| Nonlinear science is a foundational discipline which concerns the common properties of nonlinear phenomena. Chaos theory is one important subdiscipline of nonlinear science. In this paper, some problems about chaotic control, chaotic synchronization and its application in secure communication were studied by theoretical analysis and numerical simulations. The main work in this paper can be summarized as follows.Chaotic control:(1) Based on the stability theory of discrete system, state feedback method is used to stabilize unstable low-periodic orbits of the Logistic map and the coupled Logistic map, and the bifurcation points of these two system are controlled successfully. (2) Direct feedback, adaptive control and parametric perturbation are used to control Liu system, and Liu system can be guided to equilibrium, periodic motion, even hyperchaos. (3) A new hyperchaotic system is designed by adding a nonlinear controller to Lorenz chaotic system, and the relevant circuit realization is given too. (4) An universal tracking controller is designed for a class of chaotic systems, and the controller can make the output signal track all kinds of reference signals.Complete synchronization:(1) Universal adaptive synchronization controller and parameter update rule are presented for different-structure chaotic systems. With this method, a universal scheme for adaptive chaotic synchronization between uncertain systems is presented. (2) Through several examples, how to realize unidirectional coupling chaotic synchronization by less channels is discussed. (3) A feasible scheme for the impulsive synchronization of hyperchaotic Lu systems is proposed, and single channel transmitting is realized too. (4) Based on the stability theory of fractional order systems, the dynamic behavior of fractional order Liu system is studied, and a theoretical calculation method is proposed to obtain the lowest chaotic order of a fractional order system. Besides, a simple criterion for the synchronization of two identical fractional order chaotic systems is presented.Generalized synchronization:(1) The anti-synchronization of hyperchaotic Chen system is studied respectively by active control method, global control method and variable replacement method, and the comparison of these three methods is given too. (2) With variable replacement method, the projective synchronization of hyperchaotic Chen systems is realized by single channel. (3) Based on active control, a scheme is proposed to realize generalized synchronization not only of fractional order systems with same dimension, but also of systems with different dimensions. ODE systems can be seemed as particular fractional order systems, so this scheme is available to generalized synchronization of ODE systems too.Chaotic secure communication:(1) Based on the unidirectional coupling synchronization of hyperchaotic systems, a chaotic masking secure communication scheme is designed, and the relevant numerical simulation is given. (2) According to parameter modulation theory, a new secure communication scheme is proposed. The useful continuous signal can be transmitted successfully. Choose different-frequency signals as "0" and "1", then this scheme can be used to transmit digital signals via a filter. Numerical simulations show the effectiveness of the digital secure communication.This research is supported by the National Natural Science Foundation of China (No: 60573172,60973152), the Superior University doctor subject special scientific research foundation of China (No:20070141014) and the National Natural Science Foundation of Liaoning province (No:20082165). |
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