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