Skip to main content
SpringerLink
Log in

Origin of mixed-mode oscillations through speed escape of attractors in a Rayleigh equation with multiple-frequency excitations

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

Abstract

The purpose of this paper is to report a novel route to mixed-mode oscillations (MMOs), i.e., the mechanism which we call speed escape of attractors, based on a Rayleigh equation with multiple-frequency excitations. The fast subsystem exhibits two critical points, which bound the region of a periodic attractor or an equilibrium point attractor and outside which are divergent regions. We show that both the periodic attractor and equilibrium point attractor can rapidly move far away from the original place when the control parameter reaches the critical values. This helps us reveal the novel mechanism to MMOs, and two different types of MMOs, i.e., MMOs of point–point type and MMOs of cycle–cycle type, are thus obtained. Besides, the effects of excitations on the MMOs are explored. We show that the amplitudes and frequencies of excitations may have important influences on MMOs. Our results enrich the possible routes to MMOs as well as the underlying mechanisms of MMOs.

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

Similar content being viewed by others

References

  1. Sriram, K., Gopinathan, M.S.: Effects of delayed linear electrical perturbation of the Belousov–Zhabotinsky reaction: a case of complex mixed mode oscillations in a batch reactor. React. Kinet. Mech. Cat. 79, 341–349 (2003)

    Article  Google Scholar 

  2. Lu, Q.S., Gu, H.G., Yang, Z.Q., Shi, X., Duan, L.X., Zheng, Y.H.: Dynamics of fiering patterns, synchronization and resonances in neuronal electrical activities: experiments and analysis. Acta Mech. Sin. 24, 593–628 (2008)

    Article  MATH  Google Scholar 

  3. Han, X.J., Bi, Q.S.: Complex bursting patterns in Van der Pol system with two slowly changing external forcings. Sci. China Technol. Sc. 55, 702–708 (2012)

    Article  Google Scholar 

  4. Roberts, A., Widiasih, E., Wechselberger, M., Jones, C.K.R.T.: Mixed mode oscillations in a conceptual climate model. Phys. D 292–293, 70–83 (2015)

    Article  MathSciNet  Google Scholar 

  5. Ngueuteu, G.S.M., Yamapi, R., Woafo, P.: Quasi-static transient and mixed mode oscillations induced by fractional derivatives effect on the slow flow near folded singularity. Nonlinear Dyn. 78, 2717–2729 (2014)

    Article  Google Scholar 

  6. Rinzel, J.: Bursting oscillation in an excitable membrane model. In: Sleeman, B.D., Jarvis, R.J. (eds.) Ordinary and Partial Differential Equations, pp. 304–316. Springer, Berlin (1985)

    Chapter  Google Scholar 

  7. Milik, A., Szmolyan, P., Löffelmann, H., Gröller, E.: Geometry of mixed-mode oscillations in the 3-D autocatalator. Int. J. Bifurcat. Chaos 8, 505–519 (1998)

    Article  MATH  Google Scholar 

  8. Krupa, M., Popović, N., Kopell, N.: Mixed-mode oscillations in three time-scale systems: a prototypical example. SIAM J. Appl. Dyn. Syst. 7, 361–420 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brøns, M., Kaasen, R.: Canards and mixed-mode oscillations in a forest pest model. Theor. Popul. Biol. 77, 238–242 (2010)

    Article  Google Scholar 

  10. Han, X.J., Bi, Q.S.: Slow passage through canard explosion and mixed-mode oscillations in the forced Van der Pol’s equation. Nonlinear Dyn. 68, 275–283 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Benoit, E., Callot, J.L., Diener, F., Diener, M.: Chasse au canard. Collect. Math. 32, 37–119 (1981)

    MathSciNet  MATH  Google Scholar 

  12. Curtu, R.: Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network. Phys. D 239, 504–514 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Larter, R., Steinmetz, C.G.: Chaos via mixed-mode oscillations. Philos. Trans. Roy. Soc. Lond. Ser. A 337, 291–298 (1991)

