Finite switching near heteroclinic networks

TL;DR

This paper investigates the dynamics near robust heteroclinic networks, showing that infinite switching and chaos are absent when all eigenvalues are real, contrasting with cases involving complex eigenvalues.

Contribution

It establishes conditions under which complex switching cannot occur near heteroclinic networks with real eigenvalues, providing bounds on node sequences in such paths.

Findings

Infinite switching does not occur with all real eigenvalues.

Bound on the number of nodes in paths starting at multi-cycle nodes.

Contrasts with heteroclinic networks involving complex eigenvalues.

Abstract

We address the level of complexity that can be observed in the dynamics near a robust heteroclinic network. We show that infinite switching, which is a path towards chaos, does not exist near a heteroclinic network such that the eigenvalues of the Jacobian matrix at each node are all real. Furthermore, for a path starting at a node that belongs to more than one heteroclinic cycle, we find a bound for the number of such nodes that can exist in any such path. This constricted dynamics is in stark contrast with examples in the literature of heteroclinic networks such that the eigenvalues of the Jacobian matrix at one node are complex.