution. Using coupling techniques for stochastic jump processes, we prove the convergence of these particle systems to McKean-Vlasov-type jump processes. The generalized Kolmogorov equations which characterize the behavior of these processes are systems of integro-differential equations and are more amenable to analysis than the original particle systems. By analyzing these nonlocal equations, we find that our macroscale model is an elementary example of a process which oscillates in law and that our mesoscale model exhibits a rich array of ecologically relevant spatial dynamics, including traveling waves, breathing fronts and pattern formation.
This is joint work with Carla Staver (Yale), Simon Levin (Princeton) and Jonathan Touboul (Brandeis)
- October 7 No Seminar
- October 14
No Seminar - Columbus Day
- October 21 Rodrigo Treviño
(University of Maryland, College Park)
Title: Renormalization for random substitution tilings
Abstract: Aperiodic tilings such as the Penrose tiling are fascinating and beautiful objects, and they serve as models for unusual materials which go by the name quasicrystals. But they also define dynamical systems on strange spaces. As such, insight into the dynamical properties of the system reveals insight into the physical materials which they model. I will talk about how one can build a tiling space out of a tiling, the dynamics defined on it, and how one can build "moduli spaces" of tiling spaces, leading to random substitution tilings. The main goal of this talk is to convince you that a lot of the ergodic properties of the tiling systems are determined by the Lyapunov spectrum of a renormalization cocycle, that is, of a dynamical system on the "moduli space" of random tilings. Time allowing, I will mention what this has to say about some problems in mathematical physics. Some of this talk is based on work with S. Schmieding.
- October 28 Henrik Ronellenfitsch
Title: Active topolectrical circuits
Abstract: The transfer of topological concepts from the quantum world to classical mechanical and electronic systems has opened fundamentally new approaches to protected information transmission and wave guidance. A particularly promising technology are recently discovered topolectrical circuits that achieve robust electric signal transduction by mimicking edge currents in quantum Hall systems. In parallel, modern active matter research has shown how autonomous units driven by internal energy reservoirs can spontaneously self-organize into collective coherent dynamics. Here, we unify key ideas from these two previously disparate fields to develop design principles for active topolectrical circuits (ATCs) that can self-excite topologically protected global signal patterns. Building on a generic nonlinear oscillator representation, we demonstrate both theoretically and experimentally the emergence of self-organized protected edge states in ATCs. The good agreement between theory, simulations and experiment implies that ATCs can be realized in many different ways. Beyond topological protection, we also show how one can induce persistent localized bulk wave patterns by strategically placing defects in 2D lattice ATCs. These results lay the foundation for the practical implementation of autonomous electrical circuits with robust functionality in arbitrarily high dimensions.
- November 4 Ebru Toprak
Title: Dispersive Estimate and zero energy resonances for the fourth order Schrödinger Equation
- November 11 Ivan Sudakov
(University of Dayton)
Title: Nonlinear analysis of species extinction in a population competing for resources.
Abstract: The extinction of species is a core process that affects the diversity of life on Earth. One way of investigating the causes and consequences of extinctions is to build conceptual population models, and to use the dynamical outcomes of such models to provide a quantitative formalization of changes to Earth’s biosphere. In this talk, I will present a conceptual model that describes a simple and easily understandable mechanism for resource competition depending on the feedback with climate. The model explains the coexistence of many species, yet also displays the possibility of catastrophic bifurcations, where all species become extinct under the influence of climatic factors. I will show how to prove a general assertion on the existence of an attractor for this model. I will also discuss how the model admits an asymptotic solution and can be reduced to the Lotka–Volterra model.
- November 18 Raluca Tanase
Title: A tool for transporting dynamics from C to C^n
Abstract: We show how to use quasiconformal theory and tools from complex differential geometry to transport dynamical results from one complex dimension to higher dimensions. We then discuss the dynamics of germs of holomorphic diffeomorphisms of (C^n, 0) with a fixed point at the origin with exactly one neutral eigenvalue. This is based on joint work with T. Firsova, M. Lyubich, and R. Radu.
- November 25 Chongchun Zeng
Title: Steady water waves with localized vorticity
Abstract: We consider the fluid free surface problem -- the free boundary problem of the Euler equation with gravity and surface tension -- and construct finite energy small amplitude steady wave solutions with vorticity based on a bifurcation approach. Unlike those perturbed from shear flows, the vorticity of these solutions are highly concentrated, including traveling waves with compactly supported vorticity and smooth stationary waves with rapidly decaying vorticity.
- December 2 Aditi Chakrabarti
Title: Pattern Formation in Soft Elastic Solids
Abstract: Soft solids, such as hydrogels and elastomers, show a rich variety of pattern formation phenomena
that are unique to these materials. By combining the joint roles of surface tension, elasticity and
gravity as well as geometric length scales, I have explored how these soft solids behave under
different scenarios. In this presentation, I will discuss two kinds of pattern formation observed in
soft solids: first that arises when adhesion between a thin elastic film and a rigid substrate is lost
leading to interfacial patterning, and second that arises on the free surface of elastic slabs that are
subjected to their own weight in the form of elastic Rayleigh Taylor instability. I will present
experiments and scaling analyses to describe the two types of instabilities and point out the
similarities between the two problems although the relevant geometric scale, i.e. the thickness of
the films varies from microns for the adhesion driven instability to centimeters in the case of the
elastic Rayleigh Taylor. Finally, I will demonstrate a new type of dynamical instability with liquid
drops on thin elastic films that undergo spontaneous symmetry breaking and thenceforth spinning
via formation of multiple lobe chiral patterns.
- December 9 No Seminar
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