Fractional Lindstedt series

TL;DR

This paper investigates the formal perturbation expansions of resonant quasi-periodic motions in dynamical systems, demonstrating conditions for fractional power series and proving their convergence after resummation.

Contribution

It establishes conditions under which fractional Lindstedt series converge for resonant quasi-periodic motions, extending perturbation theory in dynamical systems.

Findings

Formal fractional power series can describe resonant motions.

Convergence of these series is proven after resummation.

Conditions for the applicability of fractional Lindstedt series are identified.

Abstract

The parametric equations of the surfaces on which highly resonant quasi-periodic motions develop (lower-dimensional tori) cannot be analytically continued, in general, in the perturbation parameter, i.e. they are not analytic functions of the perturbation parameter. However rather generally quasi-periodic motions whose frequencies satisfy only one rational relation ("resonances of order 1") admit formal perturbation expansions in terms of a fractional power of the perturbation parameter, depending on the degeneration of the resonance. We find conditions for this to happen, and in such a case we prove that the formal expansion is convergent after suitable resummation.