Skip to main content
SpringerLink
Log in

Complex aperiodic mixed mode oscillations induced by crisis and transient chaos in a nonlinear system with slow parametric excitation

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

Abstract

For dynamic systems with multiple time scales, its long-term behaviors usually present as mixed mode oscillations (MMOs). The formation of MMOs is closely related to the fast transitions evoked by the bifurcations of the fast sub-system. Up till now, most of the results reported in deterministic systems focus on periodic MMOs, namely the holistic structure of waveform shows periodicity, no matter whether the slow manifolds that the fast–slow flow follows are periodic or not. Moreover, since the motion of the fast–slow flow is modulated by the critical manifolds, qualitative influences of transient behaviors on the holistic structure of MMOs are less concerned. While in this paper, three patterns of aperiodic MMOs are analyzed based on a three-dimensional nonautonomous system under slow parametric excitation. According to the types of manifolds among which the fast transitions take place, these MMOs can be named as aperiodic “cycle-chaos-cycle” MMOs, aperiodic “cycle-chaos-cycle” MMOs with additional chaotic bursters, and aperiodic “chaos-cycle-point” MMOs. By combining the fast–slow analysis on certain time intervals and the probe into the transient chaos near boundary crisis, generation mechanisms of these MMOs are investigated. Our results show that the relative location between the boundary crisis value and the minimum of slow variable will qualitatively change the pattern of large amplitude oscillations (LAOs); thus, the composition of bursters can be decided by the intensity of transient chaos. On the other hand, the small amplitude oscillations (SAOs) evoked by interior crisis may contain inner structures related to bifurcation delay and sharding phenomenon. Particularly, interior crisis can lead to merging phenomena of chaotic attractors. Thus, when the chaotic LAOs stages are over, the fast–slow flow can transit to upper or lower branch in a irregular way, which results in the uncertain locations of SAOs.

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

Similar content being viewed by others

References

  1. Petrov, V., Scott, S.K., Showalter, K.: Mixed-mode oscillations in chemical systems. J. Chem. Phys. 97(9), 6191–6198 (1992)

    Google Scholar 

  2. Koper, M.: Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol–Duffing model with a cross-shaped phase diagram. Physica D 80(1–2), 72–94 (1995)

    MathSciNet  MATH  Google Scholar 

  3. Hasan, C.R., Krauskopf, B., Osinga, H.M.: Mixed-mode oscillations and twin canard orbits in an autocatalytic chemical reaction. SIAM J. Appl. Dyn. Syst. 16(4), 2165–2195 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Milik, A., Szmolyan, P., Elmann, H.L., Groller, E.: Geometry of mixed-mode oscillations in the 3-D autocatalator. Int. J. Bifurc. Chaos 8(03), 505–519 (1998)

    MATH  Google Scholar 

  5. Desroches, M.: Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh–Nagumo system. Chaos 18(1), 015107 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Ersöz, E.K., Desroches, M., Mirasso, C.R., Rodrigues, S.: Anticipation via canards in excitable systems. Chaos 29(1), 013111 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Davison, E.N., Aminzare, Z., Dey, B.: Lenord NE mixed mode oscillations and phase locking in coupled FitzHugh–Nagumo model neurons. Chaos 29(3), 033105 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Constantinescu, D., Dumbrajs, O., Igochine, V.: Bifurcations and fast–slow dynamics in a low-dimensional model for quasi-periodic plasma perturbations. Rom. Rep. Phys. 67(3), 1049–1060 (2015)

    Google Scholar 

  9. Dubbeldam, J.L.A., Krauskopf, B., Lenstra, D.: Excitability and coherence resonance in lasers with saturable absorber. Phys. Rev. E 3(60), 6580 (1999)

    Google Scholar 

  10. Kousaka, T., Ogura, T., Shimizu, K., Asahara, H., Inaba, N.: Analysis of mixed mode oscillation incrementing bifurcations generated in a nonautonomous constrained Bonhoeffer–van der Pol oscillator. Physica D 353, 48–57 (2017)

    MathSciNet  Google Scholar 

  11. Li, H., Chen, D., Gao, X., Han, Q., Wu, C.: Fast–slow dynamics of a hydropower generation system with multi-time scales. Mech. Syst. Signal Process. 110, 458 (2018)

