Periodic orbits for double regularization of 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 hyperplanes Σ 1 and Σ 2. 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 γ, converging to γ as both of the two regularization parameters go to 0 by applying implicit function theorem and geometric singular perturbation theory.

Exploring Limit Cycle Bifurcations in the Presence of a Generalized Heteroclinic Loop

“…Proof. Substituting ( 19) and ( 24) into (12) gives (26). A direct computation shows that the determinant of the following Jacobian matrix…”

“…Their study delved into the continuity of periodic orbits, particularly when the unperturbed system itself possesses such orbits [24]. Other new results about periodic orbits of piecewise smooth systems can be traced in [25][26][27].…”