Alberto Berretti, Corrado Falcolini, Guido Gentile
The shape of analyticity domains of Lindstedt series: the standard map
(405K, Postscript)

ABSTRACT.  The analyticity domains of the Lindstedt series for 
the standard map are studied numerically using Pade' 
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 KAM curve, and the natural boundary on 
the plane of the complexified perturbative parameter acquires a 
flower-like shape with 2q petals. 
We conjecture that the natural boundary has typically a fractal 
shape, which only in particular cases 
degenerates to an apparently regular curve.