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Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel'nikov method
Delshams Valdés, Amadeu; Gutiérrez Serrés, Pere; Koltsova, Oksana; Pacha Andújar, Juan Ramón
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I
We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic-hyperbolic type, having a homoclinic orbit. More precisely, we consider an (n + 2)-degree-of-freedom near integrable Hamiltonian with n centers and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the center manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel’nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).
hyperbolic KAM tori - transverse homoclinic orbits - Melnikov method
2012-05-10
Àrees temàtiques de la UPC::Matemàtiques i estadística
Classificació AMS::37 Dynamical systems and ergodic theory
Classificació AMS::70 Mechanics of particles and systems
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