Simple heteroclinic networks in ${\mathbb R}^4$

TL;DR

This paper classifies simple heteroclinic networks in four-dimensional space with symmetry group actions, identifying all possible network types and stability conditions, and illustrates findings with a numerical example.

Contribution

It provides a comprehensive classification of simple heteroclinic networks in ${\mathbb R}^4$ for symmetry groups, including stability criteria for networks admitted by ${\rm SO}(4)$.

Findings

Classified all simple heteroclinic networks in ${\mathbb R}^4$ with finite symmetry groups.

Derived necessary and sufficient conditions for stability of networks in ${\rm SO}(4)$.

Presented a numerical example demonstrating multiple stable subcycles.

Abstract

We classify simple heteroclinic networks for a $\Gamma$-equivariant system in ${\mathbb R}^4$ with finite $\Gamma \subset {\rm O}(4)$, proceeding as follows: we define a graph associated with a given $\Gamma \subset {\rm O}(n)$ and identify all so-called simple graphs associated with subgroups of ${\rm O}(4)$. Then, knowing the graph associated with a given $\Gamma$, we determine the types of heteroclinic networks that the group admits. Our study is restricted to networks that are maximal in the sense that they have the highest possible number of connections -- any non-maximal network can then be derived by deleting one or more connections. Finally, for networks of type A, i.e., admitted by $\Gamma \subset {\rm SO}(4)$, we give necessary and sufficient conditions for fragmentary and essential asymptotic stability. (For other simple heteroclinic networks the conditions for stability are known.) The results are illustrated by a numerical example of a simple heteroclinic network that involves two subcycles that can be essentially asymptotically stable simultaneously.