SCIENTIFIC PROGRAMS AND ACTIVITIES

SOUTHERN ONTARIO DYNAMICAL SYSTEMS SEMINAR

February 28, 1997

First Speaker: Dr. Ali Lari-Lavassani, The Fields Institute
Title: Bifurcation, Symmetry and Singularity Theory (Abstract below)

Second Speaker: Dr. Victor LeBlanc, University of Ottawa
Title: "Meandering of the Tip" in Spiral Waves: A Center Bundle Approach*

(Abstract below)

ABSTRACTS OF THE TALKS:

SPEAKER: Dr. Ali Lari-Lavassani, The Fields Institute

TITLE: Bifurcation, Symmetry and Singularity Theory

ABSTRACT:

This talk is intended to be partly tutorial. It is well known that (generically) in steady-state bifurcations, any zero eigenvalue is simple, the center manifold is one-dimensional, and (provided that the 1-jet of the vector field satisfies a transversality condition) a unique smooth branch of solutions bifurcates from the equilibrium.

A similar situation holds for Hopf bifurcation, but the dimension of the center manifold is then 2. In the presence of symmetry, the above generic hypothesis are violated: multiple eigenvalues can be generic. Indeed the structure of the linearization of a vector field, as well as that of the nonlinear terms is governed by symmetry.

We will review some basic group theoretical techniques and tools that enable one to organize and simplify the study of symmetric equations. Via the notions of isotropy groups and fixed point spaces, one can obtain simple and yet powerful bifurcation theorems that are generic, in the sense that they rely only on the 1-jet of the vector field. Beyond that one needs to compute the nonlinear terms and use geometric techniques, e.g. blowing up, or the powerful machinery of singularity theory. We will review these ideas via examples with simple symmetries such as the dihedral groups of a triangle or square. One of our goals will be to explicate the theory of normal forms and of the universal unfolding.

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SPEAKER: Dr. Victor LeBlanc, University of Ottawa

TITLE: Meandering of the Tip in Spiral Waves: A Center Bundle Approach*

ABSTRACT:

Meandering of a one-armed spiral tip has been observed in chemical reactions and numerical simulations. Barkley, Kness and Tuckerman show that meandering can begin by Hopf bifurcation from a rigidly rotating spiral wave (a point that is verified experimentally in a Belousov-Zhabotinsky reaction by Li, Ouyang, Petrov and Swinney). At the codimension-two point where (in an appropriate sense) the frequency of the Hopf bifurcation equals the frequency of the spiral wave, Barkley notes that spiral tip meandering can turn to linearly translating spiral tip motion. Barkley also presents a model showing that the linear motion of the spiral tip is a resonance phenomenon, and this point is verified experimentally by Li et al. and proved rigorously by Wulff.

In this talk, we suggest an alternative development of Barkley's model extending the center bundle constructions of Krupa from compact groups to noncompact groups and from finite dimensions to function spaces. This approach allows us to consider various bifurcations from a rotating wave. In particular, we can analyze in a straightforward manner the codimension-two Barkley bifurcation and the codimension-two Takens-Bogdanov bifurcation from a rotating wave. We also discuss Hopf bifurcation from a many-armed spiral showing that meandering and resonant linear motion of the spiral tip do not always occur.

*(joint work with M. Golubitsky and I. Melbourne, U. Houston)