Skip to main content

Advertisement

SpringerLink
Log in

Bounds for the fast–slow Lorenz–Stenflo system

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, the dynamical behaviors of the fast–slow Lorenz–Stenflo system are considered based on generalized Lyapunov function theory with integral inequalities. Explicit estimations of the ultimate bounds are derived. The meaningful contribution of this article is that not only do we get the ultimate boundedness of solutions of the fast–slow Lorenz–Stenflo system, but we also get the rate of the trajectories of the system going from the exterior of the trapping set to the interior of the trapping set. Computer simulation results show that the proposed method is effective.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Lorenz, E.: Deterministic non-periods flows. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  2. Kuznetsov, N., Mokaev, T., Vasilyev, P.: Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor. Commun. Nonlinear Sci. Numer. Simul. 19(4), 1027–1034 (2014)

    Article  MathSciNet  Google Scholar 

  3. Leonov, G.: Bounds for attractors and the existence of homoclinic orbits in the Lorenz system. J. Appl. Math. Mech. 65(1), 19–32 (2001)

    Article  MathSciNet  Google Scholar 

  4. Mu, C., Zhang, F., Shu, Y., Zhou, S.: On the boundedness of solutions to the Lorenz like family of chaotic systems. Nonlinear Dyn. 67(2), 987–996 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Leonov, G., Bunin, A., Koksch, N.: Attractor localization of the Lorenz system. Z. Angew. Math. Mech. 67, 649–656 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, G., Lü, J.: Dynamical Analysis. Control and Synchronization of the Lorenz Systems Family. Science Press, Beijing (2003)

  7. Leonov, G.: General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu–Morioka, Lu and Chen systems. Phys. Lett. A 376, 3045–3050 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bragin, V., Vagaitsev, V., Kuznetsov, N., Leonov, G.: Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J. Comput. Syst. Sci. Int. 50, 511–543 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Leonov, G., Kuznetsov, N.: Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos Appl. Sci. Eng. 23, 1330002 (2013)

  10. Leonov, G., Kuznetsov, N., Kiseleva, M., Solovyeva, E., Zaretskiy, A.: Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn. 77, 277–288 (2014)

    Article  Google Scholar 

  11. Liu, H., Wang, X., Zhu, Q.: Asynchronous anti-noise hyper chaotic secure communication system based on dynamic delay and state variableq switching. Phys. Lett. A 375(30–31), 2828–2835 (2011)

    Article  MATH  Google Scholar 

  12. Wang, P., Li, D., Wu, X., Lü, J., Yu, X.: Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 21, 2679–2694 (2011)

    Article  MATH  Google Scholar 

  13. Elsayed, E.: Solution and attractivity for a rational recursive sequence. Discret. Dyn. Nat. Soc. 2011, 1–17 (2011)

    Article  Google Scholar 

  14. Wang, X., Wang, M.: Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos Interdiscip. J. Nonlinear Sci. 17(3), 033106 (2007)

    Article  Google Scholar 

  15. Elsayed, E.: Solutions of rational difference system of order two. Math. Comput. Model. 55, 378–384 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bilotta, E., Chiaravalloti, F., Pantano, P.: Synchronization and waves in a ring of diffusively coupled memristor-based Chua’s circuits. Acta. Appl. Math. doi:10.1007/s10440-014-9919-7

  17. Elsayed, E.: Solution for systems of difference equations of rational form of order two. Comp. Appl. Math. doi:10.1007/s40314-013-0092-9

  18. Elsayed, E.: Behavior and expression of the solutions of some rational difference equations. J. Comput. Ana.l Appl. 15, 73–81 (2013)

    MATH  MathSciNet  Google Scholar 

  19. Zhang, F., Shu, Y., Yang, H., Li, X.: Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization. Chaos Solitons Fractals 44, 137–144 (2011)

    Article  MATH  Google Scholar 

  20. Zhang, F., Mu, C., Li, X.: On the boundedness of some solutions of the Lü system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 22, 1250015 (2012)

