Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two

Abstract

In this paper we consider an n dimensional piecewise smooth dynamical system. This system has a co-dimension 2 switching manifold Σ which is an intersection of two co-dimension one switching manifolds Σ 1 and Σ 2. We investigate the relation of periodic orbit of PWS between periodic orbit of its regularized system. If this PWS system has an asymptotically stable crossing periodic orbit γ or sliding periodic orbit, we establish conditions to ensure that also a regularization of the given system has a unique, asymptotically stable, limit cycle in a neighbourhood of γ, converging to γ as the regularization parameter goes to 0.

Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two

“…Under suitable assumptions, we have proved if the original PWS system (3) has an asymptotically stable periodic orbit, then so will the regularized system (6). This work can be seen in [30].…”

“…Regularization of two special intersecting switching manifold was considered by Llibre, da Silva and Teixeira in [28]. In our recent paper [30], we consider the relation between periodic orbit of PWS systems with a switching manifold of codimension two and limit cycles of its regularization. We consider the following system:…”

“…This work will be done in this paper. The present work can be seen as a follow-up of [30]. The study of the double regularized system (6) is important because it replaces the PWS system with a codimension two switching manifold by a continuous system.…”