Amadeu Delshams, Pere Guti\'errez, Tere M.~Seara
Exponentially small splitting for whiskered tori in Hamiltonian systems:
Flow-box coordinates and upper bounds
(533K, Postscript)
ABSTRACT. We consider a singular or weakly hyperbolic Hamiltonian, with $n+1$
degrees of freedom, as a model for the behaviour of a nearly-integrable
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\'e--Melnikov method. Due to
the exponential smallness of the Melnikov function, the size of the
error term has to be carefully controlled. So we introduce flow-box
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.