Different Types of Synchronization in Time-Delayed Systems

We investigate different types of synchronization between two unidirectionally nonlinearly coupled identical delay-differential systems related to optical bistable or hybrid optical bistable devices. This system can represent some kinds of delay-differential models, i.e. Ikeda model, Vallée model, sine-square model, Mackey--Glass model, and so on. We find existence and sufficient stability conditions by theoretical analysis and test the correctness by numerical simulations. Lag, complete and anticipating synchronization are bserved, respectively. It is found that the time-delay system can be divided into two parts, one is the instant term and the other is the delay term. Synchronization between two identical chaotic systems can be derived by adding a coupled term to the delay term in the driven system.