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

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October 2004, 11(4): 785-826. doi: 10.3934/dcds.2004.11.785

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

Amadeu Delshams ^1, , Pere Gutiérrez ^2, and Tere M. Seara ^3,

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

Received March 2003 Revised January 2004 Published September 2004

Abstract
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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 Poincaré--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.

Keywords: flow-box coordinates, Poincaré–Melnikov method, Hyperbolic KAM theory.

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

Citation: Amadeu Delshams, Pere Gutiérrez, Tere M. Seara. Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 785-826. doi: 10.3934/dcds.2004.11.785

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