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)

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.