Characterization of Non-Deterministic Chaos in Two-Dimensional Non-Smooth Vector Fields

TL;DR

This paper characterizes non-deterministic chaos in two-dimensional non-smooth vector fields, establishing conditions for chaos and entropy in Filippov systems with sliding or escaping regions.

Contribution

It proves the equivalence of topological transitivity and dense orbits in Filippov systems, extending classical results to non-smooth dynamics.

Findings

Topological transitivity is necessary and sufficient for non-deterministic chaos.

Topological transitivity implies positive topological entropy.

Dense orbits exist under transitivity, but residual set of dense orbits is not guaranteed.

Abstract

Our context is Filippov systems defined on two-dimensional manifolds having a finite number of tangency points. We prove that topological transitivity is a necessary and sufficient condition for the occurrence of non-deterministic chaos when the Filippov system has non-empty sliding or escaping regions. A fundamental result for continuous flows is the equivalence of topological transitivity and existence of a dense orbit. We prove in our setting that topological transitivity for Filippov systems is indeed equivalent to the existence of a dense Filippov orbit, although, in contrast to the continuous case, we are not able to garantee that the dense orbit implies the existence of a residual set of dense orbits. Finally we prove that, in this context, topological transitivity implies strictly positive topological entropy for the Filippov system. This calculation is made using techniques similar to those from symbolic dynamics.