A fast-manifold approach to 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 usual two-dimensional Melnikov analysis to the ''fast'' dynamics of the system which lie on an invariant manifold. It is shown that the resultant Melnikov function is the same as that obtained in the usual way involving distance functions in three dimensions [Wiggins and Holmes, 1987]. This alternative derivation provides some useful insight into the structure of the dynamical system.