Long range action in networks of chaotic elements

Abstract

We show that under certain simple assumptions on the topology (structure) of networks of strongly interacting chaotic elements a phenomenon of long range action takes place, namely, that the asymptotic (as time goes to infinity) dynamics of an arbitrarily large network is completely determined by its boundary conditions. This phenomenon takes place under very mild assumptions on local dynamics with short range interactions. However, we show that it is unstable with respect to arbitrarily weak local random perturbations.