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

In this paper we consider an \begin{document}$ n $\end{document} dimensional piecewise smooth dynamical system. This system has a co-dimension 2 switching manifold \begin{document}$ \Sigma $\end{document} which is an intersection of two hyperplanes \begin{document}$ \Sigma_1 $\end{document} and \begin{document}$ \Sigma_2 $\end{document} . We investigate the relation between periodic orbit of PWS system and periodic orbit of its double regularized system. If this PWS system has an asymptotically stable sliding periodic orbit(including type Ⅰ and type Ⅱ), we establish conditions to ensure that also a double regularization of the given system has a unique, asymptotically stable, periodic orbit in a neighbourhood of \begin{document}$ \gamma $\end{document} , converging to \begin{document}$ \gamma $\end{document} as both of the two regularization parameters go to \begin{document}$ 0 $\end{document} by applying implicit function theorem and geometric singular perturbation theory.