Exponentially small splittings of manifolds in a rapidly forced Duffing system

The splitting of the separatrices of the weakly perturbed and rapidly forced Duffing oscillator was examined. Analytic expressions for motions on the perturbed manifolds were first derived, and then asymptotically expanded in series by successive integration by parts. It was shown that the splitting of the manifolds of the rapidly forced problems becomes exponentially small as the perturbation parameter approaches zero. The expression for the splitting distance given by Melnikov theory, is a valid estimate for proving the existence of transverse homoclinic intersections.