On mean field fluctuations in globally coupled maps

Abstract

Abstract We consider the mean field fluctuations in the globally coupled map x n+1 (i) = f(x n (i)) + ϵ 1 N Σ j=1 N x n (j) with an expanding piecewise linear local map f in the thermodynamic limit N → ∞ 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 ϵ 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 ∼ ϵ2. The order of magnitude of stationary fluctuations is estimated as ∼ exp[O(ϵ−2)]. The theoretical approach used enables us to develop high-precision numerical methods, able to detect such tiny fluctuations.