Complete heteroclinic networks derived from graphs consisting of two cycles

TL;DR

This paper presents a constructive method to realize any two-cycle directed graph as a complete heteroclinic network with all unstable manifolds included, analyzing minimal modifications and stability implications.

Contribution

It introduces a systematic approach to embed two-cycle graphs into heteroclinic networks, including edge additions and phase space construction, with stability analysis.

Findings

Minimal number of edges depends on cycle length

Each added edge introduces a positive transverse eigenvalue

Implications for cycle stability are discussed

Abstract

We address the question how a given connection structure (directed graph) can be realised as a heteroclinic network that is complete in the sense that it contains all unstable manifolds of its equilibria. For a directed graph consisting of two cycles we provide a constructive method to achieve this: (i) enlarge the graph by adding some edges and (ii) apply the simplex method to obtain a network in phase space. Depending on the length of the cycles we derive the minimal number of required new edges. In the resulting network each added edge leads to a positive transverse eigenvalue at the respective equilibrium. We discuss the total number of such positive eigenvalues in an individual cycle and some implications for the stability of this cycle.