Infinite Switching in Vector Fields

Abundance of infinite switching

We describe a class of vector fields exhibiting abundant switching near a network : there exists a set of initial conditions with positive lebesgue measure realising infinite switching.For our class of vector fields, any small non-empty open ball of initial conditionsrealizes 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.