  We study the exponentially small splitting of invariant manifolds of
whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable
Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a
torus whose frequency ratio is the silver number $\Omega=\sqrt{2}-1$. We show
that the Poincar\'e-Melnikov method can be applied to establish the existence
of 4 transverse homoclinic orbits to the whiskered torus, and provide
asymptotic estimates for the tranversality of the splitting whose dependence on
the perturbation parameter $\varepsilon$ satisfies a periodicity property. We
also prove the continuation of the transversality of the homoclinic orbits for
all the sufficiently small values of $\varepsilon$, generalizing the results
previously known for the golden number.
