03-112 Amadeu Delshams, Pere Guti\'errez

Exponentially small splitting for whiskered tori in Hamiltonian systems: Continuation of transverse homoclinic orbits (460K, Postscript) Mar 13, 03

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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 $\varepsilon\to0$.

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