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.