    Article  MATH  Google Scholar 

  14. Han, X.J., Jiang, B., Bi, Q.S.: 3-torus, quasi-periodic bursting, symmetric subHopf/fold-cycle bursting, subHopf/fold-cycle bursting and their relation. Nonlinear Dyn. 61, 667–676 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Baer, S.M., Erneux, T., Rinzel, J.: The slow passage through a Hopf bifurcation: delay, memory effects, and resonance. SIAM J. Appl. Math. 49, 55–71 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Holden, L., Erneux, T.: Slow passage through a Hopf bifurcation: from oscillatory to steady state solutions. SIAM J. Appl. Math. 53, 1045–1058 (1993)

  17. Han, X.J., Bi, Q.S., Zhang, C., Yu, Y.: Delayed bifurcations to repetitive spiking and classification of delay-induced bursting. Int. J. Bifurcat. Chaos 24, 1450098 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Koper, M.T.M.: Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram. Phys. D 80, 72–94 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Channell, P., Cymbalyuk, G., Shilnikov, A.: Origin of bursting through homoclinic spike adding in a neuron model. Phys. Rev. Lett. 98, 134101 (2007)

    Article  Google Scholar 

  20. Kuehn, C.: Multiple Time Scale Dynamics. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  21. Desroches, M., Guckenheimer, J., Krauskopf, B., Kuehn, C., Osinga, H.M., Wechselberger, M.: Mixed-mode oscillations with multiple time scales. SIAM Rev. 54, 211–288 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 10, 1171–1266 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Szabelski, K., Warmiński, J.: Parametric self-excited non-linear system vibrations analysis with inertial excitation. Int. J. Nonlinear Mech. 30, 179–189 (1995)

    Article  MATH  Google Scholar 

  24. Warminski, J.: Nonlinear normal modes of a self-excited system driven by parametric and external excitations. Nonlinear Dyn. 61, 677–689 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Han, X.J., Bi, Q.S., Ji, P., Kurths, J.: Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies. Phys. Rev. E 92, 012911 (2015)

    Article  MathSciNet  Google Scholar 

  26. Han, X.J., Bi, Q.S.: Bursting oscillations in Duffing’s equation with slowly changing external forcing. Commun. Nonlinear Sci. Numer. Simul. 16, 4146–4152 (2011)

    Article  MATH  Google Scholar 

  27. Duan, L.X., Lu, Q.S., Wang, Q.Y.: Two parameter bifurcation analysis of firing activities in the Chay neuronal model. Neurocomputing 72, 341–351 (2008)

    Article  Google Scholar 

  28. Yang, Z.Q., Lu, Q.S.: Different types of bursting in Chay neuronal model. Sci. China Ser. G Phys. Mech. Astron 51, 687–698 (2008)

    Article  Google Scholar 

  29. Perc, M., Marhl, M.: Different types of bursting calcium oscillations in non-excitable cells. Chaos Solit. Fract. 18, 759–773 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Han, X.J., Bi, Q.S.: Symmetric bursting of focus-focus type in the controlled Lorenz system with two time scales. Phys. Lett. A 373, 3643–3649 (2009)

    Article  MATH  Google Scholar 

  31. Krupa, M., Vidal, A., Desroches, M., Clément, F.: Mixed-mode oscillations in a multiple time scale phantom bursting system. SIAM J. Appl. Dyn. Syst. 11, 1458–1498 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Abshagen, J., Lopez, J.M., Marques, F., Pfister, G.: Bursting dynamics due to a homoclinic cascade in Taylor–Couette flow. J. Fluid Mech. 613, 357–384 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors express their gratitude to the associate editor and the reviewers whose comments and suggestions have helped the improvements of this paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11572141, 11632008, 11472115, 11502091 and 11402226) and the Training Project for Young Backbone Teacher of Jiangsu University.

Author information

Authors and Affiliations

  1. Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, 212013, People’s Republic of China

    Xiujing Han & Fubing Xia

  2. School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, People’s Republic of China

    Chun Zhang

  3. School of Science, Nantong University, Nantong, 226007, People’s Republic of China

    Yue Yu

Authors
  1. Xiujing Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fubing Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yue Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiujing Han.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, X., Xia, F., Zhang, C. et al. Origin of mixed-mode oscillations through speed escape of attractors in a Rayleigh equation with multiple-frequency excitations. Nonlinear Dyn 88, 2693–2703 (2017). https://doi.org/10.1007/s11071-017-3403-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3403-7

Keywords

Search

Navigation