Stationary phase methods and the splitting of separatrices

Using stationary phase methods, we provide an explicit formula for the Melnikov function of the one and a half degrees of freedom system given by a Hamiltonian system subject to a rapidly oscillating perturbation. Remarkably, the Melnikov function turns out to be computable using very little information on the separatrix and in the case of non-analytic systems. This is related to a priori stable systems coupled with low regularity perturbations. A natural physical application is to perturbations controlled by wave-type equations, so in particular we also illustrate this result with the motion of charged particles in a rapidly oscillating electromagnetic field. Quasi-periodic perturbations are discussed too.A.E. and D.P.-S. are respectively supported by the ERCStartingGrants 633152 and 335079. A.L. is supported by the Knut och Alice Wallenbergs stiftelse KAW 2015.0365. This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554