Anti-synchronization of Rabinovich systems using active and adaptive controllers

The anti-synchronization of chaotic systems discusses a pair of chaotic systems called master and slave systems and seeks to design controllers (connected to the slave system) so that the states of the synchronized systems have asymptotically the same amplitude but opposite signs, i.e. the sum of the respective states of the master and slave systems can be made to converge to zero asymptotically with time. Synchronization and anti-synchronization of chaotic systems have several applications in science and engineering. In this paper, we design of active and adaptive controllers for achieving anti-synchronization of the states of identical Rabinovich chaotic systems (1979) having two parameters a and b. We apply active control method for the anti-synchronization of Rabinovich systems when the system parameters are known and prove the anti-synchronization result by Lyapunov stability theory. We apply adaptive control method for the anti-synchronization of Rabinovich systems when the system parameters are unknown. We use adaptive control theory by implementing estimates of unknown parameters and derive parameter update law for estimates of unknown parameters with the help of Lyapunov stability theory. We have illustrated the main anti-synchronization results for Rabinovich systems for active and adaptive controller designs with MATLAB plots.