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

Abstract

<p style='text-indent:20px;'>In this paper we consider an <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> dimensional piecewise smooth dynamical system. This system has a co-dimension 2 switching manifold <inline-formula><tex-math id="M2">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> which is an intersection of two hyperplanes <inline-formula><tex-math id="M3">\begin{document}$ \Sigma_1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \Sigma_2 $\end{document}</tex-math></inline-formula>. We investigate the relation between periodic orbit of PWS system and periodic orbit of its double regularized system. If this PWS system has an asymptotically stable sliding periodic orbit(including type Ⅰ and type Ⅱ), we establish conditions to ensure that also a double regularization of the given system has a unique, asymptotically stable, periodic orbit in a neighbourhood of <inline-formula><tex-math id="M5">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>, converging to <inline-formula><tex-math id="M6">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> as both of the two regularization parameters go to <inline-formula><tex-math id="M7">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> by applying implicit function theorem and geometric singular perturbation theory.</p>