TITLE:
Homoclinic orbits to invariant tori in Hamiltonian systems

AUTHORS:
Amadeu Delshams(1) and Pere Gutierrez(2)

(1) Departament de Matematica Aplicada I,
    Universitat Politecnica de Catalunya,
    Diagonal 647, 08028 Barcelona (Spain).
    E-mail: amadeu@ma1.upc.es

(2) Departament de Matematica Aplicada II,
    Universitat Politecnica de Catalunya,
    Pau Gargallo 5, 08071 Barcelona (Spain).
    E-mail: gutierrez@ma2.upc.es

ABSTRACT:
We consider a perturbation of an integrable Hamiltonian system 
which possesses invariant tori with coincident whiskers 
(like some rotators and a pendulum).
Our goal is to measure the splitting distance between the perturbed whiskers,
putting emphasis on the detection of their intersections, which
give rise to homoclinic orbits to the perturbed tori.
A geometric method is presented which takes into account the Lagrangian
properties of the whiskers. In this way, the splitting
distance is the gradient of a splitting potential.
In the regular case (also known as a priori-unstable: the Lyapunov
exponents of the whiskered tori remain fixed), the splitting potential
is well-approximated by a Melnikov potential.
This method is designed as a first step in the study of
the singular case (also known as a priori-stable: 
the Lyapunov exponents of the whiskered tori approach
to zero when the perturbation tends to zero).

KEYWORDS:
Hamiltonian systems, KAM and Nekhoroshev theory, whiskered tori,
splitting of separatrices, Arnold diffusion.

MSC numbers: 58F05, 34C37, 58F36, 34C30