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Joint Dynamics and PDE Seminar

The Boston University Department of Mathematics and Statistics, Brown University's Department of Mathematics and Division of Applied Mathematics, and the University of Massachusetts Amherst Department of Mathematics and Statistics hold joint seminars on topics in dynamics and PDE. The schedule and locations for these events can be found below. For a list of all past events of the seminar please visit the BU/Brown/UMass PDE Seminar Archive.

The organizers for the Brown/BU/UMass Seminars are Jason Bramburger, Ryan Goh, and Stathis Charalampidis.  Please contact them at [email protected], [email protected], or [email protected]ath.umass.edu.

Spring 2019

Friday, May 3 2:00pm - 6:00pm at BU

Jonathan Jaquette

In this talk we discuss a validated computational method for obtaining error estimates on the infinite dimensional stable manifold of non-trivial equilibria in parabolic PDEs. Our approach combines the parameterization method – which can provide high order approximations of finite dimensional manifolds with validated error bounds – together with the Lyapunov-Perron method – which is a powerful technique for proving the existence of (potentially infinite dimensional) invariant and inertial manifolds.

Zhiyuan Zhang

We consider the plasma confined in a general axisymmetric spatial domain with perfect conducting boundary which reflects particles specularly, and look at a certain class of equilibria, assuming axisymmetry in the problem. We prove a sharp criterion of spectral stability under these settings. Moreover, we provide several explicit families of stable/unstable equilibria using this criterion.

Matthew Durey

A droplet may ‘walk’ along the surface of a vertically vibrating fluid bath, propelled at each impact by the Faraday waves generated by all prior impacts. The longevity of this ‘path memory’ increases with the amplitude of the vibrational forcing, yielding instabilities in the system and the onset of chaos. This hydrodynamic pilot-wave system exhibits many features that were previously thought to be exclusive to the quantum realm, such as unpredictable tunnelling, emergent statistics, and quantized droplet dynamics. 

Stephanie Dodson

Spiral wave patterns observed in models of cardiac arrhythmias and chemical oscillations develop alternans and stationary line defects, both of which can be thought of as period-doubling instabilities. These instabilities are observed on bounded domains, and may be influenced by the spiral core, far-field asymptotics, and boundary conditions. In this talk, we introduce a methodology to disentangle the impacts of each region on the instabilities by analyzing spectral properties of spiral waves and boundary sinks on bounded domains with appropriate boundary conditions.

Alanna Hoyer-Leitzel

Impulsive differential equations are continuous differential equations with discrete jumps (kicks) in phase space at a deterministic sequence of times. This results in a map, but also captures the behavior of the system in between kicks. Impulsive differential equations can be used to model resilience to disturbances. In this talk I'll discuss resilience in the context of impulsive Lotka-Volterra models.

Alexandros Gelastopoulos

Neural oscillations, including rhythms in the beta1 band (12-20 Hz), are important in various cognitive functions. Often neural networks receive rhythmic input at frequencies different than their natural frequency, but very little is known about how such input affects the network's behavior. We use a simplified, yet biophysical, model of a beta1 rhythm that occurs in the parietal cortex, in order to study its response to oscillatory inputs.

Lakshminarayanan Mahadevan