**Author:**
Guido Gentile, Titus S. van Erp

**Title:** *
Breakdown of Lindstedt Expansion for Chaotic Maps
*

**Abstract: **
In a previous paper of one of us [Europhys. Lett. 59 (2002), 330--336]
the validity of Greene's method for determining the critical constant
of the standard map (SM) was questioned on the basis of some numerical
findings. Here we come back to that analysis and we provide an
interpretation of the numerical results, by showing that the conclusions
of that paper were wrong as they relied on a plausible but untrue
assumption. Hence no contradiction exists with respect to Greene's method.
We show that the previous results, based on the expansion in Lindstedt
series, do correspond to the critical constant but for a different map:
the semi-standard map (SSM). For such a map no Greene's method analogue
is at disposal, so that methods based on Lindstedt series are essentially
the only possible ones. Moreover, we study the expansion for two
simplified models obtained from the SM and SSM by suppressing the small
divisors. We call them the simplified SM and simplified SSM, respectively;
the first case turns out to be related to Kepler's equation after a proper
transformation of variables. In both cases we give an analytical solution
for the radius of convergence, that represents the singularity in the
complex plane closest to the origin. Also here, the radius of convergence
of the simplified SM turns out to be lower than that of the simplified SSM.
However, despite the absence of small divisors these two radii are lower
than those of the true maps (i.e. of the maps with small divisors) when
the winding number equals the golden mean. Finally, we study the
analyticity domain and, in particular, the critical constant for the two
maps without small divisors. The analyticity domain turns out to be a
perfect circle for the simplified SSM (as for the SSM itself), while it is
stretched along the real axis for the simplified SM, yielding a
critical constant which is larger than its radius of convergence.

**Keywords:**
Standard map; semi-standard map; Lindstedt series;
Critical constant; Radius of convergence;
Analyticity domain; Natural boundary; Small divisors.

Guido Gentile

Dipartimento di Matematica

Università di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@mat.uniroma3.it

Titus van Erp

Laboratoire de Physique

Centre Européen de Calcul
Atomique et Moléculaire

Ecole Normale Supérieure de Lyon

46 All&ecaute;e d'Italie, 69364 Lyon Cedex 07, France

e-mail: tsvanerp@cecam.fr