Exponentially small splitting of separatrices for whiskered tori in Hamiltonian systems
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hdl:2117/845
Tipus de documentArticle
Data publicació2003
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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
$\varepsilon\to0$.
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