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

TL;DR

This paper reviews a renormalisation group method for addressing the small divisor problem in the convergence of power series solutions for quasi-periodic motions in dynamical systems, applicable to both Hamiltonian and dissipative systems.

Contribution

It introduces a novel renormalisation group approach combined with multiscale techniques to handle divergence and non-analytic solutions in perturbation series.

Findings

Effective handling of small divisor problems in quasi-periodic solutions

Application to both Hamiltonian and dissipative systems

Solutions can be infinitely differentiable or defined on Cantor sets

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.