Sliding Trajectories of Generic Inelastic Piecewise-Linear Dynamical Systems on the Torus

Abstract

We consider piecewise smooth differential equations ZX-X+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{X_-X_+}$$\end{document}, where X-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_-$$\end{document} and X+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_+$$\end{document} are linear inelastic vector fields on R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document} with the torus as the discontinuity manifold. Under suitable assumptions, we classify the tangency set on the torus and describe the dynamics of the associated sliding vector field. We prove that, under generic conditions, every trajectory of the sliding vector field on the torus is closed. Finally, we establish a result on the topological equivalence of inelastic vector fields on the torus.