Generic one-parameter families of 3-dimensional Filippov Systems

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.

💡 Research Summary

The paper investigates one‑parameter families of three‑dimensional piecewise‑smooth vector fields (PSVFs) defined with respect to a switching surface M, which is the zero level set of a smooth scalar function h:ℝ³→ℝ. The authors adopt Filippov’s convention for handling the non‑uniqueness of trajectories on M and focus on local dynamics near typical codimension‑one singularities.

First, the authors recall the classical classification of smooth vector fields relative to a boundary: regular points, fold points, and cusp points, defined via the successive Lie derivatives of h along a vector field Z (i.e., Z·∇h, Z²·∇h, …). The set Σ₀ consisting of vector fields that are either regular, fold, or cusp at the origin is known to be open, dense, and structurally stable in the C^r‑topology.

The paper then moves to the bifurcation set Σ₁, which contains codimension‑one degeneracies that violate exactly one condition of Σ₀ while preserving all others (the “quasi‑generic” situation). Σ₁ is split into two main families: (a) hyperbolic equilibria whose eigenvectors are transversal to M and whose eigenvalues have simple algebraic multiplicity, further distinguished into node, saddle, and focus types; (b) points where the first two Lie derivatives of h vanish but higher‑order derivatives do not, leading to sub‑cases such as Lips, beak‑to‑beak, and slow‑tail configurations, each characterized by the sign of the Hessian of h restricted to M.

A central object of study is the two‑fold singularity, where both X⁺ and X⁻ are tangent to M at the same point. Depending on the signs of the second Lie derivatives (X⁺)²h(0) and (X⁻)²h(0), the two‑fold can be parabolic, hyperbolic, or elliptic. For elliptic two‑folds the authors construct involutions γ_{X⁺} and γ_{X⁻} associated with the local return maps of X⁺ and X⁻ on a transversal section, and define the first return map φ_X = γ_{X⁻}∘γ_{X⁺}. They show that det Dφ_X(0)=1, so the eigenvalues are a reciprocal pair β and β⁻¹. When β∈ℝ{0} the return map is of saddle type; when β = e^{iθ} (0<θ<π) it is elliptic, producing a local rotation.

The main theorem identifies a codimension‑one submanifold ℳ⊂Ω_r¹ (where Ω_r¹ = Ω_r \ (Ξ₀ ∪ Ω_T) is the space of PSVFs excluding the zero‑codimension and tangential‑both‑sides sets) that is structurally stable within Ω_r¹. ℳ is shown to be open, dense, and to consist precisely of the one‑parameter families described above. Moreover, the authors construct an explicit smooth submersion η:Ω_r→ℝ such that ℳ = η⁻¹(0). This map provides a global coordinate for the bifurcation families and guarantees that each family is a regular level set of η.

The paper is organized as follows: Section 2 collects necessary preliminaries on germs of smooth vector fields, the contact theory with a surface, and the definition of PSVFs and Filippov’s sliding vector field. Section 3 states the main results, including the openness, density, and structural stability of ℳ. Section 4 discusses the geometric nature of the codimension‑zero and codimension‑one subsets, illustrating the various normal forms. Sections 5 and 6 contain the detailed proofs of openness, density, and the construction of the submersion η. Appendix A provides auxiliary results on bifurcation theory used throughout.

In summary, the authors extend classical bifurcation theory for smooth vector fields to the piecewise‑smooth setting in three dimensions. They give a complete local classification of generic and quasi‑generic codimension‑one singularities, provide normal forms and unfoldings for each, and prove that the corresponding families form an open, dense, structurally stable codimension‑one submanifold of the space of PSVFs. This work fills a gap in the literature, offering a systematic framework for analyzing bifurcations of 3‑D PSVFs that can be applied to control systems, electrical circuits, biological models, and other applications where discontinuous dynamics arise.