    Article  MathSciNet  Google Scholar 

  21. Zhang, F., Li, Y., Mu, C.: Bounds of solutions of a kind of hyper-chaotic systems and application. J. Math. Res. Appl. 33(3), 345–352 (2013)

    MATH  MathSciNet  Google Scholar 

  22. Zhang, F., Shu, Y., Yao, X.: The dynamical analysis of a disk dynamo system and its application in chaos synchronization. Acta. Math. Appl. Sin. 36(2), 193–203 (2013)

    MATH  MathSciNet  Google Scholar 

  23. Zhang, F., Mu, C. et al.: The dynamical analysis of a new chaotic system and simulation. Math. Methods Appl. Sci. 37, 1838–1846 (2014)

  24. Zhang, F., Shu, Y., Yang, H.: Bounds for a new chaotic system and its application in chaos synchronization. Commun. Nonlinear Sci. Numer. Simul. 16, 1501–1508 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Liao, X., Fu, Y., Xie, S., Yu, P.: Globally exponentially attractive sets of the family of Lorenz systems. Sci. China Ser. F 51, 283–292 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Pogromsky, A., Santoboni, G., Nijmeijer, H.: An ultimate bound on the trajectories of the Lorenz system and its applications. Nonlinearity 16, 1597–1605 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Stenflo, L.: Generalized Lorenz equations for acoustic-gravity waves in the atmosphere. Phys. Scr. 53, 83–84 (1996)

  28. Han, X., Jiang, B., Bi, Q.: Analysis of the fast-slow Lorenz-Stenflo system. Chin. Phys. B 58(7), 4408–4414 (2009)

    MathSciNet  Google Scholar 

  29. Leonov, G., Kuznetsov, N., Vagaitsev, V.: Localization of hidden Chua’s attractors. Phys. Lett. A 375, 2230–2233 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  30. Leonov, G., Boichenko, V.: Lyapunov’s direct method in the estimation of the Hausdorff dimension of attractors. Acta Appl. Math. 26, 1–60 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  31. Leonov, G., Ponomarenko, D., Smirnova, V.: Frequency Domain Methods for Non-linear Analysis: Theory and Applications. World Scientific, Singapore (1996)

    Google Scholar 

  32. Leonov, G., Kuznetsov, N., Vagaitsev, V.: Hidden attractor in smooth Chua systems. Phys. D 241(18), 1482–1486 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  33. Leonov, G., Kuznetsov, N.: Time-varying linearization and the Perron effects. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17(4), 1079–1107 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  34. Kuznetsov, N., Leonov, G.: On stability by the first approximation for discrete systems. In: 2005 International Conference on Physics and Control, 2005. Phys Con 2005, Proceedings 2005, (1514053), pp. 596–599 (2005)

Download references

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Nos: 11371384, 61370145 and 61173183), the Fundamental Research Funds for the Central Universities (No. CDJXS11100026) and the Basic and Advanced Research Project of CQCSTC (No. cstc2014jcyjA00040). The authors wish to thank the editors and reviewers for their conscientious reading of this paper and their numerous comments for improvement which were extremely useful and helpful in modifying the paper. We also thank Guanrong Chen in City University of Hong Kong for his help with us.

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People’s Republic of China

    Fuchen Zhang

  2. Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Xingyuan Wang

  3. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Chunlai Mu

  4. School of Foreign Languages, Southwest Petroleum University, Chengdu, 610500, Sichuan, People’s Republic of China

    Guangyun Zhang

Authors
  1. Fuchen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xingyuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chunlai Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guangyun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fuchen Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, F., Wang, X., Mu, C. et al. Bounds for the fast–slow Lorenz–Stenflo system. Nonlinear Dyn 79, 539–547 (2015). https://doi.org/10.1007/s11071-014-1685-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1685-6

Keywords

Search

Navigation