Synchronization of time-delay systems

We present the linear stability analysis of synchronized states in coupled time-delay systems. There exists a synchronization threshold, for which we derive upper bounds, which does not depend on the delay time. We prove that at least for scalar time-delay systems, synchronization is achieved by transmitting a single scalar signal, even if the synchronized solution is given by a high-dimensional chaotic state with a large number of positive Lyapunov exponents. The analytical results are compared with numerical simulations of two coupled Mackey-Glass equations. [S1063-651X(98)51210-X].