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