A mechanism for switching near a heteroclinic network

TL;DR

This paper presents a stable heteroclinic network mechanism where nearby orbits exhibit irregular switching due to spiralling caused by complex eigenvalues, with implications for understanding complex dynamical systems.

Contribution

It introduces a novel mechanism for switching in heteroclinic networks using spiralling eigenvalues and develops a technique to analyze trajectories on higher-dimensional unstable manifolds.

Findings

Switching is prevalent near the heteroclinic network.

Spiralling eigenvalues induce irregular switching.

Numerical example demonstrates complex dynamics.

Abstract

We describe an example of a structurally stable heteroclinic network for which nearby orbits exhibit irregular but sustained switching between the various sub-cycles in the network. The mechanism for switching is the presence of spiralling due to complex eigenvalues in the flow linearised about one of the equilibria common to all cycles in the network. We construct and use return maps to investigate the asymptotic stability of the network, and show that switching is ubiquitous near the network. Some of the unstable manifolds involved in the network are two-dimensional; we develop a technique to account for all trajectories on those manifolds. A simple numerical example illustrates the rich dynamics that can result from the interplay between the various cycles in the network.