On Structural Stability of 3D Filippov Systems: A Semi-Local Approach

TL;DR

This paper introduces a semi-local approach to analyze the structural stability of 3D Filippov systems, providing a complete characterization and revealing complex dynamical behaviors in piecewise smooth vector fields.

Contribution

It presents a novel non-local, semi-local framework for studying the robustness of 3D Filippov systems around their switching manifolds, with a focus on geometric and qualitative analysis.

Findings

Complete characterization of semi-local structural stability.

Identification of rich dynamical behaviors in piecewise smooth systems.

New methods for qualitative geometric analysis of Filippov systems.

Abstract

The main purpose of this work is to provide a non-local approach to study aspects of structural stability of 3D Filippov systems. We introduce a notion of semi-local structural stability which detects when a piecewise smooth vector field is robust around the whole switching manifold and we give a complete characterization of such systems. In particular, we present some methods in the qualitative theory of piecewise smooth vector fields, emphasizing a geometrical analysis of the foliations generated by their orbits. Such approach displays surprisingly a rich dynamical behaviour that has been studied in details in this work. It is worth to say that this subject has not been treated recently from a non-local point of view, and we hope that the approach adopted in this work contributes to the understanding of the structural stability for piecewise-smooth vector fields in its most global sense.