    Google Scholar 

  12. Zhang, H., Chen, D., Wu, C., Wang, X.: Dynamics analysis of the fast–slow hydro-turbine governing system with different time-scale coupling. Commun. Nonlinear Sci. Numer. Simul. 54, 136 (2018)

    MathSciNet  Google Scholar 

  13. Izhikevich, E.M.: Neural excitability spiking and bursting. Int. J. Bifurc. Chaos 10(06), 061171 (2000)

    MathSciNet  MATH  Google Scholar 

  14. Kuehn, C.: Multiple Time Scale Dynamics, pp. 397–398. Springer, New York (2015)

    MATH  Google Scholar 

  15. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53 (1979)

    MathSciNet  MATH  Google Scholar 

  16. Desroches, M., Guckenheimer, J., Krauskopf, B., Kuehn, C., Osinga, H.M., Wechselberger, M.: Mixed-mode oscillations with multiple time scales. SIAM Rev. 54(2), 211–288 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Teka, W., Tabak, J., Bertram, R.: The relationship between two fast/slow analysis techniques for bursting oscillations. Chaos 22(4), 043117 (2012)

    MathSciNet  Google Scholar 

  18. Hongray, T., Balakrishnan, J., Dana, S.K.: Bursting behaviour in coupled Josephson junctions. Chaos 25(12), 123104 (2015)

    MathSciNet  Google Scholar 

  19. Desroches, M., Kaper, T.J., Krupa, M.: Mixed-mode bursting oscillations: dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster. Chaos 23(4), 046106 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Zhang, F.B., Liu, S.Q., Zhang, X.H., Wang, J., Lu, B.: Mixed-mode oscillations and bifurcation analysis in a pituitary model. Nonlinear Dyn. 94(2), 807–826 (2018)

    Google Scholar 

  21. Fallah, H.: Symmetric fold/super-Hopf bursting, chaos and mixed mode oscillations in Pernarowski model of pancreatic beta-cells. Int. J. Bifurc. Chaos 26(09), 1630022 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Rakaric, Z., Kovacic, I.: Mechanical manifestations of bursting oscillations in slowly rotating systems. Mech. Syst. Signal Process. 81, 35–42 (2016)

    Google Scholar 

  23. Nascimento, M.A., Nagao, R., Eiswirth, M., Varela, H.: Coupled slow and fast surface dynamics in an electrocatalytic oscillator: model and simulations. J. Chem. Phys. 141(23), 234701 (2014)

    Google Scholar 

  24. Bulai, I.M., Pedersen, M.G.: Stopping waves: geometric analysis of coupled bursters in an asymmetric excitation field. Nonlinear Dyn. 96(3), 1927–1937 (2019)

    Google Scholar 

  25. Sekikawa, M., Inaba, N., Yoshinaga, T., Kawakami, H.: Collapse of duck solution in a circuit driven by an extremely small periodic force. Electron. Commun. Jpn. 87(4), 51–59 (2004)

    Google Scholar 

  26. Sekikawa, M., Inaba, N., Aihara, K.: Coexisting two canards and their breakdown into chaos in the van der Pol oscillator under weak periodic perturbation. Phys. Lett. A 363(5–6), 404–410 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Yuta, N., Naohiko, I., Munehisa, S., Endo, T., Fujimoto, K.: Remarkable similarities of two pairs of stable and saddle canards in a van der Pol oscillator under extremely weak periodic perturbation. Prog. Theor. Exp. Phys. 2018(1), 013A02 (2018)

    MathSciNet  Google Scholar 

  28. Wojcik, J., Shilnikov, J.: Voltage interval mappings for activity transitions in neuron models for elliptic bursters. Physica D 240(14–15), 1164–1180 (2011)

    MATH  Google Scholar 

  29. Takahashi, H., Kousaka, T., Asahara, H., Stankevich, N.V., Inaba, N.: Mixed-mode oscillation-incrementing bifurcations and a devil’s staircase from a nonautonomous, constrained Bonhoeffer–van der Pol oscillator. Prog. Theor. Exp. Phys. 2018(10), 103A02 (2018)

