Amadeu Delshams, Marina Gonchenko, Pere Guti\'errez
Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type
(62K, LATeX 2e)
ABSTRACT. We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian sys
tems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a fast frequency vecto
r $\omega/\sqrtarepsilon$, with $\omega=(1,\Omega)$ where $\Omega$ is an irrational number of constant type, i.e. a num
ber whose continued fraction has bounded entries. Applying the Poincar\'e-Melnikov method, we find exponentially small lo
wer bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invar
iant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies.