A proof that all the sliding trajectories of generic inelastic piecewise linear dynamical systems over the torus are closed

2025/08/14 by Caldas, Mayara D. A., Martins, Ricardo M.
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.11066

Abstract

In this article, we consider piecewise smooth differential equations Z_X-X+, where X_- and X₊ are linear vector fields in dimension 3, having the torus as discontinuity manifold. We consider that Z_X-X+ is an inelastic vector field over the torus. We classify the set of tangency points on the torus for certain conditions and describe the behavior of the trajectories of the sliding vector field over the torus. We prove that, under generic conditions, all the trajectories of a piecewise smooth linear inelastic vector field over the torus are closed. We also provide a result about the topological equivalence of inelastic vector fields over the torus.

Citations

• Lyapunov stability of a rigid body with two frictional contacts
• Generic bifurcations of low codimension of planar Filippov Systems

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