FM 00-18 ; mp_arc 00-375

Author: Alberto Berretti, Corrado Falcolini, Guido Gentile

Title: The 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 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.

Keywords: Standard map, KAM invariant curves, analyticity domain, Padé approximants, critical function, natural boundary

Alberto Berretti
Dipartimento di Matematica
II Università di Roma (Tor Vergata)
Via della Ricerca Scientifica, 00133 Roma, Italy
e-mail: berretti@mat.uniroma2.it

Corrado Falcolini
Dipartimento di Matematica
II Università di Roma (Tor Vergata)
Via della Ricerca Scientifica, 00133 Roma, Italy
e-mail: falcolin@mat.uniroma2.it

Guido Gentile
Dipartimento di Matematica
Università di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@matrm3.mat.uniroma3.it