Dynamical Systems Seminars

Spring 2021

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).

- January 25 No Seminar
- February 1 Tea time
Now that everyone is settled in, feel free to come by and catch up after a long winter break!

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- February 8 Ryan Goh

Title: Growing oblique stripes

Abstract: Spatial growth plays an important role in controlling pattern formation in systems arising in many areas of science. One of the simplest mechanisms for growth is a directional "quench" which travels across a domain, suppressing patterns in one part of the domain, and exciting them in the other. In this talk we will discuss how a linear phase diffusion equation with a moving nonlinear boundary condition can be used to characterize the formation of stripes obliquely oriented to the quenching interface. We use rigorous analysis, formal asymptotics, and numerical continuation to characterize stripe selection for various quenching speeds and stripe angles. Of particular interest, we find that the slow-growth, small angle regime is governed by the glide-motion of a dislocation defect at the quench interface. Finally, we compare our results numerically to stripe formation in a quenched anisotropic Swift-Hohenberg equation.

- February 15 No seminar (Presidents Day)

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- February 22 Alice Nadeau
Cornell University

Title: Synchronization of clocks and metronomes: A perturbation analysis based on multiple timescales

Abstract: In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphase after hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform show that they often tend to synchronize in phase, not antiphase. Here we study both in-phase and antiphase synchronization in a model of pendulum clocks and metronomes and analyze their long-term dynamics with the tools of perturbation theory. Specifically, we exploit the separation of timescales between the fast oscillations of the individual pendulums and the much slower adjustments of their amplitudes and phases. By scaling the equations appropriately and applying the method of multiple timescales, we derive explicit formulas for the regimes in parameter space where either antiphase or in-phase synchronization are stable, or where both are stable. Although this sort of perturbative analysis is standard in other parts of nonlinear science, it has been applied surprisingly rarely in the context of Huygens's clocks. Unusual features of our approach include its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and both a two- and three-timescale asymptotic analysis.

- March 1 Montie Avery
University of Minnesota

Title: Universal dynamics of pulled fronts

Abstract: The formation of structure in spatially extended systems is often mediated by an invasion process, in which a pointwise stable state invades a pointwise unstable state. A fundamental goal is then to predict the speed of this invasion. The marginal stability conjecture postulates that, absent a mechanism through which the nonlinearity enhances propagation, the invasion speed is predicted by marginal linear stability of the pointwise unstable background state in a suitable norm. We introduce a set of largely model-independent conceptual assumptions under which we establish nonlinear propagation at the linear spreading speed for open classes of steep initial data, thereby resolving the marginal stability conjecture in the general case of stationary invasion. Our assumptions hold for open classes of parabolic equations, including higher order equations without comparison principles, while previous results rely on special structure of the equation and the presence of a comparison principle. Our result also establishes universality of the logarithmic in time delay in the position of the front, compared with propagation strictly at the linear speed, as predicted in generality by Ebert and van Saarloos and first established in the special case of the Fisher-KPP equation by Bramson. Our proof describes the invasion process through the interaction of a Gaussian leading edge with the pulled front in the wake. Technically, we rely on sharp linear decay estimates to control errors from this matching procedure and corrections from the initial data.

- March 8 Evelyn Sander
George Mason University

Title: Birkhoff Averages and the Breakdown of Invariant Tori in Symplectic and Volume-Preserving Maps

Abstract: The dynamics of an integrable Hamiltonian or volume-preserving systems consists of quasiperiodic motion on invariant tori. As such a system is perturbed, KAM theory guarantees that some of these tori persist, whereas others are replaced by periodic orbits, islands, and chaotic regions. The tori that persist the longest as the perturbation increases are referred to as the most robust tori, and it is of great interest to understand how to characterize the frequency vectors of these tori. In this talk, I will discuss numerical investigations in dimensions two and three. These investigations use methods for highly accurate calculation of Birkhoff averages using a weighting factor. They additionally require computational number theoretic methods for calculating whether a number (resp. vector) is rational (resp. nonresonant) up to a certain tolerance. Together, these methods make it possible to distinguish between chaotic regions, islands, and invariant tori, while simultaneously giving a highly accurate estimate of the frequency vector of each torus. We demonstrate these methods in two dimensions for variants of Chirikovâ€™s standard map. In three dimensions, we apply the methods to the three-dimensional standard map as a test case. This work is in collaboration with James Meiss.

