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
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 27: Alanna Hoyer-Leitzel (Mount Holyoke)
Title: Existence, stability, and symmetry of relative equilibria with a
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.
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