 145 Amadeu Delshams, Marina Gonchenko, Pere Guti\'errez
 Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type
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Feb 12, 14

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Abstract. We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearlyintegrable Hamiltonian sys
tems, such that the hyperbolic part is given by a pendulum. We consider a 2dimensional 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\'eMelnikov 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.
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