Abundance of infinite switching

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.