Globally Coupled Maps: Almost Analytical Treatment of Phase Transitions

Bifurcations in a system of coupled maps are investigated. Using symbolic dynamics it is proven that for coupled shift maps the well known space–time–mixing attractor becomes unstable at a critical coupling strength in favour of a synchronized state. For coupled non–hyperbolic maps analytical and numerical evidence is given that arbitrary small coupling changes the dynamical behaviour. The anomalous dependence of fluctuations on the system size is attributed to these bifurcations.