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Generic one-parameter families of 3-dimensional Filippov Systems

R. D. Euzébio, M. A. Teixeira, D. J. Tonon

TL;DR

The paper develops a local bifurcation theory for three-dimensional piecewise-smooth vector fields defined around a two-dimensional switching surface, using Filippov's framework. It identifies a codimension-one, open and dense subset $\Xi_1\subset\Omega_1^r$ consisting of six one-parameter families of PSVFs with generic or quasi-generic singularities and derives explicit normal forms and unfoldings for each family. An explicit submersion $\eta: \Omega^r\to\mathbb{R}$ is constructed to characterize these families, providing a concrete mechanism to detect codimension-one degeneracies. Together, these results deliver a systematic, 3D extension of structural stability and bifurcation analysis for PSVFs, with detailed normal forms for Lips, beak-to-beak, swallowtail, singular-regular, regular-regular saddle-node, and regular-regular Hopf cases.

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.

Generic one-parameter families of 3-dimensional Filippov Systems

TL;DR

The paper develops a local bifurcation theory for three-dimensional piecewise-smooth vector fields defined around a two-dimensional switching surface, using Filippov's framework. It identifies a codimension-one, open and dense subset consisting of six one-parameter families of PSVFs with generic or quasi-generic singularities and derives explicit normal forms and unfoldings for each family. An explicit submersion is constructed to characterize these families, providing a concrete mechanism to detect codimension-one degeneracies. Together, these results deliver a systematic, 3D extension of structural stability and bifurcation analysis for PSVFs, with detailed normal forms for Lips, beak-to-beak, swallowtail, singular-regular, regular-regular saddle-node, and regular-regular Hopf cases.

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, , is a surface embedded in . 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 , 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.
Paper Structure (21 sections, 4 theorems, 40 equations, 8 figures)

This paper contains 21 sections, 4 theorems, 40 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Basic theory
  3. Vector fields near the boundary of a 3-manifold
  4. Piecewise smooth vector fields
  5. The involution and the first return map associated to the two-fold singularities
  6. Main results
  7. Subsets of codimension zero and one of 3D PSVFs
  8. Classification of singularities of codimension zero of $X\in\Omega^r$
  9. Some subsets of codimension one of $X\in\Omega^r$
  10. Proof of Theorem \ref{['teorema-lista-cod1']}
  11. Construction of the submersion
  12. Lips and beak to beak cases.
  13. Swallowtail case.
  14. Singular point-regular case
  15. Regular-regular-saddle-node singular point case
  16. ...and 6 more sections

Key Result

Theorem 1

The subset $\Xi_1$ is a $C^{r}$ submanifold of codimension one on $\Omega^r$. In addition,

Figures (8)

  • Figure 1: Unfolding of the lips singularity.
  • Figure 2: Unfolding of the beak to beak singularity.
  • Figure 3: Unfolding of the swallowtail singularity.
  • Figure 4: Unfolding of the saddle singularity.
  • Figure 5: Unfolding of the node singularity.
  • ...and 3 more figures

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Remark 1
  • ...and 12 more