Research Some Problems In Control And Synchronization Of Chaotic Systems

Chaotic systems are well known for their very complex nonlinear systems, and have been intensively studied in various fields such as biology, physics, chemistry, engineering and information. It has been especially noticed that the researches about the application of chaos have attracted increasing attention and have become rapidly one of the development direction. Because of such chaotic systems possess certain features, for example, high randomicity, board spectra for its Fourier transform, and hyper sensitivity to initial conditions, thus the application of chaos can be especially found in secure communications, signal processing and image processing etc.. Chaos synchronization has become the key process in applying chaos. Since 1990, chaos synchronization has achieved rapid development, and there have lots of remarkable results reported about it.This dissertation studies the sufficient condition for chaos complete synchronization, and attains the generalized projective synchronization for the same or different structure chaotic, hyper-chaotic, fractional order chaotic systems using many different methods. The main contributions are listed as follows:(1) A new sufficient condition is proposed for the complete synchronization of two coupled chaotic systems.Based on the stability theory of linear time varying continuous system, the stability issue of the complete synchronization of two coupled chaotic systems is dealt with. The instantaneous eigenvalues are used as indicators of chaos synchronization quality. A sufficient condition for synchronization is attained and the eigenvalues of the error systems have negative real parts everywhere. Firstly, the analytic method is tested respectively in nonlinear coupled Rossler and nonlinear coupled unified systems. Secondly, the linear coupled unified chaotic systems and the linear coupled generalized Lorenz-like systems are taken as examples. The complete synchronization state is stable when the range of the parameters in a vector coupling function is given.(2) Two methods for generalized projective synchronization are proposed for four dimensional chaotic systems.Firstly, the nonlinear feedback controllers are designed for synchronization two same Qi chaotic systems, and this method can be applied to solve synchronization problems of Qi chaotic systems and hyper-chaotic system. Secondly, a new simple nonlinear feedback control method is proposed based on the linear stability theory. The technique can be applied to drive the synchronization two new identical hyper-chaotic systems or two different hyper-chaotic systems. The method is tested on the hyper-chaotic Liu systems and the hyper-chaotic Wang systems.(3) A general method is studied for the modified generalized projected synchronization between two identical chaotic systems with unknown parameters.Based on the Lyapunov stability theory, the controller and update rule are investigated. The technique is proved that it can make all full states of the drive system and the response system asymptotically generalized projected synchronization and adaptive synchronization with the same phase or the anti-phase for chaotic system. Although the models used in the research are the systems with the driven system's unknown parameters or the response system's unknown parameters, the method is also applicable to the systems with known parameters. The new hyper-chaotic Chen-Qi system,Lorenz system and Chen system are taken as examples, and identify chaotic systems' unknown parameters.(4)Tracking control and generalized projective synchronization are put forward for complex Dynamos systems.Based on the system stability theory, a new tracking control method is attained to make full states of a complex dynamos system and arbitrary trajectories asymptotically generalized projective synchronization. The technique can direct the scaling factor onto a desired value. Sinusoidal waves, chaotic systems, hyper-chaotic systems and fixed points are respectively taken as an example.(5) The generalized projective synchronization of the fractional order chaotic systems with the same or different structure is firstly studied.Two kind of different controllers are designed for synchronizing response system and drive system. The analytical expression on synchronization is given based on Laplace transformation theory. This method is implemented in generalized synchronization of the fractional Liu system and Lu system. So the technique provides a new thought for synchronizing a class fraction order chaotic systems and also has higher security in secure communication.Finally, some open issues on synchronization control of chaotic systems as well as the future work are pointed out.