This article is cited in 4 papers

On the 75th birthday of Professor L.P. Shilnikov

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

A. Delshams^a, P. Gutiérrez^a, O. Koltsova^b, J. R. Pacha<sup>a

a</sup> Dep. de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal 647, 08028 Barcelona, Catalonia, Spain
^b Department of Mathematics, Imperial College London, SW7 2AZ London, UK

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).

Keywords: hyperbolic KAM tori, transverse homoclinic orbits, Melnikov method.

MSC: 37J40, 37C29, 70H08

Received: 22.12.2009
Accepted: 11.01.2010

Language: English

DOI: 10.1134/S1560354710020103

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