Michael Field
Lectures on Bifurcations, Dynamics and Symmetry
Pitman Research Notes in Mathematics, September, 1996.
In the first part of these notes, we look at a large and interesting
class of examples where the Maximal Isotropy Subgroup Conjecture (MISC)
fails. For this class, we show that it is still possible to obtain
relatively complete information on the symmetry and stabilities of branches.
We also investigate the dynamics that can be spawned in
(generic) equivariant bifurcations. Even in static equivariant
bifurcation problems (the linearization of f at the bifurcation point
has zero eigenvalues), we show that it is possible to bifurcate
to periodic solutions, heteroclinic cycles, or even chaotic dynamics.
Our examples of the branches of heteroclinic cycles
that can occur in static equivariant bifurcations
provide us with phenomenological
models for the study of coupled cell systems. We give a number of
general results showing how we can stably cycle between groups of
active cells (`cycling chaos'). Heteroclinic cycles are but one instance
of equivariant transversality and in the concluding chapters
we provide an introduction to the theory of equivariant
transversality as well as an extended discussion of a
coupled system of four nonlinear oscillators
that exhibits stable, but singular, intersections of invariant
manifolds.