Skip to main content
SpringerLink
Log in

1:2 and 1:4 resonances in a two-dimensional discrete Hindmarsh–Rose model

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

Abstract

In this paper, the two-parameter bifurcations of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes 1:2 and 1:4 resonances by using a series of affine transformations and bifurcation theory. The numerical simulations including phase portraits, two-parameter bifurcation diagrams, and maximum Lyapunov exponents diagrams for two different varying parameters in a three-dimensional space, not only illustrate the theoretical analysis, but also display the interesting and complex dynamical behaviors.

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

Similar content being viewed by others

References

  1. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos—an Introduction to Dynamical Systems. Springer, New York (1996)

    MATH  Google Scholar 

  2. Bonatto, C., Gallas, J.A.C.: Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit. Phys. Rev. Lett. 101, 054101 (2008)

    Article  Google Scholar 

  3. Chen, Q.L., Teng, Z.D., Wang, L., Jiang, H.: The existence of codimension-two bifurcation in a discrete SIS epidemic model with standard incidence. Nonlinear Dyn. 71, 55–73 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, S.S., Cheng, C.Y., Lin, Y.R.: Application of a two-dimensional Hindmarsh–Rose type model for bifurcation analysis. Int. J. Bifurc. Chaos. 23, 1350055 (2013)

    Article  MathSciNet  Google Scholar 

  5. Djeundam, S.R.D., Yamapi, R., Kofane, T.C., Aziz-Alaoui, M.A.: Deterministic and stochastic bifurcations in the Hindmarsh–Rose neuronal model. Chaos 23, 033125 (2013)

    Article  Google Scholar 

  6. Dong, J., Zhang, G.J., Xie, Y., Yao, H., Wang, J.: Dynamic behavior analysis of fractional-order Hindmarsh–Rose neuronal model. Cogn. Neurodyn. 8, 167–175 (2014)

    Article  Google Scholar 

  7. Fahsi, A., Belhaq, M.: Analytical approximation of heteroclinic bifurcation in a 1:4 resonance. Int. J. Bifurc. Chaos 22(12), 1250294 (2012)

    Article  MathSciNet  Google Scholar 

  8. Freire, J.G., Gallas, J.A.C.: Cyclic organization of stable periodic and chaotic pulsations in Hartleys oscillator. Chaos Solitons Fractals 59, 129–134 (2014)

    Article  Google Scholar 

  9. FitzHugh, R.: Impulses and physiological state in theoretical models of nerve membrane. Biophys. J. 1, 445–467 (1961)

    Article  Google Scholar 

  10. Gonzàlez-Miranda, J.M.: Complex bifurcation structures in the Hindmarsh–Rose neuron model. Int. J. Bifurc. Chaos 17, 3071–3083 (2007)

    Article  MATH  Google Scholar 

  11. Guckenheimer, J.: Multiple bifurcation problems of codimension two. SIAM J. Math. Anal. 15, 1–49 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  12. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  13. He, Z.M., Lai, X.: Bifurcations and chaotic behavior of a discrete-time predator-prey system. Nonlinear Anal. Real World Appl. 12, 403–417 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hindmarsh, J.L., Rose, R.M.: A model of the nerve impulse using two first-order differential equations. Nature 296, 162–164 (1982)

    Article  Google Scholar 

  15. Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B 221, 87–102 (1984)

    Article  Google Scholar 

  16. Hizanidis, J., Kanas, V., Bezerianos, A., Bountis, T.: Chimera states in networks of nonlocally coupled Hindmarsh–Rose neuron models. Int. J. Bifurc. Chaos. 24, 1450030 (2014)

    Article  MathSciNet  Google Scholar 

  17. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)

    Article  Google Scholar 

  18. Innocentia, G., Morelli, A., Genesio, R., Torcini, A.: Dynamical phases of the Hindmarsh–Rose neuronal model: studies of the transition from bursting to spiking chaos. Chaos 17, 043128 (2007)

    Article  MathSciNet  Google Scholar 

  19. Ji, Y., Bi, Q.S.: SubHopf/Fold-Cycle bursting in the Hindmarsh–Rose neuronal model with periodic stimulation. Chin. Phys. Lett. 28, 090201 (2011)

