Quasi-periodic motions in dynamical systems. Review of a renormalisation group approach

Details Quasi-periodic motions in dynamical systems. Review of a renormalisation group approach Quasi-periodic motions in dynamical systems. Review of a renormalisation group approach

2009-10-05

arxiv.org/abs/0910.0755v1

Mathematics

Dynamical Systems

36 pages, 8 figures, review article

Scientific paper

Power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory. In the case of quasi-periodic solutions the issue of convergence of the series is plagued of the so-called small divisor problem. In this paper we review a method recently introduced to deal with such a problem, based on renormalisation group ideas and multiscale techniques. Applications to both quasi-integrable Hamiltonian systems (KAM theory) and non-Hamiltonian dissipative systems are discussed. The method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged, leading to a solution which is not analytic in the perturbation parameter: we consider explicitly examples of solutions which are only infinitely differentiable in the perturbation parameter, or even defined on a Cantor set.

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