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 UIUC)
- October 27: Semyon Dyatlov (MIT)
- November 3: Raj Prasad
(UMass, Lowell)

Title: "Infinite Measure Preserving Transformations and Tilings of the Integers Associated to them." - November 10:
- November 17: Doug Wright (Drexel)
- November 24:
- December 1: Osman
Chaudhary (BU)
- December 8: