Dynamical Systems Seminars

Fall 2021

The Dynamical Systems seminar is held on Monday afternoons at 4:00 PM in MCS B31 . There will be a joint tea with the Number Theory seminar in MCS B24 at 3:45PM.

• September 6 No Seminar

• September 13 Tea Time (no talk) Now that everyone is settled in, feel free to stop by and catch up after the summer break! Refreshments will be provided in MCS B24 and weather permitting we will gather outside in the grass area by the Communication and Life Sciences buildings
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• September 20th Ousmane Kodio (MIT)
  Title: Dynamic buckling instabilities
  Abstract: Many phenomena encountered in nature may be understood through the paradigm of buckling instability. Examples include the design of columns in structural engineering, the folding of geological formations, the collapse of blood vessels, and the fragmentation of uncooked spaghetti, to name only a few. The phenomenon of buckling is traditionally considered a nuisance, but more recently it has proven useful as al tool for patterning surfaces, particularly at small scales. Many studies have focused on the features of static buckling. In this seminar, we will discuss how dynamic buckling affects the spontaneous selection of patterns in a number of canonical problems, such as the dynamic buckling of elastic and viscous rings, and the evolution of wrinkle patterns on a lubricated layer. If time permits, we will also briefly discuss the timely problem of carbon dioxide monitoring as a tool for assessing risk of airborne disease transmission, paying particular attention to COVID-19 transmission in classrooms.
  

• September 27th Sebastian Schreiber (UC-Davis)
  Title: Species coexistence and extinction in stochastic environments
  Abstract: Environmental factors such as temperature and precipitation often exhibit stochastic fluctuations. As the survival and reproduction of plants, animals, and viruses, depend on these environmental factors, these environmental fluctuations drive fluctuations in population densities. A fundamental question in population biology is “under what conditions does environmental stochasticity hinder or facilitate the maintenance of biodiversity?” This question is particularly pressing in light of global climate models predicting increasing temporal variation in many environmental factors over the next century. One fruitful approach to tackling this question is analyzing models accounting for nonlinear feedbacks among species and environmental stochasticity. In this talk, I will present a mathematical theory for stochastic coexistence and extinction for random difference equations. I will discuss applications of the theory and open problems.
  

• October 4th Milen Ivanov (Brown University)
  Title: Truncation of Contact Defects in Reaction-Diffusion Systems
  Abstract: Solutions of reaction-diffusion systems exhibit a wide variety of patterns like spirals, stripes and Turing patterns. In particular, the Belousov-Zhabotinsky (BZ) reaction produces spiral patterns, which may undergo a period-doubling bifurcation; then a line defect is emitted from the center of the spiral and along it the pattern jumps half a period. In order to study this phenomenon, we consider the so-called contact defects, studied by Sandstede and Scheel: time-periodic functions ud(x,t), which converge (in an appropriate sense) to a periodic function as x→±∞. Of interest is the problem of truncating such a defect to a large interval, with Neumann or periodic boundary conditions. In a finite-dimensional model, obtained via Galerkin approximation, we prove the existence and uniqueness of such a truncated contact defect. Furthermore, we prove this contact defect is spectrally stable when given periodic boundary conditions, and spectrally unstable with Neumann boundary conditions. These results suggest that the observed spiral patterns with line defects are stable. A problem for future work is to extend these results to the infinite-dimensional setting.
  

• October 11th No Seminar (Indigenous People's Day)
  Title: TBA
  Abstract: TBA
  

• October 18th Mette Olufsen
  Title: The importance of oscillations in cardiovascular control
  Abstract: Blood pressure and heart rate is controlled via negative feedback and for healthy people the response to perturbations e.g., active standing, deep breathing, or head-up tilt is to system is to make the system to drive both quantities back to steady values. Recent studies have shown that patients with autonomic dysfunction, in particular POTS (Postural Orthostatic Tachycardia Syndrome) experiences excessive increase in heart rate and the emergence of low-frequency oscillations upon postural change. In this talk I will highlight how modeling can be used to explain the emergence and augmentation of these oscillations and why noise (often referred to as heart rate variability) is important for stabilizing the feedback.
  

• October 25th Cole Graham (Brown University)
  Title: Vanishing viscosity and Burgers shock formation
  Abstract: We study one facet of an old problem: the approximation of inviscid Burgers by its viscous counterpart. This vanishing viscosity limit is well understood around a fully developed shock, but less is known about the moment of shock formation. We develop a matched asymptotic expansion to describe the vanishing viscosity limit of shock formation to arbitrary precision. At the leading order, we identify universal viscous dynamics near the point of shock formation. This is joint work with Sanchit Chaturvedi.
  

