TITLE:
Splitting and Melnikov potentials in Hamiltonian systems

AUTHORS:
Amadeu Delshams, Pere Guti\'errez

Departament de Matem\`atica Aplicada I,
Universitat Polit\`ecnica de Catalunya, 
Diagonal 647, 08028 Barcelona
E-mails: amadeu@ma1.upc.es, pereg@ma1.upc.es

ABSTRACT:
We consider a perturbation of an integrable Hamiltonian system,
possessing hyperbolic invariant tori with coincident whiskers.
Following an idea due to Eliasson, 
we introduce a {\it splitting potential\/} 
whose gradient gives the splitting distance between
the perturbed stable and unstable whiskers.

The homoclinic orbits to the perturbed whiskered tori are the
critical points of the splitting potential, and therefore their
existence is ensured in both the regular (or strongly hyperbolic,
or a-priori unstable) and the singular (or weakly hyperbolic,
or a-priori stable) case. The singular case is a model of a
nearly-integrable Hamiltonian near a single resonance.

In the regular case, the {\it Melnikov potential\/} is a first order
approximation of the splitting potential, and the standard Melnikov
(vector) function is simply the gradient of the Melnikov potential.
Non-degenerate critical points of the Melnikov potential give rise to
transverse homoclinic orbits. Explicit computations are carried out
for some examples.

KEYWORDS:
Hamiltonian systems, whiskered tori, splitting potential,
Melnikov potential, Fibonacci numbers.

1991 MSC numbers:
58F05, 34C37, 58F36, 34C30, 11B39

