Steady-state bifurcations in gradient systems with symmetry
by
Christopher M. Herald
University of Nevada, Reno

In this talk I will consider the critical set of a function f on a manifold M, i.e. the set of the steady states for the corresponding equivariant gradient flow system. If f is required to be invariant under an action of a compact Lie group G on M, then its critical set will be a union of orbits.

I will discuss some transversality results on bifurcations in the critical set under deformation of the function f through a generic 1-parameter family of invariant functions. Without the group invariance condition on f, the only bifurcations that occur are saddle-node bifurcations. With the invariance condition, there are significantly more complicated bifurcations, too. Some are generalizations of the pitchfork bifurcation and still others have quite a different flavor. The latter types of bifurcations depend in a subtle way on the local structure of the manifold with group action.

Date received: February 28, 2003