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Journal of Systems Science and Complexity

, Volume 24, Issue 3, pp 433–448 | Cite as

Bidirectionally coupled synchronization of the generalized Lorenz systems

  • Juan ChenEmail author
  • Jun-an Lu
  • Xiaoqun Wu
Article

Abstract

Wu, Chen, and Cai (2007) investigated chaos synchronization of two identical generalized Lorenz systems unidirectionally coupled by a linear state error feedback controller. However, bidirectional coupling in real life such as complex dynamical networks is more universal. This paper provides a unified method for analyzing chaos synchronization of two bidirectionally coupled generalized Lorenz systems. Some sufficient synchronization conditions for some special coupling matrices (diagonal matrices, so-called dislocated coupling matrices, and so on) are derived through rigorously mathematical theory. In particular, for the classical Lorenz system, the authors obtain synchronization criteria which only depend upon its parameters using new estimation of the ultimate bounds of Lorenz system (Chaos, Solitons, and Fractals, 2005). The criteria are then applied to four typical generalized Lorenz systems in the numerical simulations for verification.

Key words

Bidirectionally-coupled chaos generalized lorenz system synchronization ultimate bound 

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References

  1. [1]
    A. Vanēček and S. Čelikovský, Control Systems: From Linear Analysis to Synthesis of Chaos, Prentice-Hall, London, 1996.zbMATHGoogle Scholar
  2. [2]
    S. Čelikovský and G. R. Chen, On a generalized Lorenz canonical form of chaotic systems, Int. J. Bifurcation and Chaos, 2002, 12: 1789–1812.zbMATHCrossRefGoogle Scholar
  3. [3]
    S. Čelikovský and G. R. Chen, On a generalized Lorenz canonical form of chaotic systems, Chaos, Solitons and Fractals, 2005, 26: 1271–1276.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    G. R. Chen and J. H. Lü, Dynamics of the Lorenz System Family: Analysis, Control and Synchronization, Science Press, Beijing, 2003.Google Scholar
  5. [5]
    E. N. Lorenz, Deterministic non-periodic flows, J. Atmos Sci., 1963, 20: 130–141.CrossRefGoogle Scholar
  6. [6]
    G. R. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation and Chaos, 1999, 9: 1465–1466.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    J. H. Lü and G. R. Chen, A new chaotic attractor coined, Int. J. Bifurcation and Chaos, 2002, 12(3): 659–661.zbMATHCrossRefGoogle Scholar
  8. [8]
    J. H. Lü, G. R. Chen, D. Z. Cheng, and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifurcation and Chaos, 2002, 12(12): 2917–2926.zbMATHCrossRefGoogle Scholar
  9. [9]
    J. H. Lü, T. Zhou, and S. Zhang, Chaos synchronization between linearly coupled chaotic system, Chaos, Solitons and Fractals, 2002, 14: 529–541.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    J. Zhou, J. Lu, and X. Wu, Linearly and nonlinearly bidirectionally coupled synchronization of hyperchaotic systems, Chaos, Solitons and Fractals, 2007, 31: 230–235.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    X. Wu, G. Chen, and J. Cai, Chao synchronization of the master-slave generalized Lorenz systems via linear state error feedback control, Physica D, 2007, 229: 52–80.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    L. Chen and J. Lu, Cluster synchronization in a complex dynamical network with two nonidentical clusters, Journal Systems Science & Complexity, 2008, 21(1): 20–33.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    X. P. Han, J. Lu, and X. Q. Wu, Synchronization of impulsively coupled systems, International Journal of Bifurcation and Chaos, 2008, 18(5): 1539–1549.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    D. Li, J. Lu, X. Wu, and G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos, Solitons and Fractals, 2005, 23: 529–534.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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