Generic one-parameter families of 3-dimensional Filippov Systems

TL;DR

This paper investigates the stability, robustness, and normal forms of one-parameter families of 3D piecewise smooth vector fields around typical singularities, using bifurcation theory and contact topology.

Contribution

It provides new results on the openness, density, and structural stability of PSVFs with generic singularities, extending classical theorems to the Filippov systems context.

Findings

Establishes conditions for structural stability of PSVFs.

Proves the openness and density of systems with generic singularities.

Provides normal forms for codimension one singularities.

Abstract

This paper addresses openness, density and structural stability conditions of one-parameter families of 3D piecewise smooth vector fields (PSVFs) defined around typical singularities. Our treatment is local and the switching set, $M$, is a $2D$ surface embedded in $\mathbb{R}^3$. In short, we analyze the robustness and normal forms of certain codimension one singularities that occur in PSVFs. The main machinery used in this paper involves the theory of contact between a vector field and $M$, Bifurcation Theory and the Topology of Manifolds. Our main result states robust mathematical statements resembling the classical Kupka-Smale Theorem in the sense that we establish the openness and density of a large class of PSVFs presenting generic and quasi-generic singularities. Due to the lack of uniqueness of certain solutions associated with PSVFs, we employ Filippov's theory as the basis of our approach throughout the paper.