TITLE:
Exponentially small splitting of separatrices under 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:
We consider fast quasiperiodic perturbations with two frequencies 
$(1/\varepsilon,\gamma/\varepsilon)$ of a pendulum, 
where $\gamma$ is the golden mean number.  
The complete system has a two-dimensional invariant torus in a
neighbourhood of the saddle point. We study the splitting
of the  three-dimensional invariant manifolds associated to this torus.
Provided that the perturbation amplitude is small enough with respect to 
$\varepsilon $, and some of its Fourier coefficients (the ones associated 
to Fibonacci numbers), are separated from zero, it is proved
that the invariant manifolds split and that the value of the splitting, 
which turns out to be exponentially small with respect to $\varepsilon $,
is correctly predicted by the Melnikov function.