Skip to main content
SpringerLink
Log in

On the hyperchaotic complex Lü system

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

Abstract

The aim of this paper is to introduce the new hyperchaotic complex Lü system. This system has complex nonlinear behavior which is studied and investigated in this work. Numerically the range of parameter values of the system at which hyperchaotic attractors exist is calculated. This new system has a whole circle of equilibria and three isolated fixed points, while the real counterpart has only three isolated ones. The stability analysis of the trivial fixed point is studied. Its dynamics is more rich in the sense that our system exhibits both chaotic and hyperchaotic attractors, as well as periodic and quasi-periodic solutions and solutions that approach fixed points. The nonlinear control method based on Lyapunov function is used to synchronize the hyperchaotic attractors. The control of these attractors is studied. Different forms of hyperchaotic complex Lü systems are constructed using the state feedback controller and complex periodic forcing.

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

Similar content being viewed by others

References

  1. Vicente, R., Daudén, J., Toral, R.: Analysis and characterization of the hyperchaos generated by a semiconductor laser subject. IEEE J. Quantum Electron. 41, 541–548 (2005)

    Article  Google Scholar 

  2. Mascolo, S., Grassi, G.: Observers for hyperchaos synchronization with application to secure communications. In: Proceedings of the 1998 IEEE Int. Conference on Control Applications, Trieste, Italy, pp. 1016–1024, 1–4 September (1998)

  3. Peng, J.H., Ding, E.J., Ging, M., Yang, W.: Synchronizing hyperchaos with a scalar transmitted signal. Phys. Rev. Lett. 76, 904–907 (1996)

    Article  Google Scholar 

  4. Matsumoto, T., Chua, L.O., Kobayashi, K.: Hyperchaos: laboratory experimental and numerical confirmation. IEEE Trans. CAS 33, 1143–1147 (1986)

    Article  MathSciNet  Google Scholar 

  5. Tamasevicius, A., Cenys, A., Mykolaitis, G., Namajunas, A., Lindberg, E.: Hyperchaotic oscillators with gyrators. IEEE Electron. Lett. 33, 542–544 (1997)

    Article  Google Scholar 

  6. Kapitaniak, T., Chua, L.O., Zhong, G.Q.: Experimental hyperchaos in coupled Chua’s circuits. IEEE Trans. CAS 41, 499–503 (1994)

    Article  Google Scholar 

  7. Barbara, C., Silvano, C.: Hyperchaotic behavior of two bi-directional Chua’s circuits. Int. J. Circuit. Theory Appl. 30, 625–637 (2002)

    Article  MATH  Google Scholar 

  8. Arena, P., Baglio, S., Fortuna, L., Manganaro, G.: Hyperchaos from cellular neural networks. Electron. Lett. 31, 250–251 (1995)

    Article  Google Scholar 

  9. Hsieh, J.Y., Hwang, C.C., Li, A.P., Li, W.J.: Controlling hyperchaos of the Rössler system. Int. J. Control 72, 882–886 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chua, L.O., Hasler, M., Moschytz, G.S., Neirynck, J.: Autonomous cellular neural networks: A unified paradigm for pattern formation and active wave propagation. IEEE Trans. CAS 42, 559–577 (1995)

    Article  MathSciNet  Google Scholar 

  11. Grassi, G., Mascolo, S.: Synchronizing high dimensional chaotic systems via eigenvalue placement with application to cellular neural networks. Int. J. Bifurc. Chaos 9, 705–711 (1999)

    Article  MATH  Google Scholar 

  12. Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979)

    Article  MathSciNet  Google Scholar 

  13. Wang, G., Zhang, X., Zheng, Y., Li, Y.: A new modified hyperchaotic Lü system. Physica A 371, 260–272 (2006)

    Article  MathSciNet  Google Scholar 

  14. Chen, A., Lu, J., Lü, J., Yu, S.: Generating hyperchaotic Lü attractor via state feedback control. Physica A 364, 103–110 (2006)

    Article  Google Scholar 

  15. Haeri, M., Dehghani, M.: Impulsive synchronization of Chen’s hyperchaotic system. Phys. Lett. A 356, 226–230 (2006)

    Article  MATH  Google Scholar 

  16. Yassen, M.T.: On hyperchaotic synchronization of hyperchaotic Lü. Nonlinear Anal. Theory Methods Appl. 68(11), 3592–3600 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Wu, X., Wang, J., Lü, J., Iu, H.H.C.: Hyperchaotic behavior in a non-autonomous unified chaotic system with continuous periodic switch. Chaos Solitons Fractals 32, 1485–1490 (2007)

