04-148 Oksana Koltsova, Lev Lerman, Amadeu Delshams, Pere Guti\'errez

Homoclinic orbits to invariant tori near a homoclinic orbit to center-center-saddle equilibrium (398K, Postscript) May 7, 04

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Abstract. We consider a perturbation of an integrable Hamiltonian vector field with three degrees of freedom with a center-center-saddle equilibrium having a homoclinic orbit or loop. With the help of the Poincar\'e map, we study the homoclinic intersections between the stable and unstable manifolds associated to persistent hyperbolic KAM tori, on the center manifold near the equilibrium. If the perturbation is such that the homoclinic loop is preserved, we establish that, in general, the manifolds intersect along 8, 12 or 16 transverse homoclinic orbits. On the other hand, in a more generic situation (the loop is not preserved) the manifolds intersect along 4 transverse homoclinic orbits, though a small neighborhood of the loop has to be excluded. In a first approximation, those homoclinic orbits can be detected as nondegenerate critical points of a Melnikov potential. We also develop an alternative Melnikov approximation in order to study the splitting of the loop itself.

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