Is there switching without suspended horseshoes?

TL;DR

This paper presents an example of an asymptotically stable homoclinic network that exhibits switching behavior without the presence of suspended horseshoes, challenging previous assumptions about the necessity of such structures for switching.

Contribution

It introduces a new example of a stable network with switching behavior that does not involve suspended horseshoes, expanding understanding of complex dynamics near networks.

Findings

Demonstrates existence of stable networks with switching without suspended horseshoes

Provides an example of sensitive dependence on initial conditions in stable networks

Challenges the assumption that suspended horseshoes are necessary for switching

Abstract

In general, infinite switching behaviour near networks is associated with the existence of suspended horseshoes. Trajectories that realize switching lie within these transitive sets. In this note, revisiting the equivariant Shilnikov scenario, we describe an attracting homoclinic network exhibiting forward switching and without suspended horseshoes in its neighbourhood. Thus we provide an example of an asymptotically stable network exhibiting sensitive dependence on initial conditions.