Skip to main content
SpringerLink
Log in

Theoretical analysis and circuit implementation of a novel complicated hyperchaotic system

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

Abstract

This article presents a new hyperchaotic system of four-dimensional quadratic autonomous ordinary differential equations, which has one equilibrium point and two quadratic nonlinearities. Some basic dynamical properties are further investigated by means of Poincaré mapping, parameter phase portraits, and calculated Lyapunov exponents and power spectra. The existence of the hyperchaotic system is verified not only by theoretical analysis but also by conducting a novel fourth-order electronic circuit experiment. Various attractors of experimental results show that this 4D hyperchaotic system is different from the historically proposed system and has good engineering application prospects.

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. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130 (1963)

    Article  Google Scholar 

  2. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1979)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Lü, J., Chen, G., Cheng, D.: Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifurc. Chaos 12(12), 2917 (2002)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Liu, C., Liu, T., Liu, K., Liu, L.: A new chaotic attractor. Chaos Solitons Fractals 22, 1031 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Liu, C., Liu, L., Liu, T., Li, P.: A new butterfly-shaped attractor of Lorenz-like system. Chaos Solitons Fractals 28, 1196–1203 (2004)

    Google Scholar 

  9. Liu, L., Su, Y., Liu, C.: A modified Lorenz system. Int. J. Nonlinear Sci. Numer. Simul. 7(2), 187–190 (2006)

    Article  MathSciNet  Google Scholar 

  10. Liu, L., Liu, C., Zhang, Y.: Experimental confirmation of a modified Lorenz system. Chin. Phys. Lett. 24, 2756–2758 (2007)

    Article  Google Scholar 

  11. Baier, G., Klein, M.: Maximum hyperchaos in generalized Henon maps. Phys. Lett. A 151, 281 (1990)

    Article  MathSciNet  Google Scholar 

  12. Chen, Z., Yang, Y., Qi, G., Yuan, Z.: A novel hyperchaos system only with one equilibrium. Phys. Lett. A 360, 696 (2007)

    Article  MathSciNet  Google Scholar 

  13. Liu, L., Zhang, Y., Liu, C.: Analysis of a novel four-dimensional hyperchaotic system. Chin. J. Phys. 46, 382 (2008)

    Google Scholar 

  14. Yu, S., Ma, Z., Qiu, S., Peng, S., Lin, Q.: Generation and synchronization of n-scroll chaotic and hyperchaotic attractors in fourth-order systems. Chin. Phys. 13, 317–328 (2004)

    Article  Google Scholar 

  15. Gamal, M.M., Mahmoud, E.E., Ahmed, M.E.: On the hyperchaotic complex Lü system. Nonlinear Dyn. 58(4), 725 (2009)

    Article  MATH  Google Scholar 

  16. Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71(2–3), 155 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Meyer, Th., Bünner, M., Parisi, J.: Hyperchaos in the generalized Rössler system. Phys. Rev. E 56, 5069–5082 (1997)

    Article  MathSciNet  Google Scholar 

  18. Nikolov, S., Clodong, S.: Hyperchaos–chaos–hyperchaos transition in modified Rössler systems. Chaos Solitons Fractals 22, 252 (2006)

    Article  MathSciNet  Google Scholar 

  19. Thamilmaran, K., Lakshmanan, M., Venkatesan, A.: Hyperchaos in a modified canonical Chua’s circuit. Int. J. Bifurc. Chaos 14(1), 221 (2004)

    Article  MATH  Google Scholar 

  20. 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 

  21. Qi, G., Wyk, M., Wyk, B., Chen, G.: On a new hyperchaotic system. Phys. Lett. A 372, 124 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Qi, G., Chen, G.: Analysis and circuit implementation of a new 4D chaotic system. Phys. Lett. A 352, 386 (2006)

    Article  MATH  Google Scholar 

  23. Liu, C., Liu, L.: Circuit implementation of a new hyperchaos in fractional-order system. Chin. Phys. B 17(08), 2829 (2008)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  26. Li, Y., Tang, K., Chen, G.: Hyperchaos evolved from the generalized Lorenz equation. Int. J. Circuit Theory Appl. 33, 235 (2005)

    Article  MATH  Google Scholar 

  27. Wang, F., Liu, C.: Hyperchaos evolved from the Liu chaotic system. Chin. Phys. 15(5), 0963 (2006)

    Article  Google Scholar 

  28. Lü, J., Chen, G., Cheng, D.: Generating multiscroll chaotic attractors: theories, methods and applications. Int. J. Bifurc. Chaos 16(4), 775–858 (2006)

    Article  MATH  Google Scholar 

  29. Wolf, A., Swift, J., Swinney, H., Vastano, J.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

    Ling Liu, Chongxin Liu & Yanbin Zhang

Authors
  1. Ling Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chongxin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yanbin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ling Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, L., Liu, C. & Zhang, Y. Theoretical analysis and circuit implementation of a novel complicated hyperchaotic system. Nonlinear Dyn 66, 707–715 (2011). https://doi.org/10.1007/s11071-011-9943-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-9943-3

Keywords

Search

Navigation