On Some Synchronization Problems Of Chaotic Systems

Chaos is a kind of special complicated nonlinear system, which is ubiquitous in nature. And it has been intensively applied in various fields such as biology, physics, chemistry, engineering and information. Because of such chaotic systems progress certain features, for example, high randomicity and hyper sensitivity to initial conditions, the application of chaos can be especially found in secure communications, signal progressing and image progressing etc. Chaos synchronization has become the key technique in secure communications. Since 1990s, chaos synchronization has achieved rapid development, and there have lots of remarkable results reported about it.The complete synchronization, generalized projective synchronization, modified projective synchronization and generalized synchronization of chaotic/hyper-chaotic systems in this dissertation. The main contributions are listed as follows:Chaos synchronization and anti-synchronization of two different chaotic systems are studied. When the parameters are known in advance, the synchronization and anti-synchronization controllers are designed by the active control method. The method of designing controller with active control is generalized. When the parameters are fully unknown, the adaptive controllers and the adaptive laws of parameters are given based on the Lyapunov stability theory and adaptive control method. The drive system and response system are interchanged during the controllers designing. The relationship of the controllers and the adaptive laws which they are interchanged are obtained. The numerical simulation results verify the effectiveness of the proposed schemes.The problem of generalized projective synchronization of different chaotic systems is investigated. Firstly, the sufficient stability conditions are presented for a nonlinear system with the special antisymmetric structure. Secondly, by using the direct design method, a controller is designed for the error system. The error system is transformed into a nonlinear system with the special antisymmetric structure under the controller. Then the state of error system is asymptotically stable at the origin. That is to say, the generalized projective synchronization between response system and drive system is realized. Finally, the corresponding numerical simulation results demonstrate the effectiveness of the proposed schemes.When the chaotic system parameters are known and unknown, the generalized projective synchronization for the chaotic systems is investigated. An active controller is designed for the chaotic system with known parameters. Then the drive system and the response system are interchanged. The relationship between the active controllers is obtained before and after interchanging the drive and response systems. Combining the adaptive control and active control method, the generalized projective synchronizations controllers of chaotic systems are designed, if the parameters are fully unknown or partially unknown. Numerical simulations are presented to demonstrate the effective-ness of the proposed schemes.The external disturbance is unavoided, when synchronization of chaotic systems is realized. When the drive system and the response system suffer disturbances, the modified projective synchronization is studied. Based on the active sliding mode control technique, the sufficient conditions for the modified projective synchronization both the non-identical and identical chaotic systems are presented. Based on the Backstepping method, the modified projective synchronization is realized by using less controllers than states of response system. The corresponding numerical simulations are provided to illuminate the effectiveness of the proposed design schemes.The generalized synchronization of chaotic systems with different order is studied. The definition of finite-time generalized synchronization is given. Based on the finite-time stability theory, two kinds of control strategies are proposed to realize the generalized synchronization of chaotic systems with different order in finite time. Besides the relation between synchronized time and the control parameterβis obtained, when the initial states of systems are given. The generalized synchronization of different order chaotic systems with fully unknown parameters is studied. Based on the adaptive control method, an adaptive controller and parameters update laws are designed. Corresponding numerical simulations are presented to show the effectiveness and feasibility of the proposed schemes.Lastly, the summary of the whole dissertation is given and the research directions in future are put forward.