On Full-State Hybrid Projective Synchronization of General Discrete Chaotic Systems

The problems of full-state hybrid projective synchronization (FSHPS) and inverse full-state hybrid projective synchronization (IFSHPS) for general discrete chaotic systems are investigated in 2D. Based on nonlinear control method and Lyapunov stability theory, new controllers are designed to study FSHPS and IFSHPS, respectively, for 2D arbitrary chaotic systems in discrete-time. Numerical example and simulations are used to validate the main results of this paper. 1. Introduction Many mathematical models of biological processes, physical processes, and chemical processes, etc., were de…ned using chaotic dynamical systems in discrete-time. Recently, more and more attention were paid to chaos synchronization in discrete-time dynamical systems [1–3]. Synchronization of discrete chaotic dynamical systems has also potential applications in secure communication [4, 5]. Since the work of Pecora and Carroll [6], various powerful methods and techniques have been proposed to investigate chaos synchronization in dynamical systems [7, 8] and different types of synchronization have been reported [9–11]. Recently, a novel type of synchronization, known as full-state hybrid projective synchronization (FSHPS), has been introduced and applied to chaotic systems in continuous-time [12], which includes projective synchronization (PS) and hybrid projective synchronization (HPS). In FSHPS, each response system state synchronizes with a linear combination of drive system states. By the same procedure we can define a new type of synchronization, called inverse full-state hybrid projective synchronization (IFSHPS), when each drive system state synchronizes with a linear combination of response system states. In this paper, based on nonlinear control method in 2D and discrete-time Lyapunov stability theory, firstly, a new synchronization controller is designed for full-state hybrid projective synchronization (FSHPS) of general chaotic systems. Secondly, a new control scheme is proposed to study the problem of inverse full-state hybrid projective synchronization (IFSHPS) for arbitrary chaotic systems. The synchronization criterions derived in this paper are established in the form of simple algebraic conditions about the linear part of the response system and the drive system, respectively, which are very convenient to verify. In order to show the effectiveness of the proposed synchronization schemes, our approach is applied to the drive Fold discrete-time system and the controlled Lorenz discrete-time system to achieve FSHPS and IFSHPS, respectively. The remainder of this paper