Self-organizing dynamics of coupled map systems

We show that the feedback from the macroscopic dynamics of a system of coupled units can synchronize the dynamics of these units. We studied the dynamics of maps coupled through their variables and control parameters. The feedback adjusted the values of the parameters of each map by using a function that depended on the difference between the Liapunov exponent of each unit and the Liapunov exponent of the mean field of the system. We showed that synchronization of the maps can be achieved under two different conditions: ~1! where the maps interact autonomously without a fixed controlling map and ~2! where the maps interact nonautomously with a single controlling map with fixed parameters. This method of feedback control may be useful in controlling more general types of parallel distributed systems. @S1063-651X~99!01303-3#