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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

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science in china(Series F) 2005年第3期48卷 304-321页

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 4.0074. 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@***) 1. department of control {2. & control engineering Huazhong University of Science & Technology 430074 Wuhan China 2. School of Automation Wuhan University of Science & Technology 430070 Wuhan China 3. School of Information Central South University of Economy Politics and Law 430064 Wuhan China 4. School of Electronics & Information engineering South China University of Technology 510640 Guangzhou China

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.

关键词： Lorenz chaotic system global attractive set positive invariant set globally exponential tracking globally exponential synchronization.

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Globally exponentially attractive sets of the family of Lorenz systems

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science in china(Series F) 2008年第3期51卷 283-292页

作者： liao xiaoxin fu {2. xie shengli YU Pei department of control {2. & engineering Huazhong University of Science & Technology Wuhan 430074 China School of Automation Wuhan University of Science & Technology Wuhan 430070 China School of Electronics & information engineering South China University of Technology Guangzhou 519640 China department of Applied Mathematics The University of Western Ontario London Ontario N6A 5B7 Canada

In this paper, the concept of globally exponentially attractive set is proposed and used to consider the ultimate bounds of the family of Lorenz systems with varying parameters. Explicit estimations of the ultimate bounds are derived. The results presented in this paper contain all the existing results as special cases. In particular, the critical cases, b→ 1^＋ and a→0^＋, for which the previous methods failed, have been solved using a unified formula.

关键词： the family of Lorenz systems globally exponentially attractive set Lagrange stability generalized Lyapunov function

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