ON MEAN-FIELD FLUCTUATIONS IN GLOBALLY COUPLED MAPS

We consider the mean field fluctuations in the globally coupled map x(n+1)(i) = f(x(n)(i)) + epsilon 1/N Sigma(j=1)(N) X(n)(j) With an expanding piecewise linear local map f in the thermodynamic limit N --> infinity using the reduced master equation (self-consistent Frobenius-Perron operator). It is shown that correlations between the elements are absent in simultaneous ensembles and only arise in time series. For epsilon small the equilibrium state (with constant mean field) is proved to be unstable, so the mean field fluctuates, corresponding to a complex attractor of the reduced master equation. It is chaotic and its Lyapunov exponent is estimated to be similar to epsilon(2). The order of magnitude of stationary fluctuations is estimated as similar to exp[O(epsilon(-2))]. The theoretical approach used enables us to develop high-precision numerical methods, able to detect such tiny fluctuations.