Melnikov method and exponentially small splitting of separatrices

Abstract

We study the splitting of separatrices for high-frequency perturbations of a planar Hamiltonian system. It is well known that in the analytic case the splitting of separatrices is exponentially small with respect to the perturbation period, 2πε, and that the constant in the exponent is related to the position of singularities of the unperturbed separatrix in the complex time plane. On the other hand, in many examples the Melnikov function can be calculated explicitly and is exponentially small with an exponent very close, but larger than the exponent from the upper estimates. So the Melnikov function appears to be smaller than the upper estimate as well as the standard estimate of the approximation error. We found a class of Hamiltonians, such that the Melnikov function gives actual asymptotics for the splitting provided the perturbation amplitude ∥ μ ∥ is less than μ 0 ε p for some p > 0.