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Regular and Chaotic Dynamics

, Volume 15, Issue 2–3, pp 222–236 | Cite as

Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel’nikov method

  • A. DelshamsEmail author
  • P. Gutiérrez
  • O. Koltsova
  • J. R. Pacha
L.P. Shilnikov-75 Special Issue

Abstract

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).

Key words

hyperbolic KAM tori transverse homoclinic orbits Melnikov method 

MSC2000 numbers

37J40 37C29 70H08 

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Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • A. Delshams
    • 1
    Email author
  • P. Gutiérrez
    • 1
  • O. Koltsova
    • 2
  • J. R. Pacha
    • 1
  1. 1.Dep. de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelona, CataloniaSpain
  2. 2.Department of MathematicsImperial College LondonLondonUK

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