DYNAMICS OF THE GLOBALLY COUPLED COMPLEX GINZBURG-LANDAU EQUATION

A discrete version of the complex Ginzburg-Landau equation is studied on a completely connected lattice of N sites. This can equivalently be described as a model of N identical globally coupled limit-cycle oscillators. The phase diagram is obtained by a combination of numerical and analytical techniques. A surprising variety of dynamical behaviors is found in the thermodynamic limit (N much greater than 1). Depending on the region of parameter space, one gets the following: (1) a simple homogeneous limit cycle; (2) a state with complete frequency locking but with no phase locking so that the global forcing term vanishes; (3) a breaking of the system into a few macroscopic clusters which can exhibit periodic or quasiperiodic dynamics; or (4) surprisingly complex states where an individual oscillator behaves in a chaotic way but in a sufficiently coherent manner so that the average complex amplitude does not vanish in the thermodynamic limit. Moreover, in this last region, the dynamics of this natural order parameter is itself chaotic.