Adaptive Control and Synchronization of the Harb System

This paper investigates the adaptive control and synchronization of the uncertain Harb system with unknown parameters. The Harb system is one of the important and simple paradigms of three-dimensional chaotic systems discovered by Harb and Zohdy in 2002. Adaptive control is a popular method in the control systems literature which is used whenever the parameters of the systems are not available. In adaptive control method, we modify the control strategies with estimates of the parameters and using Lyapunov stability theory, we derive update laws for the estimates of the system parameters. In this paper, we first design an adaptive controller so as to stabilize the uncertain Harb system to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Then we derive adaptive control laws to achieve global chaos synchronization of identical uncertain Harb systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes for the uncertain Harb system.