- 03-112 Amadeu Delshams, Pere Guti\'errez
- Exponentially small splitting for whiskered tori in Hamiltonian systems:
Continuation of transverse homoclinic orbits
Mar 13, 03
(auto. generated ps),
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Abstract. We consider an example of 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 model
consists of an integrable Hamiltonian possessing a 2-dimensional
hyperbolic invariant torus with fast frequencies
$\omega/\sqrt\varepsilon$ and coincident whiskers, plus a perturbation
of order $\mu=\varepsilon^p$. We choose $\omega$ as the golden vector.
Our aim is to obtain asymptotic estimates for the splitting, proving the
existence of transverse intersections between the perturbed whiskers for
$\varepsilon$ small enough, by applying the Poincar\'e--Melnikov method
together with a accurate control of the size of the error term.
The good arithmetic properties of the golden vector allow us to prove
that the splitting function has 4 simple zeros (corresponding to
nondegenerate critical points of the splitting potential), giving rise
to 4 transverse homoclinic orbits. More precisely, we show that a shift
of these orbits occurs when $\varepsilon$ goes across some critical
values, but we establish the continuation (without bifurcations) of the
4 transverse homoclinic orbits for all values of $\varepsilon\to0$.