 03134 Amadeu Delshams, Pere Guti\'errez, Tere M.~Seara
 Exponentially small splitting for whiskered tori in Hamiltonian systems:
Flowbox coordinates and upper bounds
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Mar 24, 03

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Abstract. We consider a singular or weakly hyperbolic Hamiltonian, with $n+1$
degrees of freedom, as a model for the behaviour of a nearlyintegrable
Hamiltonian near a simple resonance. The model consists of an integrable
Hamiltonian possessing an $n$dimensional hyperbolic invariant torus
with fast frequencies $\omega/\sqrt\varepsilon$ and coincident whiskers,
plus a perturbation of order $\mu=\varepsilon^p$. The vector $\omega$ is
assumed to satisfy a Diophantine condition.
We provide a tool to study, in this singular case, the splitting of the
perturbed whiskers for $\varepsilon$ small enough, as well as their
homoclinic intersections, using the Poincar\'eMelnikov method. Due to
the exponential smallness of the Melnikov function, the size of the
error term has to be carefully controlled. So we introduce flowbox
coordinates in order to take advantage of the quasiperiodicity
properties of the splitting. As a direct application of this approach,
we obtain quite general upper bounds for the splitting.
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