TITLE:
Splitting potential and Poincar\'e-Melnikov method
for whiskered tori 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 deal with a perturbation of a hyperbolic integrable Hamiltonian system
with $n+1$ degrees of freedom. The integrable system is assumed to have
$n$-dimensional hyperbolic invariant tori with coincident whiskers
(separatrices).

Following Eliasson, we use a geometric approach closely related to
the Lagrangian properties of the whiskers, to show that the splitting
distance between the perturbed stable and unstable whiskers is the
gradient of a periodic scalar function of $n$ phases, which we call
splitting potential. This geometric approach works for both the singular
(or weakly hyperbolic) case and the regular (or strongly hyperbolic) case,
and provides the existence of at least $n+1$ homoclinic intersections
between the perturbed whiskers.

In the regular case, we also obtain a first order approximation for the
splitting potential, that we call Melnikov potential.
Its gradient, the (vector) Melnikov function, provides
a first order approximation for the splitting distance.
Then the nondegenerate critical points of the Melnikov
potential give rise to transverse homoclinic intersections between the
whiskers. Generically, when the Melnikov potential is a Morse function,
there exist at least $2^n$ critical points.

The first order approximation relies on the $n$-dimensional
Poincar\'e-Melnikov method, to which an important part
of the paper is devoted.
We develop the method in a general setting, giving the Melnikov
potential and the Melnikov function in terms of absolutely
convergent integrals, which take into account the phase drift along the
separatrix and the first order deformation of the perturbed hyperbolic tori.
We provide formulas useful in several cases,
and carry out explicit computations
that show that the Melnikov potential is a Morse function,
in different kinds of examples.

KEYWORDS:
Hamiltonian systems, whiskered tori, splitting potential,
Poincar\'e-Melnikov method, Melnikov potential.

1991 MSC numbers:
58F05, 34C37, 58F36, 34C30, 70F15

PAC numbers:
45.20.Jj, 02.40.Vh, 05.45.-a, 95.10.Fh, 45.50.Pk