Shimizu–Morioka's chaos synchronization: An efficacy analysis of active control and backstepping methods

This research study inspects the effectiveness of synchronization methods such as active control and backstepping control from systematic design procedures of a synchronized Shimizu–Morioka system for the same parameter. It aimed to achieve synchronization between the state variables of two identical Shimizu–Morioka chaotic systems by defining the proposed varieties of the error dynamics coefficient matrix. Furthermore, this study also aimed to designed an active controller that enables the synchronization of these systems. The use of designed recursive backstepping nonlinear controllers was based on the Lyapunov function. Furthermore, it also demonstrated the stability of the synchronization of the nonlinear identical Shimizu–Morioka system. The new virtual state variable and establishment of Lyapunov functionals are used in the backstepping controller to stabilize and reduce errors between the Master (MS)/Drive (DS) systems. For comparison, the complexity of active controllers is verified to be such that the designed controller's effectiveness based on backstepping is attainable in engineering applications. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed synchronization strategy with the Runge–Kutta (RK-4) algorithm of fourth order through MatLab Simulink.