Quasiperiodic motions in dynamical systems: Review of a renormalization group approach

Power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory. In the case of quasiperiodic solutions the issue of convergence of the series is bedeviled by the so-called small divisor problem. In this paper we review a method recently introduced to deal with such a problem, based on renormalization group ideas and multiscale techniques. Applications to both quasi-integrable Hamiltonian systems [Kolmogorov-Arnold-Moser (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 C-infinity in the perturbation parameter, or even defined on a Cantor set.