- March 15 Kirill Korolev
Boston University

Title: Universality classes in the evolutionary dynamics of expanding populations

Abstract: Reaction-diffusion waves describe diverse natural phenomena from crystal growth in physics to range expansions in biology. Two classes of waves are known: pulled, driven by the leading edge, and pushed, driven by the bulk of the wave. Recently, we examined how demographic fluctuations change as the density-dependence of growth or dispersal dynamics is tuned to transition from pulled to pushed waves. We found three regimes with the variance of the fluctuations decreasing inversely with the population size, as a power law, or logarithmically. These scalings reflect distinct genealogical structures of the expanding population, which change from the Kingman coalescent in pushed waves to the Bolthausen-Sznitman coalescent in pulled waves. The genealogies and the scaling exponents are model-independent and are fully determined by the ratio of the wave velocity to the geometric mean of dispersal and growth rates at the leading edge. Our theory predicts that positive density-dependence in growth or dispersal could dramatically alter evolution in expanding populations even when its contribution to the expansion velocity is small. On a technical side, our work highlights potential pitfalls in the commonly-used method to approximate stochastic dynamics and shows how to avoid them. I will also discuss our more recent efforts to extend these results to two spatial dimensions.

- March 22 No Seminar

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- March 29 Panagiotis Kaklamanos
University of Edinburgh

Title: Bifurcations of mixed-mode oscillations in three-timescale systems.

Abstract: Many systems from the physical and the life sciences exhibit phenomena that evolve on multiple timescales; examples include reactions that occur at different rates in chemical kinetics, concentrations of ions that give rise to action potentials in neuronal firing, and sudden changes in temperature or pressure followed by long periods of almost steady states in climate models, to mention but a few. Such phenomena are often associated with trajectories that consist of alternating small- and large-amplitude oscillations in the phase space, known as mixed-mode oscillations (MMOs). In this talk, I will present an extended prototypical example that captures a geometric mechanism which encodes transitions between MMOs with different qualitative properties in three-dimensional, three-timescale systems written in the standard form of geometric singular perturbation theory (GSPT). I will then present a novel and global three-dimensional, three-timescale reduction of the four-dimensional Hodgkin-Huxley (HH) equations from Mathematical Neuroscience, which is based on GSPT. Finally, I will show how the various firing patterns that have been observed in the four-dimensional HH equations can be explained by extending results obtained for the prototypical example to this new three-dimensional, three-timescale reduction. - April 5 Roland Welter
Boston University

Title: Asymptotic approximation of fluid flows from the compressible Navier-Stokes equations

Abstract: The compressible Navier-Stokes equations generalize the well-known 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 their 1995 paper, Hoff and Zumbrun proved global existence of small energy solutions to these equations and showed that the dominant asymptotic behavior is given by diffusing Gaussian profiles advected by the sound waves of the fluid. In a joint work with Prof. Wayne and Prof. Goh, we introduce a modified compressible Navier-Stokes system, and show that this captures the dominant asymptotic behavior of the original system. We then show existence of solutions to this equation in weighted spaces, and develop a method for approximating the asymptotic behavior which extends and unifies the approach of Hoff and Zumbrun with the approach used by Gallay and Wayne to study the incompressible equations.

- April 12 Cengiz Pehlevan
Harvard University

Title: Inductive Bias of Neural Networks

Abstract: Predicting a target output from training examples is unsolvable without additional assumptions about the nature of the target function. A learnerâ€™s performance depends crucially on how its internal assumptions (or inductive bias) align with the task at hand. I will present a theory that describes the inductive bias of neural networks using kernel methods and statistical mechanics, and discuss its applications to artificial and biological neural systems and real datasets.

- April 19 No Seminar (Patriot's Day)

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- April 26 Carina Curto
The Pennsylvania State University

Title: Graphs, network motifs, and threshold-linear algebra in the brain

Abstract: Threshold-linear networks (TLNs) are commonly-used rate models for modeling neural networks in the brain. Although the nonlinearity is quite simple, it leads to rich dynamics that can capture a variety of phenomena observed in neural activity: persistent activity, multistability, sequences, oscillations, etc. Here we study competitive threshold-linear networks, which exhibit both static and dynamic attractors. These networks have corresponding hyperplane arrangements whose oriented matroids encode important features of the dynamics. We will show how the graph associated to such a network yields constraints on the set of (stable and unstable) fixed points, and how these constraints affect the dynamics. In the special case of combinatorial threshold-linear networks (CTLNs), we find an even stronger set of "graph rules" that allow us to predict emergent sequences and to engineer networks with prescribed dynamic attractors.

### Directions to BU Math Dept.

### Speakers from previous years