Perturbation methods and the Melnikov functions for slowly varying oscillators

A new approach to obtaining the Melnikov function for homoclinic orbits in slowly varying oscillators is proposed. The present method applies the Lindstedt–Poincaré method to determine an approximation of homoclinic solutions. It is shown that the resultant Melnikov condition is the same as that obtained in the usual way involving distance functions in three dimensions by Wiggins and Holmes [Homoclinic orbits in slowly varying oscillators. SIAM J Math Anal 1987;18(3):612].