Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio

TL;DR

This paper investigates the exponentially small splitting of invariant manifolds of hyperbolic tori with silver ratio in nearly-integrable Hamiltonian systems, extending previous results from golden ratio cases.

Contribution

It demonstrates the applicability of the Poincaré-Melnikov method to silver ratio tori and proves the continuation of transversality for small perturbations, generalizing prior golden ratio findings.

Findings

Existence of 4 transverse homoclinic orbits established

Asymptotic estimates for splitting transversality provided

Transversality continuation proven for all small perturbations

Abstract

We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt{2}-1$. We show that the Poincar\'e-Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the tranversality of the splitting whose dependence on the perturbation parameter $\varepsilon$ satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.