Study On Projective Synchronization Of Chaotic Systems And Its Applications

Due to its wide-scope potential applications in various scientific fields, especially in chaotic secure communication, the problem of chaos synchronization has attracted increased attention from scientists and engineers. This dissertation is devoted to study on the related problem of projective synchronization, including the control scheme and some application in chaotic security communication and parameters identification. The author's main research work and contributions are as follows:In this dissertation, the four stages of development in chaos theory are briefly introduced, some existing chaos control approaches are surveyed and the key role of chaos synchronization played in chaotic communication scheme is described.Linear adaptive feedback controller is first designed to direct the scaling factor, characterized the synchronized dynamics of projective synchronization in partially linear chaotic systems, onto a desired value. Based on the adaptive projective synchronization, a secure communication scheme is proposed and proved rigorously.Impulsive control approach is adopted to realize the chaossynchronization and projective synchronization, respectively. First, we derive some conditions for (asymptotic) stability of impulsive control systems with impulses at fixed times based on a existing theory result of impulsive differential equations, and the results are used to design impulsive synchronization for Lorenz systems. Then, we first use the impulsive control approach to control the scaling factor of projective synchronization onto any desired scale.A drive-response dynamical networks model is presented and its projective synchronization is investigated. The sufficient condition of projective synchronization in drive-response network is derived. Because the scaling factor is difficult to be estimated, pinning control techniques are used to direct the scaling factor onto the desired value.The definition of projective cluster synchronization in drive-response dynamical networks is introduced and the sufficient condition on this novel synchronization scheme is derived. For the same reason, we use pinning control methods to control the scaling factors onto the desired values.The definition of full state hybrid projective synchronization is first presented. Based on the Lyapunov stability theory, a general controller is designed for full state hybrid projective synchronization of continuous-time chaotic systems and discrete-time chaotic maps, respectively. It is worthy noting that our full state hybrid projective synchronization includes complete synchronization, anti-synchronization, full state projective synchronization, partial synchronization and mismatched synchronization as special cases.Estimating the model parameters of the drive system based on the full state hybrid projective synchronization scheme is presented. The adaptive controller is designed to realize the full state projective synchronization and to identify the unknown parameters, simultaneously. In addition, the proposed adaptive controller is quite robust against the effect of noise.The full state hybrid projective synchronization of chaotic and hyper-chaotic systems with fully unknown parameters is further investigated. A unified adaptive controller and parameters update law is designed for achieving full state hybrid projective synchronization between two chaotic systems with the same and different order based on the Lyapunov stability theory. Especially, for the case of two chaotic systems with different order, two new concepts, reduced order full state hybrid projective synchronization and increased order full state hybrid projective synchronization, are also studied.The existence of hybrid projective synchronization in a five dimensional chaotic systems is experimentally found, which shows that the definition of full state hybrid projective synchronization presented before is reasonable.The general problem of Q-S synchronization between chaotic and/or hyper-chaotic systems is studied. We propose a general controller for Q-S synchronization of chaotic and/or hyper-chaotic systems based on the Lyapunov stability theorem and the theory of generalized inverse matrix. The drive and response systems studied in this part can be strictly different dynamical systems (including different dimensional systems). In addition, the proposed controller can realize the Q-S synchronization with respect to the freely selected observable variables Qi and S_i.For each proposed theory result, we give numerical simulations to illustrate the effectiveness in this dissertation.