Dynamical Systems Seminars
Fall 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.
- September 15: No seminar; come to the BU/Keio Dynamics Workshop instead!
- September 22: Kelly
McQuighan (BU)
Title: Oscillons near Hopf bifurcations of planar forced reaction diffusion
equations
Abstract: Oscillons are planar, 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 birurcations.
It is known that small amplitude localized solutions to the planar forced
complex Ginzburg-Landau equation (fCGL) exist near onset. Using spatial
dynamics, we show that the dynamics on the center manifold of a periodically
forced reaction difuusion equation (fRD) near a Hopf bifurcation can be
captured by the fCGL. Thus, oscillon solutions to the fRD can be thought of as
a foliation over localized solutions to the fCGL. The is a work in progress,
joint with Bjorn Sandstede.
- September 29: Theo Vo (BU)
Title: Geometric Singular Perturbation Analysis of Mixed-Mode Dynamics in
Pituitary Cells
Abstract: Pseudo-plateau bursting is a type of oscillatory waveform associated
with mixed mode dynamics in slow/fast systems and commonly found in neural
bursting models. Multiple methods from dynamical systems theory have been used
to understand these bursting rhythms, which are typically treated as
2-timescale problems. In the first part of this work, we demonstrate that the
two most common analysis techniques are different unfoldings of a 3-timescale
system. Our analysis shows that canards are a key feature of these systems
that locally organise the dynamics in phase space.
Canards are closely associated with folded singularities and in the case of
folded nodes, lead to a local twisting of invariant manifolds. Folded node
canards and folded saddle canards (and their bifurcations) have been studied
extensively. The folded saddle-node (FSN) is the codimension-1 bifurcation
that gives rise to folded nodes and folded saddles. Their dynamics however,
are not well-understood. In the second part of this work, we extend canard
theory into the FSN regime by combining methods from geometric singular
perturbation theory (blow-up), and the theory of dynamic bifurcations
(analytic continuation into the plane of complex time).
- October 6: Evelyn Sander (George Mason University)
Title: The Dynamics of Nucleation
Abstract: The Cahn-Hilliard equation is one of the fundamental models to describe
phase separation dynamics in metal alloys. In this talk, I will focus on
applying traditional dynamical tools, such as bifurcation theory and
computational topology in order to gain a better understanding of the
droplet formation during nucleation for the stochastic Cahn-Hilliard
equation. I will consider different types of noise and different types
of boundary conditions.
- October 13: Columbus Day; no seminar.
- October 20: Hans Kaper
(Georgetown and MCRN)
Title: Mathematics and Climate - Challenges for Dynamical Systems
Abstract: Mathematical models and statistical arguments play a central role in
the assessment of the changes that are observed in Earth's climate
system. While much of the discussion of climate change is focused on
large-scale computational models, the theory of dynamical systems provides the
language to distinguish natural variability from change. In this talk I will
discuss some problems of current interest in climate science and indicate how,
as mathematicians, we can find inspiration for new applications.
- October 27: Semyon Dyatlov
(MIT)
Title: A microlocal toolbox for hyperbolic dynamics
Abstract: I will discuss recent applications of microlocal analysis to the
study of hyperbolic flows, including geodesic flows on negatively curved
manifolds. The key idea is to view the equation $(X+\lambda)u=f$, where $X$ is
the generator of the flow, as a scattering problem. The role of spatial
infinity is taken by the infinity in the frequency space. We will concentrate
on the case of noncompact manifolds, featuring a delicate interplay between
shift to higher frequencies and escaping in the physical space.
I will show meromorphic continuation of the resolvent of $X$; the poles, known
as Pollicott-Ruelle resonances, describe exponential decay of correlations. As
an application, I will prove that the Ruelle zeta function continues
meromorphically for flows on non-compact manifolds (the compact case, known as
Smale's conjecture, was recently settled by Giulietti-Liverani-Pollicott and a
simple microlocal proof was given by Zworski and the speaker). Joint work with
Colin Guillarmou.
- November 3: Raj Prasad
(UMass, Lowell)
Title: Infinite Measure Preserving Transformations and Tilings of the Integers
Associated to them.
Abstract: We consider sequences of integers associated to an infinite
measure preserving transformation. We describe how some of these
sequences give unexpected tilings of the integers. This talk contains
material from our book "Weakly Wandering Sequences in Ergodic Theory" written with Eigen,
Hajian and Ito.
- November 10: Pankaj Mehta (BU)
Title: Dynamical Quorum Sensing and Collective Behavior in Unicellular
Organisms
In this talk, I will present ongoing work on communication and collective
behavior in unicellular organisms. In the first half of the talk, I will show
how we are using concepts from physics and dynamical systems such as
universality to make simple, predictive models of complex biological
behaviors in cellular populations of the social amoeba
Dictyostellium. Inspired by Dictyostelium, in the second half of the talk, I will focus on the physics and mathematics of
dynamical quorum sensing. In particular, I will discuss how we can use
techniques from applied mathematics and physics to understand cell-density
dependent transitions to collective oscillations in limit-cycle oscillators
and the Kuramoto model. (see: arXiv:1406.6731; arXiv:1407.8210; Physica D:
241, 1782-1788; Chaos: 22, 043139)
- November 17: Doug Wright (Drexel)
Title: Approximation of Polyatomic FPU Lattices by KdV Equations
Abstract. Famously, the Korteweg-de Vries equation serves as a model
for the propagation of long waves in Fermi-Pasta-Ulam (FPU) lattices.
If one allows the material coefficients in the FPU lattice to vary
periodically, the -Y´classical" derivation and justification of the KdV
equation go awry. By borrowing ideas from homogenization theory, we
can derive and justify an appropriate KdV limit for this problem. This
work is joint with Shari Moskow, Jeremy Gaison and Qimin Zhang.
- November 24: No seminar.
- December 1: Osman
Chaudhary (BU)
Title: Taylor Dispersion
Abstract: When a solute is injected into a pipe or channel of moving fluid,
the combined action of diffusion and transport cause the solute to spread out
in the domain. It so happens that cross-stream variations in the fluid
velocity cause the solute to diffuse at a rate apparently much higher than one
would expect if the fluid was moving at a uniform rate across the stream. This
phenomenon is called Taylor dispersion, and we'll show how one can obtain the
apparent diffusion coefficient using a formal calculation, and obtain it
rigorously using a center manifold reduction. We'll consider both cases of a
2D channel of fluid and a 3D pipe, as well as both cases of laminar and
turbulent fluids moving through these domains. In all cases, most of the
mathematical difficulties can be represented in a greatly simplified "model"
equation, and it is through this model equation that the analysis will be
presented.
- December 8: Jae Kyoung Kim (Mathematical Biosciences Institute, Ohio State)
Title: Mathematics for complex and stochastic biochemical networks with
disparate timescales
Abstract: The functions of living cells are regulated by the complex biochemical
network, which consists of stochastic interactions among genes and
proteins. However, due to the complexity of biochemical networks and the
limit of experimental techniques, identifying entire biochemical
interaction network is still far from complete. On the other hand,
output of the networks, timecourses of genes and proteins can be easily
acquired with advances in technology. I will describe how to reveal the
biochemical network architecture with oscillating timecourse data. Next,
I will discuss how to reduce or simplify the stochastic biochemical
networks while preserving the slow timescale dynamics. Specifically, I
will show when macroscopic rate functions (e.g. Michaelis-Menten
function) describing the slow timescale dynamics of deterministic
systems can be used for stochastic simulations. Finally, I will discuss
how the network topology affects the functions and dynamics of
biochemical networks with an example of circadian (~24hr) clock.
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