• November 1 Klaus Widmayer (EPFL)
  Title: Stationary Euler flows near the Kolmogorov and Poiseuille flows
  Abstract:We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow. This situation contrasts strongly with the setting of some monotone shear flows (such as the Couette flow) and also with those of both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel. Our result has surprising consequences in the context of inviscid damping in 2D Euler and enhanced dissipation in Navier-Stokes. This is joint work with Michele Coti Zelati and Tarek Elgindi.
  

• November 8 Gabriel Ocker (BU)
  Title:Tensor decomposition of higher-order correlations by nonlinear Hebbian plasticity
  Abstract: Biological synaptic plasticity exhibits nonlinearities that are not accounted for by classic Hebbian learning rules. Here, we introduce a simple family of generalized nonlinear Hebbian learning rules. We study the computations implemented by their dynamics in the simple setting of a neuron receiving feedforward inputs. These nonlinear Hebbian rules allow a neuron to learn tensor decompositions of its higher- order input correlations. The particular input correlation decomposed and the form of the decomposition depend on the location of nonlinearities in the plasticity rule. For simple, biologically motivated parameters, the neuron learns eigenvectors of higher-order input correlation tensors. We prove that tensor eigenvectors are attractors and determine their basins of attraction. We calculate the volume of those basins, showing that the dominant eigenvector has the largest basin of attraction. We then study arbitrary learning rules and find that any learning rule that admits a finite Taylor expansion into the neural input and output also has stable equilibria at generalized eigenvectors of higher-order input correlation tensors. Nonlinearities in synaptic plasticity thus allow a neuron to encode higher-order input correlations in a simple fashion.
  

• November 15 No Seminar ()
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• November 22 Anna Vainchtein (Univ. of Pittsburgh)
  Title: Strictly supersonic solitary waves in lattices
  Abstract: We consider a nonlinear mass-spring chain with first and second-neighbor interactions and show that there is a parameter range where solitary waves in this system are strictly supersonic. In these regimes standard quasicontinuum theories, targeting long-wave limits of lattice models, are not adequate since even weak strictly supersonic solitary waves are of envelope type and crucially involve a microscopic scale in addition to the mesoscopic scale of the envelope. To capture this effect in a continuum setting it is necessary to employ unconventional, higher-order quasicontinuum approximations carrying more than one length scale. This talk is based on recent joint work with Lev Truskinovsky (ESPCI).
  

• November 29 Paul Miller (Brandeis)
  Title: Information processing in neural circuits with multiple quasi-stable attractor states
  Abstract: Circuits containing clusters of neurons with strong within-cluster excitatory connections and random between-cluster connections can possess multiple activity states, with each state defined by the set of clusters with stably high versus low neural firing rates. Noise or adaptive processes can render the states quasi-stable such that in the presence of constant input, or following successive identical inputs, the network can progress through a sequence of different activity states. Here we study the properties of such models, including the role of finite size effects in producing multistability in networks with Gaussian random cross-coupling, and we assess the extent to which such state sequences might subserve information processing and behavior in a range of cognitive tasks.
  

• December 6 Zaher Hani (U. of Michigan)
  Title: Rigorous Derivation of the wave kinetic equation
  Abstract: The kinetic theory of waves, aka wave turbulence theory, has been formulated in various fields of physics to describe the statistical behavior of interacting wave systems. This started early in the past century with the pioneering works of Peierls, Hasselman, Zakharov, and others, and developed into the highly successful and informative paradigm widely employed nowadays, both in physical theory and practice. However, for the longest time, the mathematical foundation of the theory has not been established, with all its derivations based on formal manipulations and unproven postulates. The central objects here are the "wave kinetic equation” which describes the effective dynamics of an interacting wave system in the thermodynamic limit, and the "propagation of chaos” hypothesis, which is a fundamental postulate in the field that lacks mathematical justification. This problem has attracted considerable interest in the mathematical community over the past decade or so. The culmination of this effort came recently in a series of joint works with Yu Deng (University of Southern California), in which we provide the first rigorous derivation of the wave kinetic equation, and also justify the propagation of chaos hypothesis in the same setting. The proof features a nice interplay of analysis, probability theory, combinatorics, and analytic number theory.
  

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