Cost of synchronizing different chaotic systems Original Research Article

Feedback coupling provides a general scheme for synchronizing two oscillatory chaotic systems through the intervention of a term of interaction that accounts for the difference of behaviors. We define a cost of synchronization based on a measure of the interaction term. Synchronizing different systems is not cost free and the cost increases with the requirements imposed on the synchronized behavior. We prove that many systems can reach a regime of complete synchronization at a limited, and a priory computable, cost. For identical systems, the cost of complete synchronization is zero. Some different systems can also keep a completely synchronized behavior in some of their variables at zero cost. We propose to reserve the name identical synchronization for complete synchronization at zero cost. We compute the cost for different stages of synchronization between two systems as different as the Rössler and Lorenz systems and for homochaotic cases of both families. If the response system is flexible enough to adapt to the structure of the driving system, lower synchronization cost or, eventually, identical synchronization will be possible. In this paper, we deduce adaptation laws to reach identical synchronization for any family of homochaotic systems, and we illustrate their application for the Rössler and Lorenz cases.