Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results

Abstract

In this paper we study the problem of exponentially small splitting of separatrices of one degree of freedom classical Hamiltonian systems with a non-autonomous perturbation which is fast and periodic in time. We provide a result valid for general systems which are polynomials or trigonometric polynomials in the state variables. Our result consists in obtaining a rigorous proof of the asymptotic formula for the measure of the splitting. We have obtained that the splitting has the asymptotic behavior K" e−a/" identifying the constants K, and a in terms of the features of the system. The study of our problem leads us to consider several cases. In some cases, assuming the per- turbation is small enough, it turns out that the values of K, coincides with the classical Melnikov approach. We have identified the limit size of the perturbation for which this classical theory holds true. However for the limit cases, which appear naturally both in averaging theory and bifurcation theory, we encounter that, generically, neither K nor are actually well predicted by Melnikov theory.