Melnikov method for homoclinic bifurcation in nonlinear impact oscillators

Abstract

Based on an inverted pendulum impacting on rigid walls under external periodic excitation, a class of nonlinear impact oscillators is discussed for its homoclinic bifurcation. The Melnikov method established for smooth dynamical systems is extended to be applicable to the nonsmooth one. For nonlinear impact systems, closed form solutions between impacts are generally unavailable. The absence of closed form solutions makes difficulties in estimation of the gap between the stable manifold and unstable manifold. In this paper, we give a method to compute the Melnikov functions up to the n th-order so as to obtain conditions of parameters for the persistence of homoclinic cycles which are formed via the identification given by the impact rule.