Simple heteroclinic cycles in R^4

TL;DR

This paper classifies all finite symmetry groups in four-dimensional space that can support simple, structurally stable heteroclinic cycles in generic equivariant dynamical systems, extending previous results on homoclinic cycles.

Contribution

It provides a complete list of symmetry groups in R^4 that admit simple, robust heteroclinic cycles in generic smooth systems, expanding known classifications.

Findings

Identifies all finite groups allowing simple heteroclinic cycles in R^4.

Extends previous work on homoclinic cycles to heteroclinic cycles.

Provides a comprehensive classification relevant to symmetric dynamical systems.

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 R^n 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 Gamma such that an open set of smooth Gamma-equivariant dynamical systems in R^4 possess a very 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 Gamma-orbit (in this case one speaks of homoclinic cycles).