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Abstract: Power series expansions naturally arise whenever solutions of ordinarydifferential equations are studied in the regime of perturbation theory. In thecase of quasi-periodic solutions the issue of convergence of the series isplagued of the so-called small divisor problem. In this paper we review amethod recently introduced to deal with such a problem, based onrenormalisation group ideas and multiscale techniques. Applications to bothquasi-integrable Hamiltonian systems KAM theory and non-Hamiltoniandissipative systems are discussed. The method is also suited to situations inwhich the perturbation series diverges and a resummation procedure can beenvisaged, leading to a solution which is not analytic in the perturbationparameter: we consider explicitly examples of solutions which are onlyinfinitely differentiable in the perturbation parameter, or even defined on aCantor set.



Author: Guido Gentile

Source: https://arxiv.org/



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