Abstract
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.
Similar content being viewed by others
References
Lorenz, Z. N., Deterministic non-periodic flow, J. Atoms Sci., 1963, 20: 130–141.
Lorenz, E. N., The Essence of Chaos, Washington: USA University of Washington Press, 1993.
Sparrow, C., The Lorenz Equations: Bifurcation, Chaos and Strange Attractors, New York, Berlin-Heidelberg: Springer Press, 1982.
Ruelle, D., Lorenz Attractor and Problem of Turbulence, in Lecture Notes in Mathematics, V.565, New York: Springer-Verlag, 1976.
Chen, G. R., Lü Jinhu, Dynamics Analysis, Control and Synchronization of Lorenz Systems Family (in Chinese), Beijing: Scientific Press, 2003.
Stwart, I., The Lorenz attractor exists, Nature, 2002, 406: 948–949.
Tucker, W., The Lorenz attractor exists, C. R. Acad. Sci. Paris, 1999, 328: 119–1202.
Leonov, G., Reitmann, V., Attraktoreingrenzung fur Nichtlineare System, Leipzing: Teubner-Verlag, 1987.
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.
Leonov, G., Bunin, A., Koksch, N., Attractor localization of the Lorenz system, ZAMM, 1987, 67: 649–656.
Pecora, L. M., Carroll, L. T. L., Synchronization in chaotic circuits, Phys. Rev. Lett., 1990, 64(8): 821–824.
Pecora, L. M., Carroll, L. T. L., Driving systems with chaotic signals, Phys. Rev. A, 1991, 44(4): 2374–2378.
Chen, G., Dong, X., From Chaos to Order, Singapore: World Scientific Pub. Co, 1988.
Guan Xinping, Fan Zhengping, Chen Cailian et al., Chaos Control and Its Applications in Secure Communications (in Chinese), Beijing: National Defense Industrial Press, 2002.
Brown, R., Kocorev, L., A unified definition of synchronization for dynamic systems, Chaos, 2000, 10: 344–349.
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.
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.
Liao, X. X., Chen, G. R., On feedback-controlled synchronization of chaotic systems, Int. J. of Systems Science, 2003, 34(7): 454–461.
Liao, X. X., Chen, G. R., Some new results on chaos synchronization, Control Theory & Applications, 2003, 20(2): 254–258.
Zheng, W. M., Hao, B. L., Applied Symbolic Dynamics (in Chinese), Shanghai: Shanghai Publish of Science,, Technology and Education, 1994.
Liao, X. X., Wang Jun, Global disspativity of continuous-time recurrent neural networks, Phys. Rev. E, 2003, 68: 0161181–0161187.
Lefchetz, S., Differential Equations: Geometric Theory, New York: Interscience Publishers, 1963.
Huang Lin, Basic Theory of Stability and Robustness (in Chinese), Beijing: Scientific Press, 2003.
Liao, X. X., Absolute Stability of Nonlinear Control Systems, Dordrecht: Kluwer Academic Pub., 1993.
Liao, X. X., Theory and Applications of Stability for Dynamic Systems (in Chinese), Beijing: National Defense Industrial Press, 2000.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liao, X., Fu, Y. & Xie, S. 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. Sci China Ser F 48, 304–321 (2005). https://doi.org/10.1360/04yf0087
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1360/04yf0087