A proof that all the sliding trajectories of generic inelastic piecewise linear dynamical systems over the torus are closed

TL;DR

This paper proves that, under generic conditions, all sliding trajectories of certain inelastic piecewise linear systems on the torus are closed, providing a classification of tangency points and topological equivalences.

Contribution

It establishes that all trajectories are closed in generic cases and classifies tangency points for inelastic systems on the torus.

Findings

All sliding trajectories are closed under generic conditions.

Classification of tangency points on the torus.

Topological equivalence results for inelastic vector fields.

Abstract

In this article, we consider piecewise smooth differential equations $Z_{X_-X_+}$, where $X_-$ and $X_+$ are linear vector fields in dimension 3, having the torus as discontinuity manifold. We consider that $Z_{X_-X_+}$ is an inelastic vector field over the torus. We classify the set of tangency points on the torus for certain conditions and describe the behavior of the trajectories of the sliding vector field over the torus. We prove that, under generic conditions, all the trajectories of a piecewise smooth linear inelastic vector field over the torus are closed. We also provide a result about the topological equivalence of inelastic vector fields over the torus.