On Poincar\'e-Bendixson Theorem and Non-Trivial Minimal Sets in Planar Nonsmooth Vector Fields

TL;DR

This paper extends the Poincaré-Bendixson theorem to certain nonsmooth planar vector fields and identifies a novel minimal set with a non-empty interior, highlighting differences from classical smooth systems.

Contribution

It presents a Poincaré-Bendixson theorem for nonsmooth systems without sliding regions and introduces a new minimal set with non-empty interior in Filippov systems.

Findings

A Poincaré-Bendixson theorem for nonsmooth vector fields is established.

A minimal set with non-empty interior in planar Filippov systems is demonstrated.

Differences between classical and nonsmooth limit sets are clarified.

Abstract

In this paper some qualitative and geometric aspects of nonsmooth vector fields theory are discussed. In the class of nonsmooth systems, that do not present sliding regions, a Poincar\'e-Bendixson Theorem is presented. A minimal set in planar Filippov systems not predicted in classical Poincar\'e-Bendixson theory and whose interior is non-empty is exhibited. The concepts of limit sets, recurrence and minimal sets for nonsmooth systems are defined and compared with the classical ones. Moreover some differences between them are pointed out.