Simple heteroclinic cycles in

Abstract

In generic dynamical systems heteroclinic cycles are invariant sets of codimension at least one, but they can be structurally stable in systems which are equivariant under the action of a symmetry group, due to the existence of flow-invariant subspaces. For dynamical systems in the minimal dimension for which such robust heteroclinic cycles can exist is n = 3. In this case the list of admissible symmetry groups is short and well known. The situation is different and more interesting when n = 4. In this paper, we list all finite groups Γ such that an open set of smooth Γ-equivariant dynamical systems in possesses a simple heteroclinic cycle (a structurally stable heteroclinic cycle satisfying certain additional constraints). This work extends the results which were obtained by Sottocornola in the case when all equilibria in the heteroclinic cycle belong to the same Γ-orbit (in this case one speaks of homoclinic cycles).