Breakdown of Lindstedt expansion for chaotic maps

Abstract

In a previous paper of one of us [Europhys. Lett. 59, 330–336 (2002)] 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 the conclusions of that paper were wrong as they relied on a plausible but untrue assumption. Hence no contradiction exists with respect to Greene’s method. We show that the previous results, based on the expansion in Lindstedt series, do correspond to the critical constant but for a different map: the semi-standard map (SSM). For such a map no Greene’s method analog is at disposal, so that methods based on Lindstedt series are essentially the only possible ones. Moreover, we study the expansion for two simplified models obtained from the SM and SSM by suppressing the small divisors. We call them the simplified SM and simplified SSM, respectively; 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 simplified SM turns out to be lower than that of the simplified SSM. However, despite the absence of small divisors these two radii are lower than those of the true maps (i.e., of the maps with small divisors) when the winding number equals the golden mean. Finally, we study the analyticity domain and, in particular, the critical constant for the two maps without small divisors. The analyticity domain turns out to be a perfect circle for the simplified SSM (as for the SSM itself), while it is stretched along the real axis for the simplified SM, yielding a critical constant which is larger than its radius of convergence.