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

Abstract

<p style='text-indent:20px;'>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 Σ₁ and Σ₂. 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 <i>γ</i> 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 <i>γ</i>, converging to <i>γ</i> as the regularization parameter goes to 0.