Transition from anticipatory to lag synchronization via complete synchronization in time-delay systems

The existence of anticipatory, complete, and lag synchronization in a single system having two different time delays, that is, feedback delay τ1 and coupling delay τ2, is identified. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay τ2 with a suitable stability condition is discussed. In particular, it is shown that the stability condition is independent of the delay times τ1 and τ2. Consequently, for a fixed set of parameters, all the three types of synchronizations can be realized. Further, the emergence of exact anticipatory, complete, or lag synchronization from the desynchronized state via approximate synchronization, when one of the system parameters (b2) is varied, is characterized by a minimum of the similarity function and the transition from on-off intermittency via periodic structure in the laminar phase distribution