Abundance of infinite switching

TL;DR

This paper demonstrates that certain vector fields near a network exhibit abundant infinite switching, with a positive measure of initial conditions following any prescribed infinite path, supported by the existence of large strange attractors.

Contribution

It introduces a class of vector fields with abundant switching near networks and characterizes their global dynamics for almost all parameters.

Findings

Infinite switching occurs in the studied vector fields.

Large strange attractors are present near heteroclinic tangles.

The global dynamics are characterized for a specific family of differential equations.

Abstract

In this article, we describe a class of vector fields exhibiting abundant switching} near a network: for every neighbourhood of the network and every infinite admissible path, the set of initial conditions within the neighbourhood that follows the path has positive Lebesgue measure. The proof relies on the existence of "large'' strange attractors in the terminology of Broer, Sim\'o and Tatjer (Nonlinearity, 667--770, 1998) near a heteroclinic tangle unfolding an attracting network with a two-dimensional heteroclinic connection. For our class of vector fields, any small non-empty open ball of initial conditions realizes infinite switching. We illustrate the theory with a specific one-parameter family of differential equations, for which we are able to characterise its global dynamics for almost all parameters.