TITLE:
Singular separatrix splitting and the Poincare-Melnikov
method for area preserving maps

AUTHORS:
Amadeu Delshams, Rafael Ramirez-Ros

Departament de Matematica Aplicada I
Universitat Politecnica de Catalunya
Diagonal 647, 08028 Barcelona, Spain
E-mails: amadeu@ma1.upc.es, rafael@vilma.upc.edu

ABSTRACT:
The splitting of separatrices of area preserving maps close to the
identity is one of the most paradigmatic examples of an
{\it exponentially small} or {\it singular} phenomenon.
The intrinsic small parameter is the characteristic
exponent $h>0$ of the saddle fixed point.
A standard technique to measure the splitting of separatrices
is the so-called Poincar\'e-Melnikov method,
which has several specific features in the case
of analytic planar maps.
The aim of this talk is to compare the predictions for the splitting
of separatrices provided by the Poincar\'e-Melnikov method,
with the analytic and numerical results in a simple example
where computations in multiple-precision arithmetic are performed.

KEYWORDS:
Area-preserving map, singular separatrix splitting, Poincare-Melnikov method,
numerical experiments

MSC numbers: 34C37, 34E05, 34E15, 65L12