Adaptive Single Input Control for Synchronization of a 4D Lorenz-Stenflo Chaotic System

For general nonlinear four-dimensional (4D) chaotic systems, synchronization basically cannot be achieved by applying single input control. The purpose of this paper is to study the chaotic synchronization between two nearly identical 4D Lorenz-Stenflo systems by utilizing the adaptive single input controllers with one state variable. To complete the previous results in the literature, two cases of chaotic synchronization are considered. First, the drive system is assumed to be given in advance. The other case is set that system parameters of the drive system are uncertain but invariable with respect to time for the response system. Based on the Lyapunov theorem of stability, two kinds of adaptive single input controllers feeding by one variable associated with corresponding system parameter updated laws are developed to make the drive and response systems asymptotically synchronized. Numerical studies are presented to demonstrate the effectiveness of proposed schemes.