Guido Gentile and Titus S. van Erp
Breakdown of Lindstedt Expansion for Chaotic Maps
(764K, postscript)
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 no
contradiction is found with respect to Greene's method.
We show that the previous results based on the expansion in
Lindstedt series do correspond to the transition value
but for a different map: the semi-standard map (SSM).
Moreover, we study the expansion obtained from the SM and SSM
by suppressing the small divisors.
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 SM's analogue turns out to be
lower than the one of the SSM. However, despite the absence
of small denominators these two radii are lower than the ones of
the true maps for golden mean winding numbers.
Finally, the analyticity domain and, in particular,
the critical constant for the two maps without small divisors
are studied analytically and numerically.
The analyticity domain appears to be an perfect circle for the SSM
analogue, while it is stretched along the real axis for the SM analogue
yielding a critical constant that is larger than its radius of convergence.