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Link to original content: https://doi.org/10.1007/s11432-008-0024-2
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Globally exponentially attractive sets of the family of Lorenz systems

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Abstract

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.

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References

  1. Lorenz E N. Deterministic non-periodic flow. J Atoms Sci, 1963, 20: 130–141

    Article  Google Scholar 

  2. Lorenz E N. The essence of Chaos. Washington: Univ of Washington Press, 1993

    MATH  Google Scholar 

  3. Sparrow C. The Lorenz equations. Bifurcation, Chaos and Strange Attractors. New York: Springer-Verlag, 1976

    Google Scholar 

  4. Stewart I. The Lorenz attractor exists. Nature, 2002, 406: 948–949

    Article  Google Scholar 

  5. Chen G, Lü J. Dynamics Analysis, Control and Synchronization of Lorenz Families (in Chinese). Beijing: Science Press, 2003

    Google Scholar 

  6. Tucker W. A rigorous ODE solver and Smale’s 14th problem. Found Comput Math, 2002, 2: 53–117

    MATH  MathSciNet  Google Scholar 

  7. Leonov G A, Bunin A L, Kokxh N. Attractor localization of the Lorenz system. ZAMM, 1987, 67: 649–656

    Article  MATH  Google Scholar 

  8. Leonov G A. Bound for attractors and the existence of Homoclinic orbits in the Lorenz system. J Appl Math Mech, 2001, 65(1): 19–32

    Article  MathSciNet  Google Scholar 

  9. Liao X X, Fu Y, Xie S. On the new results of global attractive sets and positive invariant sets of the Lorenz chaotic system and the applications to chaos control and synchronization. Sci China Ser F-Inf Sci, 2005, 48(3): 304–321

    Article  MATH  MathSciNet  Google Scholar 

  10. Yu P, Liao X X. New estimates for globally attractive and positive invariant set of the family of the Lorenz systems. Int J Bifurcation & Chaos, 2006, 16(11) (in press)

  11. Li D, Lu J, Wu X, et al. Estimating the bounded for the Lorenz family of chaotic systems. Chaos, Solitons and Fractals, 2005, 23: 529–534

    Article  MATH  MathSciNet  Google Scholar 

  12. Leonov G A. On estimates of attractors of Lorenz system. Vestnik leningradskogo universiten matematika, 1988, 21(1): 32–37

    MathSciNet  Google Scholar 

  13. Yu P, Liao X X. Globally attractive and positive invariant set of the Lorenz system. Int J Bifurcation & Chaos, 2006, 16(3): 757–764

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Control Science & Engineering, Huazhong University of Science & Technology, Wuhan, 430074, China

    Liao XiaoXin

  2. School of Automation, Wuhan University of Science & Technology, Wuhan, 430070, China

    Liao XiaoXin

  3. School of Electronics & Information Engineering, South China University of Technology, Guangzhou, 519640, China

    Fu YuLi & Xie ShengLi

  4. Department of Applied Mathematics, The University of Western Ontario London, Ontario, N6A 5B7, Canada

    Yu Pei

Authors
  1. Liao XiaoXin
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  2. Fu YuLi
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  3. Xie ShengLi
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  4. Yu Pei
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Corresponding author

Correspondence to Fu YuLi.

Additional information

Supported partly by the National Natural Science Foundation of China (Grant Nos. 60474011 and 60274007), the National Natural Science Foundation of China for Excellent Youth (Grant No. 60325310), the Guangdong Province Science Foundation for Program of Research Team (Grant No. 04205783), the Natural Science Fund of Guangdong Province, China (Grant No. 05006508), and the Natural Science and Engineering Research Council of Canada (Grant No. R2686A02)

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Liao, X., Fu, Y., Xie, S. et al. Globally exponentially attractive sets of the family of Lorenz systems. Sci. China Ser. F-Inf. Sci. 51, 283–292 (2008). https://doi.org/10.1007/s11432-008-0024-2

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  • DOI: https://doi.org/10.1007/s11432-008-0024-2

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