Transversality in Dynamical Systems with Generalized Symmetry

Abstract

We prove that a generic $k$-parameter bifurcation of a dynamical system with a monoid symmetry occurs along a generalized kernel or center subspace of a particular type. More precisely, any (complementable) subrepresentation $U$ is given a number $K_U$ and a number $C_U$. A $k$-parameter bifurcation can generically only occur along a generalized kernel isomorphic to $U$ if $k \geq K_U$. It can generically only occur along a center subspace isomorphic to $U$ if $k \geq C_U$. The numbers $K_U$ and $C_U$ depend only on the decomposition of $U$ into indecomposable subrepresentations. In particular, we prove that a generic one-parameter steady-state bifurcation occurs along one absolutely indecomposable subrepresentation. Likewise, it follows that a generic one-parameter Hopf bifurcation occurs along one indecomposable subrepresentation of complex or quaternionic type, or along two isomorphic absolutely indecomposable subrepresentations. In order to prove these results, we show that the set of endomorphisms with generalized kernel (or center subspace) isomorphic to $U$ is the disjoint union of a finite set of conjugacy invariant submanifolds of codimension $K_U$ and higher (or $C_U$ and higher). The results in this article hold for any monoid, including non-compact groups.