05-337 Giovanni Gallavotti, Guido Gentile, and Alessandro Giuliani

Fractional Lindstedt series (703K, postscript) Sep 26, 05

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Abstract. The parametric equations of the surfaces on which highly resonant quasi-periodic motions develop (lower-dimensional tori) cannot be analytically continued, in general, in the perturbation parameter $\varepsilon$, i.e. they are not analytic functions of $\varepsilon$. However rather generally quasi-periodic motions whose frequencies satisfy only one rational relation (``resonances of order $1$') admit formal perturbation expansions in terms of a fractional power of $\varepsilon$ depending on the degeneration of the resonance. We find conditions for this to happen, and in such a case we prove that the formal expansion is convergent after suitable resummation.

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