Classification and stability of simple homoclinic cycles in

Abstract

Heteroclinic cycles, unions of equilibria and connection trajectories, can be structurally stable in a Γ-equivariant system due to the existence of invariant subspaces. A structurally stable heteroclinic cycle is called simple if all connecting trajectories are one-dimensional. Heteroclinic cycles where equilibria are related by a symmetry γ ∈ Γ are called homoclinic. This paper presents a complete study of simple homoclinic cycles in . We find all symmetry groups Γ such that a Γ-equivariant dynamical system in can possess a simple homoclinic cycle. We introduce a classification of simple homoclinic cycles in based on the action of the system symmetry group. For systems in , we list all classes of simple homoclinic cycles. For each class, we derive necessary and sufficient conditions for asymptotic stability and fragmentary asymptotic stability in terms of eigenvalues of linearization near the steady state involved in the cycle. For any action of the groups Γ which can give rise to a simple homoclinic cycle, we list classes to which the respective homoclinic cycles belong, thus determining the conditions for the asymptotic stability of these cycles.