Exponentially small splitting of separatrices for whiskered tori in Hamiltonian systems

Abstract

We study the existence of transverse homoclinic orbits in a singular or weakly hyperbolic Hamiltonian, with $3$ degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The example considered consists of an integrable Hamiltonian possessing a $2$-dimensional hyperbolic invariant torus with fast frequencies $\omega/\sqrt\varepsilon$ and coincident whiskers or separatrices, plus a perturbation of order $\mu=\varepsilon^p$, giving rise to an exponentially small splitting of separatrices. We show that asymptotic estimates for the transversality of the intersections can be obtained if $\omega$ satisfies certain arithmetic properties. More precisely, we assume that $\omega$ is a quadratic vector (i.e.~the frequency ratio is a quadratic irrational number), and generalize the good arithmetic properties of the golden vector. We provide a sufficient condition on the quadratic vector $\omega$ ensuring that the Poincar\'e--Melnikov method (used for the golden vector in a previous work) can be applied to establish the existence of transverse homoclinic orbits and, in a more restrictive case, their continuation for all values of $\varepsilon\to0$.