Construction of heteroclinic networks in $\mathbb{R}^4$

TL;DR

This paper classifies all possible simple robust heteroclinic networks in four-dimensional space, focusing on their construction, types, and stability properties, and introduces new network types not previously documented.

Contribution

It provides a complete classification of heteroclinic networks in , including new types of networks involving type A cycles and analyzes their stability.

Findings

Few ways to join simple heteroclinic cycles into networks in 

Complete list of all such networks in 

Stability distinctions between type A and type B networks

Abstract

We study heteroclinic networks in $\mathbb{R}^4$, made of a certain type of simple robust heteroclinic cycle. In simple cycles all the connections are of saddle-sink type in two-dimensional fixed-point spaces. We show that there exist only very few ways to join such cycles together in a network and provide the list of all possible such networks in $\mathbb{R}^4$. The networks involving simple heteroclinic cycles of type A are new in the literature and we describe the stability of the cycles in these networks: while the geometry of type A and type B networks is very similar, stability distinguishes them clearly.