TITLE:
From the H\'enon conservative map to the Chirikov standard map for large
parameter values

AUTHORS:
   Narc\'{\i}s Miguel, Carles Sim\'o and Arturo Vieiro

   Dept. de Matem\`atica Aplicada i An\`alisi,
   Universitat de Barcelona, Gran Via 585, 08007,
   Barcelona, Catalunya

E-mail: narcis@maia.ub.es, carles@maia.ub.es, vieiro@maia.ub.es
 
ABSTRACT:
In this paper we consider conservative quadratic H\'enon maps and
Chirikov's standard map, and relate them in some sense.

First, we present a study of some dynamical properties of
orientation-preserving and orientation-reversing quadratic H\'enon maps
concerning the stability region, the size of the chaotic zones, its
evolution with respect to parameters and the splitting of the
separatrices of fixed and periodic points plus its role in the
preceding aspects.

Then the phase space of the standard map, for large values of the
parameter, $k$, is studied. There are some stable orbits which appear
periodically in $k$ and are scaled somehow. Using this scaling, we show
that the dynamics around these stable orbits is the one of above H\'enon
maps plus some small error, which tends to vanish as $k\to\infty$.
Elementary considerations about diffusion properties of the standard map
are also presented.