**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.

#### 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)