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

TL;DR

This paper develops a method to derive asymptotic estimates for the exponentially small splitting of separatrices near whiskered tori with quadratic irrational frequencies in perturbed Hamiltonian systems, highlighting the influence of frequency arithmetic.

Contribution

It introduces a novel methodology for asymptotic analysis of separatrix splitting in systems with quadratic frequency ratios, emphasizing the role of arithmetic properties.

Findings

Asymptotic estimates depend periodically on the perturbation parameter.

The behavior of splitting estimates is strongly influenced by the arithmetic nature of the frequencies.

The methodology applies to 3-degree-of-freedom Hamiltonian systems with hyperbolic invariant tori.

Abstract

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.