Dynamical Systems Seminars

Spring 2023

The Dynamical Systems seminar is held on Monday afternoons at 4:00 PM in CCDS 548 . There will be a brief tea in CCDS 548 at 3:45PM.

• January 23 -- No seminar
  

• January 30 Bjorn de Rijk (Karlsruhe Institute of Technology)
  Title: Regularity Control in the Nonlinear Stability Analysis of Periodic Waves
  Abstract: A standard issue in the nonlinear stability analysis of periodic waves in pattern-forming systems is the loss of regularity arising by the introduction of a spatio-temporal phase modulation. That is, the modulated perturbation variable typically satisfies a quasilinear equation. Nonlinear damping estimates, effectively controlling the H^s-norm of the modulated perturbation by its L^2-norm, have proven to be a successful tool to regain the lost regularity. However, outside of standard parabolic settings, such nonlinear damping estimates can be difficult to obtain and, in general, their existence is not guaranteed. In addition, being L^2-energy estimates, nonlinear damping estimates are naturally incompatible with merely bounded perturbations, which satisfy no localization or periodicity assumptions. In this talk I will provide an alternative method to control regularity in the nonlinear argument, motivated by work establishing the nonlinear stability of Lugiato-Lefever periodic waves against localized perturbations, on the one hand, and of reaction-diffusion periodic waves against bounded perturbations, on the other hand.

• February 6 TBD
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• February 13 Ricardo Carretero (San Diego State University)
  Title: Dynamical reductions for solitonic filaments
  Abstract: We describe the dynamical reduction for the 2D evolution of solitonic and kink stripes in archetypal nonlinear wave models such as the nonlinear Schrödinger (NLS) equation and the Klein-Gordon (KG) equation. The reductions, based on adiabatic invariants and variational approximations, are developed to capture the statics, stability, and dynamics of single stripes. Extensions to breather stripes in the sine-Gordon model and dark soliton shells in the 3D NLS will also be covered. We will also extend our considerations to the interaction dynamics between stripes. Under appropriate validity bounds, all reductions are favorably compared against the original full dynamical evolution.

• February 20 No Seminar - President's Day
  

• February 27 Ingrid Daubechies (Duke)
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• March 6 No Seminar - Spring Break
  

• March 13 Erik Bergland (Brown)
  Title: Multiple-Timescale Dynamics in the Fitzhugh-Nagumo Equation
  Abstract: In 2018, Carter and Sandstede analyzed parametric pulse replication in the Fitzhugh-Nagumo (FHN) equation, unfolding a one-parameter family of pulses dubbed the homoclinic banana. In a follow-up paper with Rademacher in 2021 they numerically observed temporal pulse replication: the evolution of an initial condition close to a one-pulse in the banana traced out the family of solutions dynamically, transitioning to a two-pulse despite having fixed parameter values. The authors conjectured that the parametric transition described by the banana generates a nearby invariant manifold, which the temporal dynamics drift along slowly. The rate of this drift is believed to be governed by the size of the deviation from the parameter values that define the banana.
  
  In this talk, we shall examine the challenges to making this proposal rigorous. Chief among these is the linear instability of the intermediate solutions along the homoclinic banana. These complications lead one to consider a simpler regime, namely the stability of solutions to the doubly-diffusive FHN equation with constant initial data close to a normally-attracting portion of the critical manifold of the accompanying fast-slow ODE. Strategies to prove this result using pointwise estimates will be discussed.

• March 20 Mary Lou Zeeman (Bowdoin College)
  Title: Reactivity, Resilience and Flow-Kick Dynamics
  Abstract: Resilience questions make us look at familiar mathematics through a new lens. Resilience is a slippery concept that has different meanings in different contexts. It is often described as the ability of a system to absorb change and disturbance while maintaining its basic structure and function. There is, therefore, an inherent interplay between transient dynamics and perturbation in resilience questions, especially when the perturbations are repeated.

  In the first part of this talk, we subject the flow of an autonomous system of ODEs to regular shocks ("kicks") of constant size, direction, and frequency, representing repeated, discrete disturbances. The resulting flow-kick systems occupy a surprisingly under-explored area between deterministic and stochastic dynamics. We describe some examples with applications to ecology and epidemiology.

  A flow-kick system is in equilibrium when the disturbance and transient dynamics balance. The stability of a flow-kick equilibrium is determined by the accumulated stability (variational equation) along the transient dynamic. This leads to a focus on reactivity (short term response to disturbance) rather than eigenvalues (long term response) in linearizations. In the second part of the talk, we describe a new framework for analyzing the radial and tangential dynamics of two-dimensional linear systems that exploits a dual relationship between reactivity and eigenvalues to more explicitly capture its reactivity properties. This is joint work with BU graduate student Alanna Haslam.

