We study the splitting of invariant manifolds of whiskered tori with two orthree frequencies in nearly-integrable Hamiltonian systems,such that the hyperbolic part is given by a pendulum.We consider a 2-dimensional torus witha frequency vector $omega=(1,Omega)$, where $Omega$ is a quadraticirrational number, or a 3-dimensional torus with a frequency vector$omega=(1,Omega,Omega^2)$, where $Omega$ is a cubic irrational number.Applying the Poincaré--Melnikov method, we find exponentially smallasymptotic estimates for the maximal splitting distance between the stable andunstable manifolds associated to the invariant torus, and we show that suchestimates depend strongly on the arithmetic properties of the frequencies. Inthe quadratic case, we use the continued fractions theory to establish acertain arithmetic property, fulfilled in 24 cases, which allows us to provideasymptotic estimates in a simple way. In the cubic case, we focus our attentionto the case in which $Omega$ is the so-called cubic golden number (the realroot of $x^3+x-1=0$), obtaining also asymptotic estimates. We point out thesimilitudes and differences between the results obtained for both the quadraticand cubic cases.
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