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

  • Xiaoxin Liao
  • Yuli Fu
  • Shengli Xie


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.


Lorenz chaotic system global attractive set positive invariant set globally exponential tracking globally exponential synchronization 


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  1. 1.
    Lorenz, Z. N., Deterministic non-periodic flow, J. Atoms Sci., 1963, 20: 130–141.CrossRefGoogle Scholar
  2. 2.
    Lorenz, E. N., The Essence of Chaos, Washington: USA University of Washington Press, 1993.MATHGoogle Scholar
  3. 3.
    Sparrow, C., The Lorenz Equations: Bifurcation, Chaos and Strange Attractors, New York, Berlin-Heidelberg: Springer Press, 1982.Google Scholar
  4. 4.
    Ruelle, D., Lorenz Attractor and Problem of Turbulence, in Lecture Notes in Mathematics, V.565, New York: Springer-Verlag, 1976.Google Scholar
  5. 5.
    Chen, G. R., Lü Jinhu, Dynamics Analysis, Control and Synchronization of Lorenz Systems Family (in Chinese), Beijing: Scientific Press, 2003.Google Scholar
  6. 6.
    Stwart, I., The Lorenz attractor exists, Nature, 2002, 406: 948–949.CrossRefGoogle Scholar
  7. 7.
    Tucker, W., The Lorenz attractor exists, C. R. Acad. Sci. Paris, 1999, 328: 119–1202.MathSciNetGoogle Scholar
  8. 8.
    Leonov, G., Reitmann, V., Attraktoreingrenzung fur Nichtlineare System, Leipzing: Teubner-Verlag, 1987.Google Scholar
  9. 9.
    Leonov, G. A., Abramovich, S. M., Bunin, A. I., Problems of nonlinear and turblent process in physics, Proc. of Second International Working Group, Kiev (in Russian), 1985, Part II, 75–77.Google Scholar
  10. 10.
    Leonov, G., Bunin, A., Koksch, N., Attractor localization of the Lorenz system, ZAMM, 1987, 67: 649–656.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Pecora, L. M., Carroll, L. T. L., Synchronization in chaotic circuits, Phys. Rev. Lett., 1990, 64(8): 821–824.CrossRefMathSciNetGoogle Scholar
  12. 12.
    Pecora, L. M., Carroll, L. T. L., Driving systems with chaotic signals, Phys. Rev. A, 1991, 44(4): 2374–2378.CrossRefGoogle Scholar
  13. 13.
    Chen, G., Dong, X., From Chaos to Order, Singapore: World Scientific Pub. Co, 1988.Google Scholar
  14. 14.
    Guan Xinping, Fan Zhengping, Chen Cailian et al., Chaos Control and Its Applications in Secure Communications (in Chinese), Beijing: National Defense Industrial Press, 2002.Google Scholar
  15. 15.
    Brown, R., Kocorev, L., A unified definition of synchronization for dynamic systems, Chaos, 2000, 10: 344–349.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Liao, X. X., Chen, G. R., Wang, H. O., One global synchronization of chaotic systems, dynamic continuous, Discrete and Impulsive Systems, Ser. B., 2003, 10(6): 865–872.MATHMathSciNetGoogle Scholar
  17. 17.
    Liao, X. X., Chen, G. R., Chaos synchronization of general Lurie systems via time-lag feedback control, Int. J. Bifurcation and Chaos, 2003, 13(1): 207–213.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Liao, X. X., Chen, G. R., On feedback-controlled synchronization of chaotic systems, Int. J. of Systems Science, 2003, 34(7): 454–461.MathSciNetGoogle Scholar
  19. 19.
    Liao, X. X., Chen, G. R., Some new results on chaos synchronization, Control Theory & Applications, 2003, 20(2): 254–258.MathSciNetGoogle Scholar
  20. 20.
    Zheng, W. M., Hao, B. L., Applied Symbolic Dynamics (in Chinese), Shanghai: Shanghai Publish of Science,, Technology and Education, 1994.Google Scholar
  21. 21.
    Liao, X. X., Wang Jun, Global disspativity of continuous-time recurrent neural networks, Phys. Rev. E, 2003, 68: 0161181–0161187.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Lefchetz, S., Differential Equations: Geometric Theory, New York: Interscience Publishers, 1963.Google Scholar
  23. 23.
    Huang Lin, Basic Theory of Stability and Robustness (in Chinese), Beijing: Scientific Press, 2003.Google Scholar
  24. 24.
    Liao, X. X., Absolute Stability of Nonlinear Control Systems, Dordrecht: Kluwer Academic Pub., 1993.MATHGoogle Scholar
  25. 25.
    Liao, X. X., Theory and Applications of Stability for Dynamic Systems (in Chinese), Beijing: National Defense Industrial Press, 2000.Google Scholar

Copyright information

© Science in China Press 2005

Authors and Affiliations

  1. 1.Department of Control Science & Control EngineeringHuazhong University of Science & TechnologyWuhanChina
  2. 2.School of AutomationWuhan University of Science & TechnologyWuhanChina
  3. 3.School of InformationCentral South University of Economy, Politics and LawWuhanChina
  4. 4.School of Electronics & Information EngineeringSouth China University of TechnologyGuangzhouChina

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