## Dynamical Systems Seminars Spring 2017

The Dynamical Systems seminar is held on Monday afternoons at 4:00 PM in MCS 148. Tea beforehand is at 3:45 PM in MCS 144.

• January 23: Mike Todd (St Andrews)
Title: Stability of measures in interval dynamics
Abstract: Given a family of interval maps, each map possessing a `physical' measure (an invariant measure absolutely continuous w.r.t. Lebesgue), we have a weak form of stability if these measures change continuously through the family. Even for uniformly hyperbolic dynamical systems this stability can fail. I†¢ll give minimal conditions for a class of non-uniformly hyperbolic interval maps to satisfy this stability property. This work forms part of a paper with Neil Dobbs, where more general thermodynamic properties are proved to be stable (entropy, pressure, equilibrium states), and I†¢ll give some indication of the general approach there.

• January 30: Eric Chang (BU)
Title: The Sierpinski Mandelbrot spiral for the rational map
Abstract: We investigate the parameter plane for the family of maps $$F(z) = z^n + \lambda/z^d$$ where $$n \geq 4$$ is even, $$d \geq 3$$ is odd, and $$\lambda$$ is a complex parameter. Concentrating on $$F(z) = z^4 + \lambda / z^3$$, we prove the existence of two structures in the parameter plane: a Sierpinski Mandelbrot arc consisting of infinitely many alternating Sierpinski holes and Mandelbrot sets, as well as a Sierpinski Mandelbrot spiral consisting of infinitely many SM arcs. We also show that there are infinitely many SM spirals in the parameter plane.

• February 6: Osman Chaudhary (BU)
Title: Rigorous Justification of Taylor Dispersion via Center Manifold Theory
Abstract: Imagine fluid moving through a long pipe or channel, and we inject dye or solute into this pipe. Initially, the dye just moves along downstream with the fluid. However, it is also slowly diffusing down the pipe and towards the edges as well. It turns out that after a long time, the combined effect of transport via the fluid and this slow diffusion results in what is effectively a much more rapid diffusion process, lengthwise down the stream. If 0 < \nu << 1 is the slow diffusion coefficient, then the effective longitudinal diffusion coefficient is inversely proportional to \nu, i.e. much larger. This phenomenon is called Taylor Dispersion, first studied by GI Taylor in the 1950s, and studied subsequently by many authors since, such as Aris, Chatwin, Smith, Roberts, and others. I'll propose a dynamical systems explanation of this phenomenon: specifically, I'll explain how one can use a Center Manifold reduction to obtain Taylor Dispersion as the dominant term in the long-time limit, and also explain how this Center Manifold can be used to provide any finite number of correction terms to Taylor Dispersion as well.

• February 13: No seminar planned; please consider attending the workshop at ICERM on The Dynamics of Small Scale Fluids.

• February 20: Holiday, No Seminar

• February 27: Alanna Hoyer-Leitzel (Mount Holyoke)
Title: Existence, stability, and symmetry of relative equilibria with a dominant vortex
Abstract: We will consider the problem of point vortices in the plane from the perspective of Hamiltonian systems, in particular we will analyze existence, stability, and symmetry of point vortex relative equilibria with one dominant vortex and N vortices with infinitesimal circulations. Previous results for the problems of existence and stability focused on the case where the N small vortices had the same circulation. In this talk, I'll present a new proof to generalize to the case where the N vortices can have different circulations, and then apply these results to find examples of stable asymmetric relative equilibria in the case when N=3. We'll also see the use of some computational algebraic geometry to find bifurcations in families of relative equilibria when N=3, and some beginning analysis for N=4.

• March 6: Spring Break, No Seminar

• March 13: TBA
Title: TBA
Abstract: TBA

• March 20: No seminar planned; please consider attending the workshop at ICERM on Making a Splash - Droplets, Jets and Other Singularities.

• March 27: Patrick Cummings (BU)
Title: NLS Approximations for PDEs with Quadratic and Quasilinear Terms
Abstract: We discuss approximation results in general. In particular, their motivation, history, and general theory. We will discuss the main methods used to justify approximation results and the common difficulties stemming from those methods. We then introduce two methods to overcome those difficulties. We finish by considering two physical models that can be approximated using the nonlinear Schr\"odinger equation. The first is a model of the water wave equations. By working with a modified energy functional motivated by Craig and Hunter et al., we are able to avoid some of the restrictions on previous results of Schneider and Wayne. We then consider the Klein-Gordon-Zakharov system and prove an approximation result using analytic norms.

• April 3: Noé Cuneo (McGill)
Title: An overview of nonequilibrium steady states for chains of oscillators and rotors
Abstract: I will talk about chains of oscillators and rotors interacting with stochastic heat baths at different temperatures. I will introduce these very simple models in the framework of the (yet unsolved!) problem of heat conduction. Then, we will focus on a much more elementary question: the existence of an invariant measure (called non-equilibrium steady state), which has been proved only in some specific cases over the past 20 years. I will explain how distinct models lead to distinct difficulties, and sketch some of the ideas used to overcome them.

• April 10: Darko Volkov (WPI)
Title: An all-frequency boundary integral equation for Maxwell†¢s equations in dielectric media. Formulation and error analysis.
Abstract: In this talk we will discuss a new boundary integral equation (BIE) formulation whose solution pertains to numerical simulations of propagating time-harmonic electromagnetic waves in three dimensional dielectric media. We developed and analyzed this BIE which is governed by an operator that is of the classical identity plus compact form. A novel feature of that second-kind BIE is that, when augmented with two stabilization equations, the corresponding reformulation does not suffer from spurious resonances or low-frequency breakdown. This BIE formulation provides a tool for developing a new class of efficient and high-order algorithms (with numerical analysis) for simulation of the three dimensional dielectric media model from low- to high-frequencies. We will outline the proof of spectral numerical convergence; there are special challenges in that proof due to the use of specific quadrature rules. In particular, the typical two-dimensional case proof of convergence method (found, for example, in a textbook by Colton and Kress), is not applicable in our case.

• April 17: Holiday, No Seminar

• April 24: No seminar planned; please consider attending the workshop at ICERM on Water Waves.

• May 1: Maxim Olshanii (UMass Boston)
Title: The Inverse Linearization Problem
Abstract: We investigate the relationship between the nonlinear partial differential equations (PDEs) of mathematical physics and the their linearizations around localized stationary solutions. It turns out that for some classes of PDEs, it is possible to solve the Inverse Linearization Problem, i.e. given the linearization, to restore the original PDE. Of a particular interest are the instances of transparency of the former that are shown to hint on the possible integrability of the latter.