Hybrid Complex Projective Synchronization of Complex Chaotic Systems Using Active Control Technique with Nonlinearity in the Control Input

Abstract

In this paper, the active control method based on Lyapunov function is used to study hybrid complex projective synchronization (HCPS). In the complex space, the response system is asymptotically synchronized up to the drive system by the state transformation by using a complex scale matrix. An extension of projective synchronization from the field of real numbers to the field of complex numbers has been done in this paper; i.e., the scaling factors are complex. The unpredictability of the scaling factors in the proposed synchronization scheme can additionally increase the security of communication. This synchronization method is studied between two non-identical complex chaotic nonlinear 3-dimensional systems. We take Lorenz system as the driving system and Chen system as the response system. In order to demonstrate the robustness of the proposed control method, dead-zone nonlinearity input is imposed to the control input. The closed loop stability conditions based on Lyapunov function and Barbalat lemma are derived. Finally, numerical simulations are presented to verify the results of the proposed scheme.