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

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.