Breakdown of Lindstedt Expansion for Chaotic Maps

TL;DR

This paper revisits Lindstedt series expansions for chaotic maps, clarifying their relation to Greene's method and analyzing the convergence properties and analyticity domains of standard and semi-standard maps without small divisors.

Contribution

It provides an analytical and numerical study of the radius of convergence and analyticity domains for simplified map models, clarifying their relation to the true maps and previous methods.

Findings

The Lindstedt expansion corresponds to the transition value for the semi-standard map.

The radius of convergence is lower for the standard map analogue than for the semi-standard map.

The analyticity domain is a perfect circle for the semi-standard map analogue.

Abstract

In a previous paper of one of us [Europhys. Lett. 59 (2002), 330--336] the validity of Greene's method for determining the critical constant of the standard map (SM) was questioned on the basis of some numerical findings. Here we come back to that analysis and we provide an interpretation of the numerical results by showing that no contradiction is found with respect to Greene's method. We show that the previous results based on the expansion in Lindstedt series do correspond to the transition value but for a different map: the semi-standard map (SSM). Moreover, we study the expansion obtained from the SM and SSM by suppressing the small divisors. The first case turns out to be related to Kepler's equation after a proper transformation of variables. In both cases we give an analytical solution for the radius of convergence, that represents the singularity in the complex plane closest to the origin. Also here, the radius of convergence of the SM's analogue turns out to be lower than the one of the SSM. However, despite the absence of small denominators these two radii are lower than the ones of the true maps for golden mean winding numbers. Finally, the analyticity domain and, in particular, the critical constant for the two maps without small divisors are studied analytically and numerically. The analyticity domain appears to be an perfect circle for the SSM analogue, while it is stretched along the real axis for the SM analogue yielding a critical constant that is larger than its radius of convergence.