Skip to main content
SpringerLink
Log in

Qualitative behaviors of the continuous-time chaotic dynamical systems describing the interaction of waves in plasma

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

Abstract

In this paper, a new Lorenz-like chaotic system describing the interaction of three resonantly coupled waves in plasma is studied. Explicit ultimate boundedness and global attraction domain are derived according to stability theory of dynamical systems. The innovation of the paper is that this paper not only proves this chaotic system is globally bounded for the parameters of this system but also gives a family of mathematical expressions of global exponential attractive sets for this system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained. Finally, numerical localization of attractor is presented.

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

Similar content being viewed by others

References

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

    Article  Google Scholar 

  2. Leonov, G.: Necessary and sufficient conditions of the existence of homoclinic trajectories and cascade of bifurcations in Lorenz-like systems: birth of strange attractor and 9 homoclinic bifurcations. Nonlinear Dyn. 84(2), 1055–1062 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kuznetsov, N., Alexeeva, T., Leonov, G.: Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations. Nonlinear Dyn. 85(1), 195–201 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Leonov, G., Kuznetsov, N., Mokaev, T.: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Top. 224(8), 1421–1458 (2015)

    Article  Google Scholar 

  5. 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  MathSciNet  MATH  Google Scholar 

  6. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Leonov, G.: Existence criterion of homoclinic trajectories in the Glukhovsky–Dolzhansky system. Phys. Lett. A 379(6), 524–528 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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  MathSciNet  MATH  Google Scholar 

  10. 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  MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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 

  13. Zhang, F., Lin, D., X, M., Li, H.: Dynamical behaviors of the chaotic brushless DC motors model. Complexity 21(4), 79–85 (2016)

    Article  MathSciNet  Google Scholar 

  14. Zhang, F., Mu, C., Zhou, S., Zheng, P.: New results of the ultimate bound on the trajectories of the family of the Lorenz systems. Discrete Continuous Dyn. Syst. Ser. B 20(4), 1261–1276 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Leonov, G., Kuznetsov, N.: On differences and similarities in the analysis of Lorenz, Chen, and Lu systems. Appl. Math. Comput. 256, 334–343 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Leonov, G.: The Tricomi problem for the Shimizu–Morioka dynamical system. Dokl. Math. 86(3), 850–853 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Wang, X., Wang, M.: A hyperchaos generated from Lorenz system. Phys. A 387(14), 3751–3758 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  21. Zhang, Y., Wang, X.: A new image encryption algorithm based on non-adjacent coupled map lattices. Appl. Soft Comput. 26, 10–20 (2015)

    Article  Google Scholar 

  22. Zhang, Y., Wang, X.: A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice. Inf. Sci. 273, 329–351 (2014)

    Article  Google Scholar 

  23. Niu, Y., Wang, X., Wang, M., Zhang, H.: A new hyperchaotic system and its circuit implementation. Commun. Nonlinear Sci. Numer. Simul. 15(11), 3518–3524 (2010)

    Article  Google Scholar 

  24. Wang, X., Song, J.: Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control. Commun. Nonlinear Sci. Numer. Simul. 14(8), 3351–3357 (2009)

    Article  MATH  Google Scholar 

  25. Wang, X., He, Y.: Projective synchronization of fractional order chaotic system based on linear separation. Phys. Lett. A 372(4), 435–441 (2008)

    Article  MATH  Google Scholar 

  26. Yu, P., Liao, X., Xie, S., Fu, Y.: A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family. Commun. Nonlinear Sci. Numer. Simul. 14(7), 2886–2896 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, F., Zhang, G.: Further results on ultimate bound on the trajectories of the Lorenz system. Qual. Theory Dyn. Syst. 15(1), 221–235 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, F., Mu, C., Zheng, P., Lin, D., Zhang, G.: The dynamical analysis of a new chaotic system and simulation. Math. Methods Appl. Sci. 37(12), 1838–1846 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang, F., Liao, X., Zhang, G.: On the global boundedness of the Lü system. Appl. Math. Comput. 284, 332–339 (2016)

    MathSciNet  Google Scholar 

  30. Pikovsky, A., Rabinovich, M., Trakhtengerts, V.: Appearance of stochasticity on decay confinement of parametric instability. JTEF 74, 1366–1374 (1978)

    Google Scholar 

  31. Rabinovich, M.: Stochastic self-oscillations and turbulence. Soviet Physics Uspekhi 21(5), 443 (1978)

    Article  Google Scholar 

  32. 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  MathSciNet  MATH  Google Scholar 

  33. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, Berlin (1990)

  34. Hirsch, M., Smale, S., Devaney, R.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Elsevier, Singapore (2008)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant Nos. 11426047, 11501064), the Basic and Advanced Research Project of CQCSTC (Grant No. cstc2014jcyjA00040), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ1500605), the Research Fund of Chongqing Technology and Business University (Grant No. 2014-56-11), China Postdoctoral Science Foundation (Grant No. 2016M590850) and the Program for University Innovation Team of Chongqing (Grant No. CXTDX201601026). We thank professors Min Xiao in the College of Automation, Nanjing University of Posts and Telecommunications and Gaoxiang Yang at the Department of Mathematics and Statistics of Ankang University for their help with us. 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.

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 & Guangyun Zhang

  2. College of Mathematics and Statistics, Southwest University, Chongqing, 400716, People’s Republic of China

    Fuchen Zhang

  3. College of Electronic and Information Engineering, Southwest University, Chongqing, 400716, People’s Republic of China

    Xiaofeng Liao

Authors
  1. Fuchen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaofeng Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. 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., Liao, X. & Zhang, G. Qualitative behaviors of the continuous-time chaotic dynamical systems describing the interaction of waves in plasma. Nonlinear Dyn 88, 1623–1629 (2017). https://doi.org/10.1007/s11071-017-3334-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3334-3

Keywords

Search

Navigation