  Systems of a large number N of globally coupled maps have become popular as a
relatively simple prototype of high-dimensional dynamics, showing many
interesting and typical phenomena like synchronisation, cluster formation and
multistability, and having potential applications in systems like Josephson
junction arrays or in biophysical models. There exists a wealth of numerical
investigations of globally coupled maps. While much progress has been made in
the explanation of the macroscopic behaviour of such systems in the limit of
infinite N, there is still need for a sound theory about the asymptotic
behaviour of finite-N systems as N approaches infinity. This article introduces
a method by which it is possible to obtain asymptotic estimates for long-term
deviations from the thermodynamic limit behaviour. This method is based upon
the concept of quasipotentials, originally developed by Freidlin, Wentzell, and
others for describing the influence of small random perturbations on the
long-term behaviour of dynamical systems. The problems of explicitly computing
quasipotentials in the present context and potential approximation schemes are
discussed. All the concepts described in this article are illustrated with a
simple example.
