Dynamical Systems Seminars

Spring 2014

The Dynamical Systems seminar is held on Monday afternoon at 4:00 PM in MCS 148. Tea beforehand at 3:45 PM in MCS 153.

- January 27: Kelly McQuighan (Brown)

Title: Oscillons near Hopf Bifurcations of Planar Reaction Diffusion Equations

Abstract: Oscillons are spatially localized, temporally oscillating, radially symmetric structures. They have been observed in various experimental contexts, including fluid systems, granular systems, and chemical systems. Oscillons often arise near forced Hopf bifurcations, which are modeled mathematically with the forced complex Ginzburg-Landau equation. I present a proof of the existence of small amplitude oscillons in the planar forced complex Ginzburg-Landau equation through a geometric blow-up analysis. The analysis in complemented by a numerical continuation study of oscillons away from onset using Matlab and AUTO. This is joint work with Bjorn Sandstede at Brown University.

- February 3: Jiayin Jin
(Michigan State)

Title: The dynamics of boundary droplets for the mass-conserving Allen-Cahn equation.

Abstract: We establish the existence of a global invariant manifold of bubble states for the mass-conserving Allen-Cahn Equation in two space dimensions and give the dynamics for the center of the bubble.

- February 10: Greg Faye (Minnesota)

Title: Existence of traveling pulse solutions in excitable media with nonlocal coupling

Abstract: In this talk, we prove the existence of fast traveling pulses for a class of FitzHugh-Nagumo equations with nonlocal diffusion. Unlike the dynamical systems approach via geometric singular perturbation theory (Fenichel's theorem and Exchange Lemma), our proof relies on matched asymptotics techniques and Fredholm properties of differential operators on suitable Banach spaces (Spectral Flow and Nonlocal Exchange Lemma). This is joint work this Arnd Scheel.

- February 17: No seminar - President's Day

- February 24: Mei Yin (Brown)

Title: Phase transitions in the edge-triangle exponential random graph model

Abstract: The edge-triangle exponential random graph model has been a topic of continued research interest. We review recent developments in the study of this classic model and concentrate on the phenomenon of phase transitions. We first describe the asymptotic feature of the model along general straight lines. We show that as we continuously vary the slopes of these lines, a typical graph exhibits quantized behavior, jumping from one complete multipartite structure to another, and the jumps happen precisely at the normal lines of an infinite polytope. We then turn to exponential models where certain constraints are imposed and capture another interesting type of jump discontinuity. This expository talk is based on recent joint work with Alessandro Rinaldo and Sukhada Fadnavis and current joint work in progress with Richard Kenyon. We will point out that many questions/issues raised in this talk are actually studied under different names or from different directions in dynamics.

- March 3: Professional development session on research journals for
graduate students

- March 10: No seminar - Spring Break

- March 17: Johanna
Mangahas (Brown)

Title: List-and-check algorithms for classifying mapping classes and outer automorphisms

Abstract: Fixing a generating set for the mapping class group MCG(S) of a surface, it's easy to see that stretch factors of pseudo-Anosov elements grow at most exponentially with word length. Non-pseudo-Anosov mapping classes have canonical fixed curve sets, and it turns out the lengths of these curves also grow at most exponentially with word length. I'll share a simple computation for such an upper bound, which leads to one of the algorithms of the title. This is joint with Thomas Koberda. Mapping class groups are a frequent point of reference for understanding Out(F), the group of outer automorphisms of the free group F. The "fully irreducible" elements of Out(F) are its version of pseudo-Anosov mapping classes. I'll describe an analogous list-and-check method for identifying fully irreducible elements of Out(F). This is joint with Matt Clay and Alexandra Pettet.

- March 24: No seminar.

- March 31: Ava Mauro
(Boston University)

Title: Numerical Methods and Stochastic Simulation Algorithms for Reaction-Drift-Diffusion Systems

Abstract: We have developed a new numerical method for simulating stochastic reaction-drift-diffusion systems, in which the drift arises from spatially varying potential fields. Such potential fields are useful for modeling the spatially heterogeneous environment within a biological cell. The method combines elements of the First-Passage Kinetic Monte Carlo (FPKMC) method for simulating reaction-diffusion systems and the Wang-Peskin-Elston lattice discretization of drift-diffusion. In this combined method, which we call Dynamic Lattice FPKMC, each molecule undergoes a continuous-time random walk on its own lattice, and the lattices change adaptively over time. We will describe the method, summarize results demonstrating the convergence and accuracy of the method, and show applications motivated by cell biology.

