Isochronous Attainable Manifolds for Piecewise Smooth Dynamical Systems

Abstract

Considering the concept of attainable sets for differential inclusions, we introduce the isochronous manifolds relative to a piecewise smooth dynamical systems in \(\mathbb {R}^{2}\) and \(\mathbb {R}^{3}\), and study how analytical and topological properties of such manifolds are related to sliding motion and to partially nodal attractivity conditions on the discontinuity manifolds. We also investigate what happens to isochronous manifolds at tangential exit points, where attractivity conditions cease to hold. In particular, we find that isochronous curves in \(\mathbb {R}^{2}\), which are closed simple curves under attractivity regime, become open curves at such points.