Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits

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Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds

October 2004, 11(4): 757-783. doi: 10.3934/dcds.2004.11.757

Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits

Amadeu Delshams ^1, and Pere Gutiérrez ^2,

1.	Departament de Matemàtica Aplicada I, ETSEIB-Universitat Politècnica de Catalunya, Diagonal 647, E-08028 Barcelona, Spain
2.	Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028, Spain

Received December 2002 Revised October 2003 Published September 2004

Abstract
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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 Poincaré-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 $\varepsilon\to0$.

Keywords: arithmetic properties of frequencies, transverse homoclinic orbits., Poincaré-Melnikov method.

Mathematics Subject Classification: 70H08, 34C37, 34C30, 70K4.

Citation: Amadeu Delshams, Pere Gutiérrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 757-783. doi: 10.3934/dcds.2004.11.757

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