TITLE:
Lower and upper bounds for the splitting of separatrices of the
pendulum under a fast quasiperiodic forcing

AUTHORS:
Amadeu Delshams(1), Vassili Gelfreich(2), \`Angel Jorba(1), Tere M. Seara(1)

(1): Departament de Matem\`atica Aplicada I, 
Universitat Polit\`ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
E-mails: amadeu@ma1.upc.es, angel@tere.upc.es, tere@ma1.upc.es

(2): Departament de Matem\`atica Aplicada i An\`alisi,
Universitat de Barcelona, Gran via 585, 08007 Barcelona, Spain,
and
Chair of Applied Mathematics, 
St.Petersburg Academy of Aerospace Instrumentation,
Bolshaya Morskaya 67, 190000, St. Petersburg, Russia
E-mail: gelf@maia.ub.es, gelf@misha.usr.saai.ru

ABSTRACT:
Quasiperiodic perturbations with two frequencies
$(1/\varepsilon ,\gamma /\varepsilon )$ of a pendulum are considered,
where $\gamma $ is the golden mean number.
We study the splitting of the three-dimensional invariant manifolds
associated to a two-dimensional invariant torus in a neighbourhood
of the saddle point of the pendulum.
Provided that some of the Fourier coefficients of the perturbation
(the ones associated to Fibonacci numbers) are separated from zero,
it is proved that the invariant manifolds split for $\varepsilon $
small enough.
The value of the splitting, that turns out to be
${\rm O} (\exp (-{\rm const} /\sqrt{\varepsilon }) )$,
is correctly predicted by the Melnikov function.