Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity

Abstract

We consider an $n$ dimensional dynamical system with discontinuous right-hand side (DRHS), whereby the vector field changes discontinuously across a co-dimension 1 hyperplane \begin{document}$S$\end{document} . We assume that this DRHS system has an asymptotically stable periodic orbit \begin{document}$γ$\end{document} , not fully lying in \begin{document}$S$\end{document} . In this paper, we prove that also a regularization of the given system has a unique, asymptotically stable, periodic orbit, converging to \begin{document}$γ$\end{document} as the regularization parameter goes to $0$.