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| We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian systems which come from the three-dimensional RTBP. It is known that in a neighborhood of the elliptic equilibrium $L_4$ there are KAM tori as well as single resonance zones where lower dimensional (2D) hyperbolic tori appear. We consider a 2-dimensional torus with a fast frequency vector $\omega/\sqrt\epsilon$, with $\omega=(1,\Omega)$, where $\Omega$ is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincar\'{e}-Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies. Such lower bounds imply the existence of splitting between the invariant manifolds, which provides a strong indication of the non-integrability of the system near the given torus, and opens the door to the application of topological methods for the study of Arnold diffusion in such systems. |
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