Splitting of Separatrices for Rapid Degenerate Perturbations of the Classical Pendulum

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1159-1198, June 2024.
Abstract.In this work we study the splitting distance of a rapidly perturbed pendulum [math] with [math] a [math]-periodic function and [math]. Systems of this kind undergo exponentially small splitting, and, when [math], it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided [math]. Our study focuses on the case [math], and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as [math]. On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency [math] in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with [math] where, for most [math], the perturbation satisfies [math]. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton–Jacobi formalism. The leading exponentially small term appears at order [math], where [math] is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.