On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization

作     者：LIAO Xiaoxin 1, 2, 3 , FU Yuli 4 & XIE Shengli 4 1. Department of Control Science & Control Engineering, Huazhong University of Science & Technology, Wuhan 430074, China 2. School of Automation, Wuhan University of Science & Technology, Wuhan 430070, China 3. School of Information, Central South University of Economy, Politics and Law, Wuhan 430064, China 4. School of Electronics & Information Engineering, South China University of Technology, Guangzhou 510640, China Correspondence should be addressed to Liao Xiaoxin (email: xiaoxin_liao@***) 

作者机构：Department of Control Science & Control Engineering Huazhong University of Science & Technology Wuhan China School of Automation Wuhan University of Science & Technology Wuhan China School of Information Central South University of Economy Politics and Law Wuhan China School of Electronics & Information Engineering South China University of Technology Guangzhou China 

出 版 物：《Science in China(Series F)》 (中国科学（F辑英文版）)

年 卷 期：2005年第48卷第3期

页      面：304-321页

核心收录：

学科分类：11[军事学] 0810[工学-信息与通信工程] 1105[军事学-军队指挥学] 08[工学] 081002[工学-信号与信息处理] 110503[军事学-军事通信学] 

基　　金：This work was supported by the National Natural Science Foundation of China(Grant Nos.60274007,60474011) the Guangdong Povince Science Foundation for Program of Research Team(Grant No.04205783) 

主　　题：Lorenz chaotic system global attractive set positive invariant set globally exponential tracking globally exponential synchronization. 

摘      要：Constructing a family of generalized Lyapunov functions, a new method is proposed to obtain new global attractive set and positive invariant set of the Lorenz chaotic system. The method we proposed greatly simplifies the complex proofs of the two famous estimations presented by the Russian scholar Leonov. Our uniform formula can derive a series of the new estimations. Employing the idea of intersection in set theory, we extract a new Leonov formula-like estimation from the family of the estimations. With our method and the new estimation, one can confirm that there are no equilibrium, periodic solutions, almost periodic motions, wandering motions or other chaotic attractors outside the global attractive set. The Lorenz butterfly-like singular attractors are located in the global attractive set only. This result is applied to the chaos control and chaos synchronization. Some feedback control laws are obtained to guarantee that all the trajectories of the Lorenz systems track a periodic solution, or globally stabilize an unstable (or locally stable but not globally asymptotically stable) equilibrium. Further, some new global exponential chaos synchronization results are presented. Our new method and the new results are expected to be applied in real secure communication systems.