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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.

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