{"paper":{"title":"On non-smooth vector fields having a torus or a sphere as the sliding manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ricardo Miranda Martins","submitted_at":"2012-07-02T14:19:22Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.0397","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}