Collective chaos and noise in the globally coupled complex Ginzburg-Landau equation

We study a globally coupled version of the complex Ginzburg-Landau equation (GC-CGLE) which consists of a large number N of identical two-dimensional oscillators coupled through their mean amplitude, Depending on parameter values, different dynamical regimes are attained. We focus particularly on an interesting regime where the individual oscillators follow erratic motion but in a sufficiently coherent way so that the average motion does not vanish when N becomes large and is also chaotic. A simple description of this state is proposed by considering the motion of a single forced two-dimensional system which has both a limit cycle and a fixed point as stable attractors. Determining which of these two deterministic attractors is selected by a weak noise and how this depends on the parameter of the reduced system allows us to determine self-consistently the average amplitude and dominant frequency of the collective behaviour of the full system. Finally, we show that adding a small noise to the GC-CGLE transforms the chaotic collective behaviour into a purely periodic one.