The Orbit Space Approach for Piecewise Smooth Vector Fields

TL;DR

The paper tackles nonuniqueness in Filippov PSVFs by introducing an orbit-space framework $\tilde{M}$ that carries a continuous flow $\tilde{\Phi}$ and a meaningful metric $\tilde{d}$, inspired by inverse-limit techniques. It proves that, under finite tangency and boundedness conditions, the orbit space is separable, Baire, and perfect, enabling a clean transitivity analysis via Birkhoff's theorem and linking dynamics in the original system to its orbit-space counterpart. The authors demonstrate the framework on canonical transitive two-dimensional PSVFs (the bean model BCE and the sphere model EJV), showing transitivity in the orbit space and establishing conditions under which orbit-space transitivity mirrors Filippov transitivity. They provide a explicit five-step construction to build a countable dense subset of orbits, which yields a practical approach to analyze general PSVFs and to explore the extent to which orbit-space transitivity reflects the global dynamics of Filippov systems.

Abstract

In this work we develop a well-defined theory of orbit spaces for piecewise smooth vector fields (PSVFs). This approach is inspired by the techniques already used in the study of endomorphisms, namely inverse limit analysis, and has been used before for PSVFs. We then apply the construction of our theory to understanding transitivity in PSVFs. Our results prove that the known examples of transitive PSVFs in the literature, the bean model and the sphere model, are indeed transitive in the orbit space.

Figures (10)

• Figure 1: The bean and sphere models, PSVFs that are transitive. The highlighted points are the tangency points. They play a fundamental role in the topological transitivity of these systems by connecting stable and unstable sliding regions.
• Figure 2: Graphical representation of \ref{['lemma:close.orbit.points.imply.close.orbits', 'lemma:close.orbits.imply.close.orbit.points']}.
• Figure 3: Graphical representation of \ref{['lemma:approximation.orbits']}.
• Figure 4: A dense set of points $E^\mathrm{t}$ is taken on the orbit segments arriving and leaving each tangency point $p$.
• Figure 5: The sets of points $E^\mathrm{ss}$ and $E^\mathrm{us}$ (dense on $\Sigma^{\mathrm{ss}}$ and $\Sigma^{\mathrm{us}}$, respectively) are taken by flowing the points $p \in E_1$ along their orbits until they reach the sliding region at the points $p^\mathrm{ss}$ and $p^\mathrm{us}$.

Theorems & Definitions (45)

• Theorem A
• Theorem B
• Theorem C
• Remark 1
• Theorem 2.1: Birkhoff transitivity theorem
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3