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

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.