Title: High-precision computations of divergent asymptotic series and
       homoclinic phenomena

Authors: V. Gelfreich (1) and C. Sim\'o (2)

(1) Mathematics Institute, University of Warwick, Coventry CV4 7AL,
    United Kingdom.
(2) Dept. de Matem\`atica Aplicada i An\`alisi, Univ. de Barcelona,
    Gran Via 585, 08007 Barcelonar, Spain.
E-mail: gelf@maths.warwick.ac.uk, carles@maia.ub.es

Abstract

We study asymptotic expansions for the exponentially small splitting of
separatrices of area preserving maps combining analytical and numerical points
of view. Using analytic information, we conjecture the basis of functions of an
asymptotic expansion and then extract actual values of the coefficients of the
asymptotic series numerically. The computations are performed with
high-precision arithmetic, which involves up to several thousands of decimal
digits. This approach allows us to obtain information which is usually
considered to be out of reach of numerical methods. In particular, we use our
results to test that the asymptotic series are Gevrey-1 and to study positions
and types of singularities of their Borel transform. Our examples are based on
generalisations of the standard and H\'enon maps.