Local bifurcations in k-symmetric dynamical systems

Abstract

A map is called a (reversing) k-symmetry of the dynamical system represented by the map if k is the smallest positive integer for which U is a (reversing) symmetry of the kth iterate of L. We study generic local bifurcations of fixed points that are invariant under the action of a compact Lie group that is a reversing k-symmetry group of the map L, on the basis of a normal form approach. We derive normal forms relating the local bifurcations of k-symmetric maps to local steady-state bifurcations of symmetric flows of vector fields. Alternatively, we also discuss the derivation of normal forms entirely within the framework of Taylor expansions of maps. We illustrate our results with some examples.