Dynamical Systems Seminars
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
Title: The dynamics of boundary droplets for the mass-conserving Allen-Cahn
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
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
- March 10: No seminar - Spring Break
- March 17: Johanna
Title: List-and-check algorithms for classifying mapping classes and outer
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
Title: Numerical Methods and Stochastic Simulation Algorithms for
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
Title: Rational Maps: The structure of Julia sets from accessible Mandelbrot
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
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
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.
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