    Article  Google Scholar 

  20. Jing, Z.J., Chang, Y., Guo, B.L.: Bifurcation and chaos in discrete FitzHugh–Nagumo system. Chaos Solitons Fractals 21, 701–720 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Krauskopf, B.: Bifurcations at \(\infty \) in a model for 1:4 resonance. Ergod. Theory Dyn. Syst. 17, 899–931 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York (2004)

    Book  MATH  Google Scholar 

  23. Li, B., He, Z.M.: Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model. Nonlinear Dyn. (2014). doi:10.1007/s11071-013-1161-8

  24. Li, Y.L., Xiao, D.M.: Bifurcations of a predator-prey system of Holling and Leslie types. Chaos Solition Fractals 34, 606–620 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Liu, X.L., Xiao, D.M.: Complex dynamic behaviors of a discrete-time predator-prey system. Chaos Solition Fractals 32, 80–94 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Liu, X.L., Liu, S.Q.: Codimension-two bifurcations analysis in two-dimensional Hindmarsh–Rose model. Nonliear Dyn. 67, 847–857 (2012)

    Article  MATH  Google Scholar 

  27. Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)

    Article  Google Scholar 

  28. Ott, E.: Chaos in Dynamical Systems, 2nd edn. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  29. Peng, M.S.: Multiple bifurcations and periodic “bubbling” in a delay population model. Chaos Solitons Fractals 25, 1123–1130 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Polyanin, A.D., Chernoutsan, A.I.: A Concise Handbook of Mathematics, Physics, and Engineering Science. CRC Press, New York (2011)

    Google Scholar 

  31. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2nd edn. CRC Press, Boca Raton (1999)

    MATH  Google Scholar 

  32. Ruan, S.G., Xiao, D.M.: Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61, 1445–1472 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Shilnikov, A., Kolomiets, M.: Methods of the qualitative theory for the Hindmarsh–Rose model: a case study, a tutorial. Int. J. Bifurc. Chaos 18(8), 2141–2168 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Storace, M., Linaro, D., Lange, E.: The Hindmarsh–Rose neuron model: bifurcation analysis and piecewise-linear approximations. Chaos 18, 033128 (2008)

    Article  MathSciNet  Google Scholar 

  35. Svetoslav, N.: An alternative bifurcation analysis of the Rose–Hindmarsh model. Chaos Solitons Fractals 23, 1643–1649 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  36. Tsuji, S., Ueta, T., Kawakami, H., Fujii, H., Aihara, K.: Bifurcations in two-dimensional Hindmarsh–Rose type model. Int. J. Bifurc. Chaos 17, 985–998 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  37. Vandermeer, J.: Period ‘bubbling’ in simple ecological models: pattern and chaos formation in a quartic model. Ecol Model. 95, 311–317 (1997)

    Article  Google Scholar 

  38. Viana, E.R., Rubinger, R.M., Albuquerque, H.A., Dias, F.O., De Oliveira, A.G., Ribeiro, G.M.: Periodicity detection on the parameter-space of a forced Chuas circuit. Nonlinear Dyn. 67, 385–392 (2012)

    Article  Google Scholar 

  39. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  40. Yi, N., Zhang, Q., Liu, P., Lin, Y.: Codimension-two bifurcations analysis and tracking control on a discrete epidemic model. J. Syst. Sci. Complex. 24, 1033–1056 (2011)

  41. Zheng, Y.G.: Delay-induced dynamical transitions in single Hindmarsh–Rose system. Int. J. Bifurc. Chaos. 23, 1350150 (2013)

    Article  Google Scholar 

Download references

Acknowledgments

The authors thank the editor and the referees for their valuable suggestions and comments which lead to improvement of the manuscript.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Bo Li & Zhimin He

Authors
  1. Bo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhimin He
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhimin He.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, B., He, Z. 1:2 and 1:4 resonances in a two-dimensional discrete Hindmarsh–Rose model. Nonlinear Dyn 79, 705–720 (2015). https://doi.org/10.1007/s11071-014-1696-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1696-3

Keywords

Search

Navigation