iBet uBet web content aggregator. Adding the entire web to your favor.
iBet uBet web content aggregator. Adding the entire web to your favor.



Link to original content: https://unpaywall.org/10.1007/S11432-010-4032-7
Sufficient and necessary conditions for Lyapunov stability of Lorenz system and their application | Science China Information Sciences Skip to main content
Springer Nature Link
Log in

Sufficient and necessary conditions for Lyapunov stability of Lorenz system and their application

  • Research Papers
  • Published:
Science China Information Sciences Aims and scope Submit manuscript

Abstract

The sufficient and necessary conditions for Lyapunov stability of the zero equilibrium point of Lorenz system are discussed, and some brief criteria are presented for globally exponential stability, globally asymptotical stability and instability. Furthermore, the behavior of the non-zero equilibrium point of Lorenz system is also investigated, and several sufficient and necessary conditions are provided for locally exponential stability and instability. The established theorems in this paper develop and extend the existing achievements on Lyapunov stability of Lorenz system. In conclusion, by applying the obtained results, some less conservative feedback-control laws are derived to guarantee the globally exponential stability of the chaos control of Chen system, Lü system, Yang-Chen system and Yu-Xia Li system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  Google Scholar 

  2. Lorenz E N. The Essence of Chaos. Washington: University of Washington Press, 1993

    Book  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  5. Chen G R, Lv J H. Dynamics Analysis, Control and Synchronization of Lorenz System (in Chinese). Beijing: Science Press, 2003

    Google Scholar 

  6. Yang W L, Wang T N. Theory and Application of Nonlinear Dynamics (in Chinese). Beijing: National Defense Industry Press, 2007

    Google Scholar 

  7. Warwick T. A rigorous ODE solver and Smale’s 14th problem. Found Comput Math, 2002, 2: 53–117

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  9. Leonov G A. Bound for attractors and the existence of homoclinic orbits in the Lorenz system. J Appl Math Mech, 2001, 65: 19–32

    Article  MathSciNet  Google Scholar 

  10. Liao X X, Fu Y L, Xie S L. 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-Inf Sci, 2005, 48: 304–321

    Article  MATH  MathSciNet  Google Scholar 

  11. Li D M, Lu J A, Wa X Q, et al. Estimating the bounded for the Lorenz family of chaotic systems. Chaos Solut Fract, 2005, 23: 529–534

    Article  MATH  Google Scholar 

  12. Yu P, Liao X X. New estimations for globally attractive and positive invariant set of the family of the Lorenz systems. Int J Bifur Chaos, 2006, 16: 3383–3390

    Article  MATH  MathSciNet  Google Scholar 

  13. Liao X X, Fu Y L, Xie S L, et al. Globally exponentially attractive sets of family of Lorenz systems. Sci China Ser F-Inf Sci, 2008, 51: 283–292

    Article  MATH  MathSciNet  Google Scholar 

  14. Li Y X. Research on Anticontrol of Hyperchaos for Continuous-time Systems (in Chinese). Ph.d. Dissertation, Gruangzhou: Guangdong University of Technology, 2005

    Google Scholar 

  15. Yang Q, Chen G. A chaotic system with one saddle and two stable node-foci. Int J Bifur Chaos, 2008, 18: 1393–1414

    Article  MATH  Google Scholar 

  16. Liao X X. Talking on the theory, methods and application of Lyapunov stability (in Chinese). J Nanjing Univ Inf Sci Tech: Nat Sci Ed, 2009, 1: 1–15

    Google Scholar 

  17. Liu Y G. Global stabilization by output feedback for a class of nonlinear systems with uncertain control coefficients and unmeasured states dependent growth. Sci China Ser F-Inf Sci, 2008, 51: 1508–1520

    Article  MATH  Google Scholar 

  18. Luo Q, Deng F Q, Mao X R, et al. Theory and application of stability for stochastic reaction diffusion systems. Sci China Ser F-Inf Sci, 2008, 51: 158–170

    Article  MATH  MathSciNet  Google Scholar 

  19. Chen Y Y, Luo Q. Global exponential stability in Lagrange sense for a class of neural networks (in Chinese). J Nanjing Univ Inf Sci Tech: Nat Sci Ed, 2009, 1: 50–58

    Google Scholar 

  20. Luo Y P, Xia W H, Liu G R, et al. W1,2(Ω)-and X1,2(Ω)-stability of reaction-diffusion cellular neural networks with delay. Sci China Ser F-Inf Sci, 2008, 51: 1980–1991

    Article  MathSciNet  Google Scholar 

  21. Liao X X. Theory and Application of Stability. Wuhan: Huazhong Normal University Press, 2001

    Google Scholar 

  22. Liao X X. Theory, Method and Application of Stability. 2nd ed. Wuhan: Huazhong University of Science & Technology Press, 2010

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Information and Control, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Qi Luo

  2. Department of Control Science and Engineering, Huazhong University of Science and Technology, Image Processing and Intelligent Control Key Laboratory of Education Ministry of China, Wuhan, 430074, China

    XiaoXin Liao & ZhiGang Zeng

Authors
  1. Qi Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. XiaoXin Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. ZhiGang Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qi Luo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luo, Q., Liao, X. & Zeng, Z. Sufficient and necessary conditions for Lyapunov stability of Lorenz system and their application. Sci. China Inf. Sci. 53, 1574–1583 (2010). https://doi.org/10.1007/s11432-010-4032-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11432-010-4032-7

Keywords

Search

Navigation