- April 1 (special Tuesday seminar, 3:30pm in MCS 148): Elizabeth Fitzgibbon
(Boston University)

Title: Rational Maps: The structure of Julia sets from accessible Mandelbrot sets

Abstract: We consider the family of singularly perturbed complex polynomials, F(z) = z^n + c/z^d. Many small copies of the well-known Mandelbrot set are visible in the parameter plane. An infinite number of these are located around the boundary of the connectedness locus. Maps taken from the main cardioids of these accessible Mandelbrot sets have attracting periodic cycles. A method for constructing models of the Julia sets corresponding to such maps is described. These models are then used to explore the existence of dynamical conjugacies between maps taken from distinct accessible Mandelbrot sets in the upper halfplane.

- April 7: Rocio Gonzalez Ramirez
(Boston University)

Title: Existence and stability of traveling waves in a biologically constrained model of seizure wave propagation

Abstract: Epilepsy - the condition of recurrent, unprovoked seizures - manifests in brain voltage activity with characteristic spatio-temporal patterns including traveling waves. To characterize these waves, we analyze clinical data recorded in vivo from human cortex during a seizure. Using a mean-field approach we model the neuronal population activity and obtain traveling wave solutions for this model. We employ the observed clinical data to constrain the model, and obtain parameter configurations that support traveling waves with features consistent with the observed waves. We also study the stability of the traveling wave solutions. To do so, we locate the essential spectrum and construct an Evans function to study the point eigenvalues of the linearization of the system about the wave.

- April 9 (special Wednesday seminar, 4pm in MCS B21): Dan Cuzzocreo
(Boston University)

Title: Dynamical Invariants and Parameter Space Structures for Rational Maps

Abstract: For parametrized families of dynamical systems, two major goals are classifying the systems up to topological conjugacy, and understanding the structure of the bifurcation locus. The family $F_\lambda = z^n + \lambda/z^d$ gives a $1$-parameter, $n+d$ degree family of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial $z^n$. This work presents several results related to these goals for the family $F_\lambda$, particularly regarding a a structure of ``necklaces" in the $\lambda$ parameter plane. This structure consists of infinitely many simple closed curves which surround the origin, and which contain postcritically finite parameters of two types: superstable parameters and escape time Sierpi\'nski parameters. We prove the existence of a deeper fractal system of ``subnecklaces," wherein the escape time Sierpi\'nski parameters on the previously known necklaces are themselves surrounded by infinitely many necklaces. We also derive a dynamical invariant to distinguish the conjugacy classes among the superstable parameters on a given necklace, and to count the number of conjugacy classes.

- April 14: Jacob Bedrossian
(Courant Institute, NYU)

Title: Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

Abstract: We prove asymptotic stability of shear flows close to the planar, periodic Couette flow in the 2D incompressible Euler equations. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically mixed to small scales by an almost linear evolution and in general enstrophy is lost in the weak limit. Joint work with Nader Masmoudi.

- April 18 (special Friday seminar; 4pm in MCS 148): Sarah Koch (Michigan)

Title: Eigenvalues and Thurston's theorem.

Abstract: Given a postcritically finite rational map on the Riemann sphere, there are several dynamical systems that correspond to it, which naturally arise in the setting of Thurston's topological characterization of rational maps. Associated to each of these dynamical systems is a corresponding linear operator. In this talk we discuss the sets of eigenvalues of these operators and explore connections between them.

- April 21: No seminar - Patriot's Day

- April 28: Miles Wheeler (Brown)

Title: Large-amplitude solitary water waves with vorticity

Abstract: The water wave equations describe the motion of an incompressible inviscid fluid under the influence of gravity which is bounded above by a free surface under constant (atmospheric) pressure. In this talk, we will construct exact solitary water waves of large amplitude and with an arbitrary distribution of vorticity. Starting from a shear flow with a flat free surface, we use a degree-theoretic continuation argument to construct a global connected set of symmetric solitary waves of elevation, whose profiles decrease monotonically on either side of a central crest. We will also discuss solitary waves generated by a non-constant pressure on the free surface.

- May 2 (special Friday seminar, 1-2pm in MCS B33): Graham Cox (UNC Chapel Hill)

Title: A Morse index theorem for elliptic operators on bounded domains

Abstract: The Maslov index is a symplecto-geometric invariant that counts signed intersections of Lagrangian subspaces. It was recently shown that the Maslov index can be used to compute Morse indices of Schrodinger operators on star-shaped domains. We extend these results to general selfadjoint, elliptic operators on domains with arbitrary boundary geometry, and discuss some applications. (Joint work with C. Jones and J. Marzuola.)