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Active control is a simple and effective method to realize chaotic synchronization. The existing active control techniques often have at least one of the following limitations: (1) the design of the controller relies on the Routh-Hurwitz criterion which forces the error system to be linear; (2) the dimension of the controller must be equal to the dimension of the controlled response system, often leading to a high cost of the controller; and (3) the drive system and the response system sometimes must be identical, which limits the application in situations where synchronization between different systems is required. Therefore, in the present study, we proposed a different active control method which can overcome the above limitations at the same time. This proposed active control method is based on an asymptotic stability theorem for nonlinear systems in which an asymptotic stability can be achieved when the coefficient matrix satisfies a specific configuration. The proof of the stability theorem is given, and the corresponding numerical simulation was provided to demonstrate the realization and the validity of the proposed method.

Generalized Projective Synchronization for Chaotic Systems via Active Control with Reduced Order