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.
February 4 No seminar
February 11 Shugan Ji
Title: Time periodic solutions of nonlinear wave equation with x-dependent coefficients
Abstract: In this talk, we consider the time periodic solutions of nonlinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of an inhomogeneous string and the propagation of seismic waves in nonisotropic media. We shall talk about some existence results on the time periodic solutions of such a model with different types of nonlinearity.
March 4 Michael Bialy
(Tel Aviv University); Note: The tea and seminar will be
held in MCS B33.
Title: Around Birkhoff's conjecture for convex and other
Abstract: Birkhoff's conjecture states that the only integrable billiards in the plane are ellipses. I am going to give a survey of recent progress in this conjecture and to discuss geometric results and questions around it. Based on joint works with Andrey E. Mironov, Maxim Arnold.
No prior knowledge of the subject will be assumed.
March 18 Jonathan Jacquette
Title: Computer assisted proofs of Wright's and Jones' Conjectures: Counting and discounting slowly oscillating periodic solutions to a delay differential equation
A classical example of a nonlinear delay differential equations is
Wright's equation: $y'(t) = \alpha y(t-1) [1+y(t)]$, considering $\alpha >0$ and $y(t)>-1$. This talk discusses two conjectures associated with this equation: Wright's conjecture, which states that the origin is the global attractor for all $ \alpha \in ( 0 , \pi /2 ]$; and Jones' conjecture, which states that there is a unique slowly oscillating periodic solution for $ \alpha > \pi / 2$.
To prove Wright's conjecture our approach relies on a careful
investigation of the neighborhood of the Hopf bifurcation
occurring at $\alpha = \pi /2 $. Using a rigorous numerical
integrator we characterize slowly oscillating periodic solutions
and calculate their stability, proving Jones' conjecture for
$\alpha \in [1.9,6.0]$ and thereby all $\alpha\geq 1.9$. We complete the proof of Jones conjecture using global optimization methods, extended to treat infinite dimensional problems.
March 25 Matt Holzer
(George Mason University)
Title: Invasion fronts in spatially extended systems
Abstract: Invasion fronts refer to fronts propagating into unstable
states. An important characteristic of such fronts is the speed at
which they propagate. In this talk, I will review wavespeed selection
principles and discuss two recent research projects related to wavespeed
selection. In the first, we study the emergence of locked fronts in a
class of two component reaction diffusion equations. In the second, a
model of global epidemics is considered and predictions for arrival
times are obtained based upon the linearization about the unstable
April 1 Margaret Beck
Title: Spectral Stability, Spatial Dynamics, and the Maslov Index
Abstract: Understanding the stability of solutions to PDEs is important, because it is typically only stable solutions which are observable. For many PDEs in one spatial dimension, stability is well-understood, largely due to a formulation of the problem in terms of so-called spatial dynamics, where one views the single spatial variable as a time-like evolution variable. This allows for many powerful techniques from the theory of dynamical systems to be applied. In higher spatial dimensions, this perspective is not clearly applicable. In this talk, I will discuss recent work that suggests both that the Maslov index could be a important tool for understanding stability when the system has a symplectic structure, particularly in the multi-dimensional setting, and also suggests a possible analogue of spatial dynamics in the multi-dimensional setting.
April 8 Cynthia A. Bradham
Title: The Molecular Basis of Skeletal Patterning
During embryonic development, a single cell, the fertilized egg, gives rise to a complete animal.
Understanding how tissues are sculpted and patterned during embryonic development is
important for the prevention of birth defects and to inform tissue engineering and regenerative
medicine, but the mechanisms that drive developmental patterning processes remain largely
unknown due to the complexity of the question. Because the mechanisms underlying
development are well-conserved in evolution, much can be learned from studying the patterning
mechanisms of simple organisms such as sea urchins. The sea urchin larval skeleton serves as
a simple model of patterning. The skeleton is secreted by primary mesenchymal cells (PMCs),
while the patterning information is contained within the ectoderm, and sensed by the migrating
PMCs. Using high throughput sequencing, we performed a screen to identify novel skeletal
patterning cues that are expressed by the ectoderm. Collectively, we’ve discovered conserved
cues required for patterning most of the major skeletal regions. We are now addressing how the
PMCs interpret the ectodermal patterning cues using high throughput sequencing at the single
Luigi Chierchia (University of Roma Tre)
Title: Generic topological and geometrical structure of nearly-integrable
Hamiltonian systems at simple resonances
Abstract: As well known, the dynamics of nearly-integrable Hamiltonian systems
is strongly affected by resonances. Particularly relevant (e.g., in the
study of Arnold diffusion, or in the metric theory of regular motions)
are simple resonances.
We describe the analytic, geometric and topological structure
of simple resonances for nearly-integrable analytic Hamiltonian systems,
establishing, in particular, "sharp" analytic normal forms for
generic holomorphic perturbations.
April 29 Bastian Hilder (Universität
Title: Modulating traveling fronts in pattern forming systems with conserved quantities
Abstract: In this talk, I will consider a Swift-Hohenberg equation coupled to a conservation law. As a parameter increases this system undergoes a Turing bifurcation and small periodic solutions emerge. The first part of the talk is concerned with the existence of modulating traveling fronts near the bifurcation. These fronts describe an invasion of the unstable ground state by the periodic pattern and provide a mechanism of pattern formation. In the second part, I will discuss the stability of fronts in the corresponding amplitude equation, a modified Ginzburg-Landau system with a conservation law. This is a natural first step towards a stability result for the full problem.
Please also consider attending the
Brown/BU/UMassAmherst seminar in PDE and Dynamics on May 3rd.