Splitting of separatrices for rapid degenerate perturbations of the classical pendulum (Q6564688)

The problem of the exponentially small splitting of separatrices for Hamiltonian systems with a non-autonomous perturbation is a classical one in the theory of dynamical systems. This is mainly due to the special role played by transversal intersections between stable and unstable invariant manifolds in the appearance of chaos and the celebrated Arnold diffusion. As is well known, for non-fast perturbations, classical perturbation theory provides the so-called Melnikov function, providing an analytical expression for the distance between these invariant manifolds up to first order in the perturbation parameter. It turns out, however, that the Melnikov theory does not allow one to conclude how is this distance when the perturbation is fast in time, unless the perturbation parameter is exponentially small in the time-scale, and thus other techniques are required to analyze this situation.\N\NThe paper under review address this problem in a special degenerate case, when the Melnikov function seems to give the asymptotic expression of the splitting distance but the non-degeneracy conditions are not satisfied. To do so, new tools are required to find a formula based on special solutions of the inner equation (a particular partial differential equation associated to the model analyzed) and the use of the Hamilton-Jacobi formalism. The main theorems provide explicit expressions for the first two terms in the asymptotic expansion of the splitting distance and an algorithm for computing any other term. Although the system analyzed corresponds to the rapidly perturbed classical pendulum, the methodology used in the paper is independent of the particular form of the Hamiltonian, so that it can be applied in principle to any Hamiltonian system with one a half degrees of freedom with a homoclinic orbit and a non-generic fast perturbation.