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We study the splitting of invariant manifolds of whiskered tori with twofrequencies in nearly-integrable Hamiltonian systems, such that the hyperbolicpart is given by a pendulum. We consider a 2-dimensional torus with a fastfrequency vector $\omega/\sqrt\epsilon$, with $\omega=(1,\Omega)$ where$\Omega$ is an irrational number of constant type, i.e. a number whosecontinued fraction has bounded entries. Applying the Poincar\'e-Melnikovmethod, we find exponentially small lower bounds for the maximal splittingdistance between the stable and unstable invariant manifolds associated to theinvariant torus, and we show that these bounds depend strongly on thearithmetic properties of the frequencies.

Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type