On non-smooth vector fields having a torus or a sphere as the sliding manifold

TL;DR

This paper investigates non-smooth 3D vector fields with torus or sphere as the sliding manifold, analyzing tangencies and the behavior of trajectories, which are mainly closed, in the context of inelastic vector fields.

Contribution

It characterizes the tangencies and trajectory behavior of non-smooth vector fields with specific manifolds as sliding surfaces, focusing on cases with inelastic vector fields.

Findings

Trajectories over the sliding manifold are mainly closed.

Tangencies of the vector field with the manifold are thoroughly described.

Behavior of the sliding vector field is characterized in both torus and sphere cases.

Abstract

In this paper we consider a non-smooth vector field $Z=(X,Y)$, where $X,Y$ are linear vector fields in dimension 3 and the discontinuity manifold $\Sigma$ of $Z$ is or the usual embedded torus or the unitary sphere at origin. We suppose that $\Sigma$ is a sliding (stable/unstable) manifold with tangencies, by considering $X,Y$ inelastic over $\Sigma$. In each case, we study the tangencies of the vector field $Z$ with $\Sigma$ and describe the behavior of the trajectories of the sliding vector field over $\Sigma$: they are basically closed.