Dynamical Systems Seminars

Fall 2020

The Dynamical Systems seminar is held on Monday afternoons at 4:00 PM (as always!) but remotely via Zoom. Furthermore, the seminar for this fall will be local and somewhat more informal. Each speaker will give a 20-30 minute talk on an interesting research topic/problem, or something they find interesting and are learning about. All (including students) are welcome to participate and present. If you'd like to attend the seminar please register using the following link: https://bostonu.zoom.us/meeting/register/tJwrdOChqjgtHNGrOY-qutAX6XzLyRlrOd8o. If you'd like to give a talk, please email Ryan Goh (rgoh "at" bu "dot" edu).

- September 7 No seminar (Labor day)

Consider attending the One world dynamics seminar on Sept. 11th - September 14 Glen Hall

Title: A lunar theory for a moon we don't have

Abstract: "Celestial Mechanics is the study of one system of differential equations, but solutions of that system are so complicated that we have to be happy with studying special cases and approximations. We discuss a ""lunar theory"" which is an approximate to the three-body problem for which one of the bodies (the moon) is very small and that body is very close to one of the other bodies (the planet) that is orbiting the third body (the star). The tools are all from MA 225 and MA 226."

- September 21 Ryan Goh

Title: Dynamics and PDE applications of matrix Riccati equations

Abstract: Matrix Riccati equations (MREs) are systems of differential equations posed in a phase space of matrices with a quadratic polynomial vector-field. Originally finding use in the study of optimal control problems, they have more recently arisen as a possible tool for studying stability of special solutions of nonlinear PDEs. In this talk I will discuss some general characteristics of MREs from a dynamical systems viewpoint, connections between MREs and compactification of linear flows on a Grassmannian manifold, and how MREs are used to study nonlinear PDEs. Throughout the talk I will aim to illustrate the concepts through examples.

- September 28 Jonathan Jaquette

Title: A short survey on computer assisted proofs in dynamics.

Abstract: One of the lessons from Lorenz’s butterfly is that the smallest changes can have large effects. This lesson casts a pall over numerical analysis, where miniscule errors accrue with every step. This talk will survey several computer assisted proofs in dynamics, wherein the cumulative effect of these numerical errors are quantified and leveraged to prove mathematically precise theorems

- October 5 Ricardo Carretero

Title: Vortices and vortex rings in quantum superfluids: a quasi-particle approach

Abstract: Motivated by recent experiments studying 2D vortex dynamics in Bose-Einstein condensates (BECs), we illustrate that, by considering these vortices as quasi-particles, such systems can be accurately described by reduced models of coupled ordinary differential equations. It is then possible to study in detail the dynamics, stability, and bifurcations of vortex configurations and match the ensuing results to experimental observations. We will also explore some extensions of the quasi-particle approach for 3D vortex rings which are formed when a vortex line (a "twister") is looped back onto itself creating a close ring that carries vorticity. We first showcase how vortex rings are commonplace in a wide range of fluids. We then focus on the occurrence of vortex rings in BECs and their mutual interactions, collisions, and scattering scenarios. We also briefly discuss an efficient computational implementation for solving the full, original, partial differential equations using GPU accelerated codes in real time!

- October 12 (Columbus day)

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- October 19 Margaret Beck

Title: A brief introduction to exponential dichotomies

Abstract:In this talk, I will introduced the notion of and precisely define an exponential dichotomy. Several key examples will be shown, and it will be discussed why exponential dichotomies are useful in the analysis of both finite and infinite-dimensional dynamical systems. The so-called Roughness Theorems, used to prove that exponential dichotomies exist, will also be stated and discussed.

- October 26 Gabriel Ocker

Title: Tensor decompositions by biologically motivated nonlinear learning dynamics

Abstract: It is a classic result in neural learning theory that a simple neuron, receiving linearly weighted stochastic inputs, will learn principal components of the input. It learns them through Hebbian dynamics that are essentially bilinear in the neuron’s input and output. Biological synaptic plasticity exhibits nonlinearities that are not accounted for by this classic model. We examine the dynamics of a simple nonlinear generalization of Hebbian learning, and show that it allows a neuron to learn tensor decompositions of 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 parameter choices, the steady states are tensor eigenvectors of the input correlation. We characterize these solutions and their stability for some tractable models of uncorrelated inputs.

- November 2 C. Eugene Wayne

Title: Breathers as metastable states for weakly damped lattices of Hamiltonian oscillators

Abstract: We discuss the flow of energy in a lattice of Hamiltonian oscillators with weak damping at one end of the lattice. We derive bounds on the rate of dissipation when the initial energy in the lattice is localized in a spatially distant part of the lattice. For a special model, we exhibit a family of breather solutions for the undamped problem and show that the rate of energy dissipation can be explained by a very slow drift along this family of breathers. This is joint work with Noé Cuneo (Univ. of Paris 7), Jean-Pierre Eckmann (Univ. of Geneva) and Daniel Caballero (Boston Univ.)

- November 9 Tasso Kaper

Title: Slow passage through Hopf bifurcations

Abstract: How to delay the onset of oscillations, if they are unwanted instabilities, using MA412 (complex analysis) and MA225 (hemispheres)

- November 16 Samuel Isaacson

Title: The stochastic dynamics of chemical kinetics models.

Abstract: Classical ODE models for chemical systems and population processes have had an enormous impact in modeling of biological systems, infectious disease spread, and general chemical dynamics. However, for many physical systems stochastic jump processes provide a more realistic, and microscopic, representation of their dynamics. I’ll give an introduction to the jump process formulation of stochastic chemical kinetics, and illustrate what is known about the rigorous relationship between such processes and more macroscopic ODE models. Rigorous results relating the two formulations are primarily for finite time intervals, and it is an open problem to understand their agreement or disagreement over unbounded intervals. I’ll illustrate through examples some cases where the two representations show significantly different qualitative behavior on long-time scales. Time permitting, I’ll also discuss extensions of such models to include spatial transport, leading to reaction-diffusion PDE models and stochastic particle systems.

- November 23 Xiaoxuan Wu

Model Reduction For Multi-Scale Partial Differential Equations

Abstract: A new method is developed for model reduction in multi-scale systems of coupled reaction-diffusion equations. The new method uses scaling variables which are naturally suggested by the classical diffusion operator. The main differential operators have a spectral gap and an infinite basis of natural modes associated to the point spectrum. The first principal results consist of establishing the existence of nonlinear slow modes for reaction-diffusion systems and of rigorously studying the algebraic and exponential decay of general solutions toward them. It is illustrated on two prototypical examples, the Davis-Skodje model and the Michaelis-Menten-Henri (MMH) model with diffusion of both species. The second principal result consists of introducing a new class of multi-scale reaction-diffusion equations that possess closed-form, low-dimensional, invariant manifolds. This new class of PDEs is of interest in its own right, and it provides a useful set of benchmark problems for testing and comparing numerical methods for model reduction in nonlinear PDEs.

- November 30 Roland Welter

Title: Asymptotic approximation of a modified compressible Navier-Stokes system

Abstract: The compressible Navier-Stokes equations generalize the usual incompressible Navier-Stokes to the case when gradients in the density are large or when the velocities under study are close to the speed of sound of the fluid. In a joint work with Gene Wayne and Ryan Goh, we build on Hoff and Zumbrun's asymptotic approximation for a modified compressible Navier-Stokes system by using techniques developed by Wayne and Gallay to study the incompressible equations.

- December 7 No Seminar today

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### Directions to BU Math Dept.

### Speakers from previous years