Switching in heteroclinic networks

TL;DR

This paper investigates the dynamics near heteroclinic networks with real eigenvalues, showing how the global transition maps influence switching behavior, with examples illustrating complex dynamics when certain geometric conditions are broken.

Contribution

It analyzes how the structure of global transition maps affects switching in heteroclinic networks, providing specific examples in five-dimensional space.

Findings

Switching occurs along the common connection in the House network.

Switching occurs along a cycle in the Bowtie network.

Global transition map structure determines the type of switching.

Abstract

We study the dynamics near heteroclinic networks for which all eigenvalues of the linearization at the equilibria are real. A common connection and an assumption on the geometry of its incoming and outgoing directions exclude even the weakest forms of switching (i.e. along this connection). The form of the global transition maps, and thus the type of the heteroclinic cycle, plays a crucial role in this. We look at two examples in $\mathbb{R}^5$, the House and Bowtie networks, to illustrate complex dynamics that may occur when either of these conditions is broken. For the House network, there is switching along the common connection, while for the Bowtie network we find switching along a cycle.