  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.
