Bifurcation methods of periodic orbits for piecewise smooth systems

Abstract

It is known that the Melnikov function method is equivalent to the averaging method for studying the number of limit cycles of planar analytic (or C∞) near-Hamiltonian differential systems. In this paper, we study piecewise smooth near-integrable systems and establish the Melnikov function method and the averaging method for finding limit cycles. We also show the equivalence of the two methods even for systems in high dimensional space. Particularly, we obtain the formula of the second order Melnikov function for planar piecewise near-Hamiltonian systems. We finally provide an application example.