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Synchronized states in chaotic systems coupled indirectly through a dynamic environment

We consider synchronization of chaotic systems coupled indirectly through a common environmnet where the environment has an intrinsic dynmics of its own modulated via feedback from the systems. We find that a rich vareity of synchronization behavior, such as in-phase, anti-phase,complete and anti- synchronization is possible. We present an approximate stability analysis for the different synchronization behaviors. The transitions to different states of synchronous behaviour are analyzed in the parameter plane of coupling strengths by numerical studies for specific cases such as Rossler and Lorenz systems and are characterized using various indices such as correlation, average phase difference and Lyapunov exponents. The threshold condition obtained from numerical analysis is found to agree with that from the stability analysis.