Adaptive synchronization of two time-delayed fractional-order chaotic systems with different structure and different order

In this paper, the chaos synchronization of time-delayed fractional-order (FO) Lorenz chaotic systems and time-delayed FO Chen chaotic systems with different fractional derivative order is considered. First, by use of the fractional calculus techniques, we give the phase trajectories of the time-delayed FO Lorenz and time-delayed FO Chen chaotic systems. Then, combined the active control and adaptive control theory, a novel synchronization scheme, which includes the compensation controller and the optimal controller, is proposed for two time-delayed FO chaotic systems with different structure and different order in presence of parameter uncertainty. Based on the new version of FO Lyapunov stability theory, we design the compensation controllers and optimal controllers, respectively. Finally, numerical simulations have shown to illustrate the effectiveness and validation of the proposed method.