Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio

TL;DR

This paper investigates the exponentially small splitting of invariant manifolds of whiskered tori with quadratic frequency ratios in nearly-integrable Hamiltonian systems, providing explicit asymptotic estimates linked to the arithmetic properties of the ratios.

Contribution

It offers explicit asymptotic estimates for the splitting and transversality of invariant manifolds in systems with quadratic irrational frequency ratios, highlighting their dependence on continued fraction properties.

Findings

Splitting distance is exponentially small in the perturbation parameter.

Transversality holds for most small parameter values, with exceptions near bifurcation points.

Estimates are explicitly constructed from the continued fraction of the quadratic ratio.

Abstract

The splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied. We consider a torus with a fast frequency vector $\omega/\sqrt\varepsilon$, with $\omega=(1,\Omega)$ where the frequency ratio $\Omega$ is a quadratic irrational number. Applying the Poincar\'e-Melnikov method, we carry out a careful study of the dominant harmonics of the Melnikov potential. This allows us to provide an asymptotic estimate for the maximal splitting distance, and show the existence of transverse homoclinic orbits to the whiskered tori with an asymptotic estimate for the transversality of the splitting. Both estimates are exponentially small in $\varepsilon$, with the functions in the exponents being periodic with respect to $\ln\varepsilon$, and can be explicitly constructed from the continued fraction of $\Omega$. In this way, we emphasize the strong dependence of our results on the arithmetic properties of $\Omega$. In particular, for quadratic ratios $\Omega$ with a 1-periodic or 2-periodic continued fraction (called metallic and metallic-colored ratios respectively), we provide accurate upper and lower bounds for the splitting. The estimate for the maximal splitting distance is valid for all sufficiently small values of $\varepsilon$, and the transversality can be established for a majority of values of $\varepsilon$, excluding small intervals around some transition values where changes in the dominance of the harmonics take place, and bifurcations could occur.