FM 11-2009 (mp_arc 09-186; arXiv:0910.0755 [math.DS])

Author: Guido Gentile

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

Keywords: Renormalisation group; Multiscale analysis; Quasi-periodic solutions; Diophantine vectors; Bryuno vectors; KAM theory; Dissipative systems; Quasi-periodically forced systems; Perturbation series; Lindstedt series; Divergent series; Resummation.

Guido Gentile
Dipartimento di Matematica
UniversitÓ di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy