# American Institute of Mathematical Sciences

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

 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

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.
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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