How many points converge to a heteroclinic network in an aperiodic way?

TL;DR

This paper investigates the conditions under which points in a dynamical system converge to heteroclinic networks in an aperiodic manner, introducing a new stability concept for sequences along such networks.

Contribution

It introduces a novel notion of asymptotic stability for general sequences on heteroclinic networks and analyzes the limitations on aperiodic convergence.

Findings

Uncountably many aperiodic sequences cannot attract a set of nontrivial measure.

Provides examples illustrating when aperiodic convergence is expected or not.

Discusses open questions related to stability and convergence in heteroclinic networks.

Abstract

Homoclinic and heteroclinic connections can form cycles and networks in phase space, which organize global phenomena in dynamical systems. On the one hand, stability notions for (omni)cycles give insight into how many initial conditions approach the network along a single given (omni)cycle. On the other hand, the term switching is used to describe situations where there are trajectories that follow any possible sequence of heteroclinic connections along the network. Here we give a notion of asymptotic stability for general sequences along a network of homoclinic and heteroclinic connections. We show that there cannot be uncountably many aperiodic sequences that attract a set with nontrivial measure. Finally, we discuss examples where one may or may not expect aperiodic convergence towards a network and conclude with some open questions.