Shape of analyticity domains of Lindstedt series: the standard map

Abstract

The analyticity domains of the Lindstedt series for the standard map are studied numerically using Padé approximants to model their natural boundaries. We show that if the rotation number is a Diophantine number close to a rational value p/q, then the radius of convergence of the Lindstedt series becomes smaller than the critical threshold for the corresponding Kol'mogorov-Arnol'd-Moser curve, and the natural boundary on the plane of the complexified perturbative parameter acquires a flower-like shape with 2q petals.