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

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 $Σ$ of $Z$ is or the usual embedded torus or the unitary sphere at origin. We suppose that $Σ$ is a sliding (stable/unstable) manifold with tangencies, by considering $X,Y$ inelastic over $Σ$. In each case, we study the tangencies of the vector field $Z$ with $Σ$ and describe the behavior of the trajectories of the sliding vector field over $Σ$: they are basically closed.

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