A methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies

The aim of this work is to provide asymptotic estimates for the splitting of separatrices in a perturbed 3-degree-of-freedom Hamiltonian system, associated to a 2-dimensional whiskered torus (invariant hyperbolic torus) whose frequency ratio is a quadratic irrational number. We show that the dependence of the asymptotic estimates on the perturbation parameter is described by some functions which satisfy a periodicity property, and whose behavior depends strongly on the arithmetic properties of the frequencies. First, we describe the Hamiltonian system to be studied. It is also considered in [6], as a generalization of the famous Arnold’s example [1], and provides a model for the behavior of a nearly-integrable Hamiltonian system in the vicinity of a single resonance (see [3] for a motivation). In canonical coordinates (x, y, φ, I) ∈ T × R × T2 × R2, we consider a perturbed Hamiltonian