97-495 Rudnev M., Wiggins S.

Separatrix Splittings Near Resonance in Perturbations of Integrable, A-Priori Stable Hamiltonian Systems with Three or More Degrees-of-Freedom (181K, LaTeX) Sep 15, 97

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Abstract. In this note we consider the problem of the splitting of separatrices near resonances in perturbations of a-priori stable, integrable Hamiltonian systems with three or more degrees-of-freedom. For a model problem that is an $n$ degree-of-freedom generalization of Arnold's original model we show that the exponents in the exponentially small (with respect to the perturbation parameter) upper bound for the measure of the transversality of the splitting correspond to the exponents arising in the Nekhoroshev theorem which describes the evolution of the action variables near a resonant torus. These exponents are given by $\frac{1}{2(n-m)}$, where $n$ is the number of degrees-of-freedom and $m$ is the multiplicity of the resonance. In appropriate coordinates, the problem of separatrix splitting near a resonant torus can be viewed as the problem of the splitting of separtrices of a torus with a certain number of ``fast'' and ``slow'' frequencies (the number of slow frequencies plus one is the multiplicity of the resonance). We show that a splitting direction corresponding to a slow frequency gives rise to a splitting distance that has an algebraic dependence on the perturbation parameter, a splitting direction corresponding to a fast frequency gives rise to a splitting distance that has an exponentially small dependence on the perturbation parameter, and if there is {\em at least one} fast frequency, then the measure of transversality of the splitting will be exponentially small with respect to the perturbation parameter.

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