Skip to main content
SpringerLink
Log in

Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system

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

Abstract

In this paper, a new fractional-order hyperchaotic system based on the Lorenz system is presented. The chaotic behaviors are validated by the positive Lyapunov exponents. Furthermore, the fractional Hopf bifurcation is investigated. It is found that the system admits Hopf bifurcations with varying fractional order and parameters, respectively. Under different bifurcation parameters, some conditions ensuring the Hopf bifurcations are proposed. Numerical simulations are given to illustrate and verify the results.

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
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

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

    Article  Google Scholar 

  2. Rössler, O.E.: Continuous chaos; four prototype equations. Ann. N. Y. Acad. Sci. 316, 376–392 (1979)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Lü, J., Chen, G.: Dynamical analysis of a new chaotic attractor. Int. J. Bifurc. Chaos 12, 1001–1015 (2002)

    Article  MATH  Google Scholar 

  5. Liu, C., Liu, T., Liu, L., Liu, K.: A new chaotic attractor. Chaos Solitons Fractals 22, 1031–1038 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Leonov, G.A., Kuznetsov, N.V.: Time-varying linearization and the Perron effects. Int. J. Bifurc. Chaos 17(4), 1079–1107 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kuznetsov, N.V., Leonov, G.A.: On stability by the first approximation for discrete systems. In: 2005 International Conference on Physics and Control, PhysCon 2005. Proceedings vol. 2005, pp. 596–599. (2005). doi:10.1109/PHYCON.2005.1514053

  9. Sun, H.H., Abdelwahab, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional order. IEEE Trans. Autom. Control 29, 441–444 (1984)

    Article  MATH  Google Scholar 

  10. Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of noninteger order transfer functions for analysis of electrode process. J. Electroanal. Chem. 33, 253–265 (1971)

    Article  Google Scholar 

  11. Kusnezov, O.: Electromagnetic Theory. Chelsea, New York (1971)

    Google Scholar 

  12. Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, 034101 (2003)

    Article  Google Scholar 

  13. Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. I 42, 485–490 (1995)

    Article  Google Scholar 

  14. Mohammad, S.T., Mohammad, H.: A necessary condition for double scroll attractor existence in fractional-order systems. Phys. Lett. A 367, 102–113 (2007)

    Article  MATH  Google Scholar 

  15. Varsha, D.G., Sachin, B.: Chaos in fractional ordered Liu system. J. Comput. Appl. Math. 59, 1117–1127 (2010)

    Article  MATH  Google Scholar 

  16. Hassard, B., Kazarinoff, N., Wan, Y.: Theory and Application of Hopf Bifurcation. Cambridge University Press, Cambridge (1982)

  17. Abdelouahab, M.S., Hamri, N.E., Wang, J.: Hopf bifurcation and chaos in fractional-order modified hybrid optical system. Nonlinear Dyn. 69, 275–284 (2011)

    Article  MathSciNet  Google Scholar 

  18. Li, X., Wu, R.C.: Dynamics of a new hyperchaotic system with only one equilibrium point. J. Math. 2013, 935384 (2013)

    Article  Google Scholar 

  19. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967)

  20. Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Proceedings of the International IMACS IEEE-SMC Multiconference on Computational Engineering in Systems Applications, vol. 2. Lille, France (1996)

  21. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 13, 3–22 (2002)

    Article  MathSciNet  Google Scholar 

  22. Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Phys. Lett. A 358, 1–4 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kuznetsov, N.V., Leonov, G.A., Vagaitsev, V.I.: Analytical-numerical method for attractor localiza- tion of generalized Chua’s system. In: IFAC Proceedings Volumes (IFAC-PapersOnline), vol. 4(1). (2010). doi:10.3182/20100826-3-TR-4016.00009

  24. Leonov, G.A., Kuznetsov, N.V.: 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 23(1), 1–69 (2013)

    Article  MathSciNet  Google Scholar 

  25. Kuznetsov, N., Kuznetsov, O., Leonov, G., Vagaitsev, V.: Analytical numerical localization of hidden attractor in electrical Chua’s circuit, Lecture Notes in Electrical Engineering LNEE, vol. 174, pp. 149–158. Springer, Berlin (2013)

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time sereis. Phys. D 16(3), 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mishina, A.P., Proskuryakov, I.V.: Higher Algebra. Nauka, Moscow (1965)

    MATH  Google Scholar 

  29. Tavazoei, M.S., Haeri, M., Attari, M., Bolouki, S., Siami, M.: More details on analysis of fractional-order Van der Pol oscillator. J. Vib. Control 15(6), 803–819 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tavazoei, M.S., Haeri, M., Attari, M.: A proof for nonexistence of periodic solutions in time invariant fractional order systems. Automatica 45(8), 1886–1890 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tavazoei, M.S.: A note on fractional-order derivatives of periodic functions. Automatica 46, 945–948 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the reviewers and the editor for their valuable comments and suggestions. This work is supported by the Program of National Natural Science Foundation of China (No. 51275229), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20093401120001).

Author information

Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing , 210016, China

    Xiang Li

  2. School of Mathematics, Anhui University, Hefei , 230039, China

    Ranchao Wu

Authors
  1. Xiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ranchao Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiang Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X., Wu, R. Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system. Nonlinear Dyn 78, 279–288 (2014). https://doi.org/10.1007/s11071-014-1439-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1439-5

Keywords

Search

Navigation