• March 27 Trevor Norton (Boston University)
  Title: Kink-like Solutions for the FPUT Lattice and the mKdV as a Modulation Equation
  Abstract: The Fermi-Pasta-Ulam-Tsingou (FPUT) lattice became of great mathematical interest when it was observed that it exhibited a near-recurrence of its initial condition, despite it being a nonlinear system. This behavior was explained by showing that the Korteweg-de Vries (KdV) equation has soliton solutions and serves as a continuum limit for the FPUT. Much work has been done into analyzing the solitary wave solutions of the FPUT and the relationship between the lattice and its continuum limit. For certain potentials the modified KdV (mKdV) instead serves as the continuum limit for the FPUT. However, there has been little research done to examine how the defocusing mKdV can be used a modulation equation for the FPUT or how the kink solutions of the mKdV relate to solutions of the FPUT. In this talk, we will begin to address these topics. We will see that the existence of the kink-like solutions can be proved using a center manifold argument. The profiles of these solutions can be approximated well by the profile of kink solutions of the mKdV. We will also see that the mKdV can be used as a modulation equation for small-amplitude, long-wavelength solutions of the FPUT. Lastly, a strategy for showing asymptotic stability of the kink-like solutions will be discussed.

• April 3 Kathryn Lindsey (Boston College)
  Title: Rational maps built by mixing two polynomials
  Abstract: A mating of two quadratic polynomials (with connected Julia sets) is, roughly speaking, a rational map that is built by ``gluing'' the two filled Julia sets together along their boundaries to form a topological sphere, and then defining the map on this sphere to be whichever polynomial is defined. Together with Sarah Koch and Tom Sharland, I introduced a new and related way to view some rational maps as the result of ``gluing'' together two filled Julia sets: a mixing. In contrast to a traditional mating of Julia sets (in which the two filled Julia sets are invariant), a mixing intertwines the dynamics on the two Julia sets, in such a way that the second iterate of the mixing coincides with the second iterate of the mating. In order to form this definition, we prove theorems describing pairs of bicritical rational maps that have shared iterates.

• April 10 Konstantin Mischaikow (Rutgers University)
  Title: Identifying Nonlinear Dynamics with High Confidence from Sparse Data
  Abstract: There are a variety of statistical techniques that given sufficient time series identify explicit models, e.g. differential equations or maps, that are then evaluated to predict dynamics. However, chaotic dynamics and bifurcation theory implies sensitivity with respect to small errors in data and parameters, respectively. This suggests a potential inherent instability in going directly from data to models. We propose a novel method, combining Conley theory and Gaussian Process surrogate modeling with uncertainty quantification, through which it is possible to characterize local and global dynamics, e.g., existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics, with lower bounds on the confidence that this characterization of the dynamics is correct. Furthermore, numerical experiments indicate that it is possible to identify nontrivial dynamics with high confidence with surprisingly small data sets.

• April 17 No Seminar - Patriot's Day
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• April 24 Stephanie Dodson (Colby College)
  Title: Behavior of Spiral Wave Spectra with a Rank-Deficient Diffusion Matrix
  Abstract: Spiral waves emerge in numerous pattern forming systems and are commonly modeled with reaction-diffusion systems. Some systems used to model biological processes, such as ion-channel models, fall under the reaction-diffusion category and often have one or more non-diffusing species which results in a rank-deficient diffusion matrix. Previous theoretical research focused on spiral spectra for strictly positive diffusion matrices. In this research, we use a general two-variable reaction-diffusion system to compare the essential and absolute spectra of spiral waves for strictly positive and rank-deficient diffusion matrices. We show that the essential spectrum is not continuous in the limit of vanishing diffusion in one component. Moreover, we predict locations for the absolute spectrum in the case of a non-diffusing slow variable. Predictions are confirmed numerically for the Barkley and Karma models. This work is joint with Bjorn Sandstede.

• May 1 Braden Brinkman (Stony Brook University)
  Title: Non-perturbative renormalization group analysis of collective behavior in networks of spiking neurons
  Abstract: The critical brain hypothesis posits that neural circuits may operate close to critical points of a phase transition, which has been argued to have functional benefits for neural computation. In physics, our modern understanding of the collective behavior that emerges near such phase transitions is provided by the renormalization group. However, the all-or-nothing nature of neural spikes transmitted across a web of synaptic connections make it difficult to apply the ideas of the renormalization group directly to models of leaky integrate-and-fire networks commonly used in neuroscience. In this talk I will discuss recent work adapting the non-perturbative renormalization group (NPRG) method to network models of stochastic spiking neurons, providing a means of investigating the nature of different types of collective behavior in neural circuitry.

• May 8 No Seminar - Finals
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