Abstract
The hybrid synchronization problem of a class of chaotic systems is investigated in this paper. Firstly, the existence of hybrid synchronization problems in such systems is proved theoretically by a proposed necessary and sufficient condition. That is, the hybrid synchronization problem is equivalent to solve a group of nonlinear algebraic equations about α . It is interesting that one value of α indicates one type of synchronization. Secondly, all solutions for the hybrid synchronization problem are obtained by finding solutions of all the above equations about α . Thirdly, an universal control method is proposed to realize such hybrid synchronization problems. Finally, illustrative examples are provided to verify the validity and effectiveness of the obtained results.
Highlights
It is believed that the strange attractor and butterfly effect proposed by Lorenz in 1963 could help people begin to keep an eye on chaotic dynamics and even be used in nonlinear science.From on, chaotic behavior has been extensively analyzed in many fields, e.g., engineering, ecology, biology, economics, and so on, even in the social sciences
The main contributions are summarized as follows: (1) A new unified definition named hybrid synchronization is presented; (2) A necessary and sufficient condition is proposed, which is used to proof the existence of the hybrid synchronization problem of the chaotic systems
The hybrid synchronization problem is equivalent to solve a group of nonlinear algebraic equations about α, i.e., one value of α indicates one type of synchronization problem; (3) All solutions for the hybrid synchronization problem are obtained by finding solutions of all algebraic equations about α; (4) An universal control method is designed to ensure the realization the hybrid synchronization problem
Summary
It is believed that the strange attractor and butterfly effect proposed by Lorenz in 1963 could help people begin to keep an eye on chaotic dynamics and even be used in nonlinear science. As is stated in [9], the existence of hybrid synchronization can effectively enhance security in communication Due to these reasons, to design the proper controller to reach the hybrid synchronization has become an important problem. The main contributions are summarized as follows: (1) A new unified definition named hybrid synchronization is presented; (2) A necessary and sufficient condition is proposed, which is used to proof the existence of the hybrid synchronization problem of the chaotic systems. Based on this condition, the hybrid synchronization problem is equivalent to solve a group of nonlinear algebraic equations about α, i.e., one value of α indicates one type of synchronization problem;.
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