    Article  MATH  Google Scholar 

  18. Jia, Q.: Hyperchaos generated from the Lorenz chaotic system and its control. Phys. Lett. A 366, 217–225 (2007)

    Article  Google Scholar 

  19. Barboza, R.: Dynamics of a hyperchaotic Lorenz system. Int. J. Bifurc. Chaos 17(12), 4285–4294 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Thomas, R., Basios, V., Eiswirth, M., Kruel, T., Rössler, O.E.: Hyperchaos of arbitrary order generated by a single feedback circuit, and the emergence of chaotic walks. Chaos 14(3), 669–674 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12(3), 659–661 (2002)

    Article  MATH  Google Scholar 

  22. Vanecek, A., Celikovsky, S.: Control Systems: From Linear Analysis to Synthesis of Chaos. Prentice-Hall, New York (1996)

    MATH  Google Scholar 

  23. Lorenz, E.N.: Deterministic non-periodic flows. J. Atoms Sci. 20(1), 130–141 (1963)

    Article  MathSciNet  Google Scholar 

  24. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9(7), 1465–1466 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lü, J., Chen, G., Cheng, D., Celikovsky, S.: Bridge the gap between the Lorenz and Chen systems. Int. J. Bifurc. Chaos 12(12), 2917–2926 (2002)

    Article  MATH  Google Scholar 

  26. Lü, J., Chen, G., Cheng, D.: A new chaotic system and beyond: The generalized Lorenz-like system. Int. J. Bifurc. Chaos 14(5), 1507–1537 (2004)

    Article  MATH  Google Scholar 

  27. Mahmoud, G.M., Bountis, T., Mahmoud, E.E.: Active control and global synchronization of the complex Chen and Lü systems. Int. J. Bifurc. Chaos 17(12), 4295–4308 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rauh, A., Hannibal, L., Abraham, N.: Global stability properties of the complex Lorenz model. Physica D 99, 45–58 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  29. Mahmoud, G.M., Aly, S.A., Al-Kashif, M.A.: Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system. Nonlinear Dyn. 51, 171–181 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Mahmoud, G.M., Al-Kashif, M.A., Farghaly, A.A.: Chaotic and hyperchaotic attractors of a complex nonlinear system. J. Phys. A: Math. Theor. 41(5), 055104 (2008) (12 pp.)

    Article  MathSciNet  Google Scholar 

  31. Mahmoud, G.M., Al-Kashif, M.A., Aly, S.A.: Basic properties and chaotic synchronization of a complex Lorenz system. Int. J. Mod. Phys. C 18, 253–265 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Mahmoud, G.M., Bountis, T., AbdEl-Latif, G.M., Mahmoud, E.E.: Chaos synchronization of two different chaotic complex Chen and Lü systems. Nonlinear Dyn. 55(1–2), 43–53 (2009). doi:10.1007/s11071-008-9343-5

    Article  MATH  MathSciNet  Google Scholar 

  33. Mahmoud, G.M., Bountis, T., Al-Kashif, M.A., Aly, S.A.: Dynamical properties and synchronization of complex nonlinear equations for detuned lasers. Dyn. Syst. 24(01), 63–79 (2009). doi:10.1080/14689360802438298

    Article  MATH  MathSciNet  Google Scholar 

  34. Mahmoud, G.M., Ahmed, M.E., Mahmoud, E.E.: Analysis of hyperchaotic complex Lorenz systems. Int. J. Mod. Phys. C 19(10), 1477–1494 (2008)

    Article  MATH  Google Scholar 

  35. Mahmoud, G.M., Mahmoud, E.E.: Synchronization and control of hyperchaotic complex Lorenz system. In: Dynamics Days Europe 2008 Conference, Delft, The Netherlands, 25–29 August (2008)

  36. Mahmoud, G.M., Aly, S.A., Farghaly, A.A.: On chaos synchronization of a complex two-coupled dynamos system. Chaos Solitons Fractals 33, 178–187 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  37. Mahmoud, G.M., Bountis, T.: The dynamics of systems of complex nonlinear oscillators: A review. Int. J. Bifurc. Chaos 14(11), 3821–3846 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Assiut University, Assiut, 71516, Egypt

    Gamal M. Mahmoud & Mansour E. Ahmed

  2. Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

    Emad E. Mahmoud

Authors
  1. Gamal M. Mahmoud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emad E. Mahmoud
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mansour E. Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gamal M. Mahmoud.

Additional information

A part of the results of this manuscript has been accepted for publication in the CHAOS 2008 Proceedings, Chania, Crete, Greece, June 3–6, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mahmoud, G.M., Mahmoud, E.E. & Ahmed, M.E. On the hyperchaotic complex Lü system. Nonlinear Dyn 58, 725–738 (2009). https://doi.org/10.1007/s11071-009-9513-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-009-9513-0

Keywords

Search

Navigation