  30. Belykh, V.N., Belykh, I.V., Colding-JØrgensen, M., Mosekilde, E.: Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models. Eur. Phys. J. E 3(3), 205–219 (2000)

    Google Scholar 

  31. Channell, P., Cymbalyuk, G., Shilnikov, A.: Origin of bursting through homoclinic spike adding in a neuron model. Phys. Rev. Lett. 98(13), 134101 (2007)

    Google Scholar 

  32. Guckenheimer, J., Lizarraga, I.: Shilnikov homoclinic bifurcation of mixed-mode oscillations. SIAM J. Appl. Dyn. Syst. 14(2), 764–786 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Burke, J., Desroches, M., Granados, A., Kaper, T.J., Krupa, M., Vo, T.: From canards of folded singularities to torus canards in a forced van der Pol Equation. J. Nonlinear Sci. 26(2), 405–451 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Ju, H.W., Neiman, A.B., Shilnikov, A.L.: Bottom-up approach to torus bifurcation in neuron models. Chaos 28(10), 106317 (2018)

    MathSciNet  Google Scholar 

  35. Wechselberger, M.: Existence and bifurcation of canards in \({\mathbb{R}}^{3}\) in the case of a folded node. SIAM J. Appl. Dyn. Syst. 4(1), 101–139 (2005)

    MathSciNet  MATH  Google Scholar 

  36. Guckenheimer, J.: Return maps of folded nodes and folded saddle-nodes. Chaos 18(1), 015108 (2008)

    MathSciNet  Google Scholar 

  37. Mujica, J., Bernd, K., Osinga, H.M.: Tangencies between global invariant manifolds and slow manifolds near a singular Hopf bifurcation. SIAM J. Appl. Dyn. Syst. 17(2), 1395–1431 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Desroches, M., Krauskopf, B., Osinga, H.M.: The geometry of slow manifolds near a folded node. SIAM J. Appl. Dyn. Syst. 7(4), 1131–1162 (2008)

    MathSciNet  MATH  Google Scholar 

  39. Guckenheimer, J., Scheper, C.: A geometric model for mixed-mode oscillations in a chemical system. SIAM J. Appl. Dyn. Syst. 10(1), 92–128 (2011)

    MathSciNet  MATH  Google Scholar 

  40. Kuehn, C.: On decomposing mixed-mode oscillations and their return maps. Chaos 21(3), 033107 (2011)

    MathSciNet  MATH  Google Scholar 

  41. Hasan, C.R., Krauskopf, B., Osinga, H.M.: Saddle slow manifolds and canard orbits in \({\mathbb{R}}^{4}\) and application to the full Hodgkin–Huxley model. J. Math. Neurosci. 8, 5 (2018)

    MATH  Google Scholar 

  42. Guckenheimer, J.: Singular Hopf bifurcation in systems with two slow variables. SIAM J. Appl. Dyn. Syst. 7(7), 1355–1377 (2008)

    MathSciNet  MATH  Google Scholar 

  43. Guckenheimer, J., Meerkamp, P.: Unfoldings of singular Hopf bifurcation. SIAM J. Appl. Dyn. Syst. 11(4), 1325–1359 (2012)

    MathSciNet  MATH  Google Scholar 

  44. Xie, W., Xu, J., Cai, L., Jin, Y.: Dynamics and geometric desingularization of the multiple time scale FitzHugh–Nagumo–Rinzel model with fold singularity. Commun. Nonlinear Sci. Numer. Simul. 63, 322–338 (2018)

    MathSciNet  Google Scholar 

  45. Feudel, U., Neiman, A., Pei, X., Wojtenek, W.M., Moss, F.: Homoclinic bifurcation in a HodgkinHuxley model of thermally sensitive neurons. Chaos 10(1), 231239 (2000)

    MATH  Google Scholar 

  46. Cymbalyuk, G., Shilnikov, A.L.: Coexistence of tonic spiking oscillations in a leech neuron model. J. Comput. Neurosci. 18(3), 255–263 (2005)

    MathSciNet  Google Scholar 

  47. Shilnikov, A.L.: Complete dynamical analysis of a neuron model. Nonlinear Dyn. 68(3), 305–328 (2012)

    MathSciNet  MATH  Google Scholar 

  48. Kawczyński, A.L., Strizhak, P.E.: Period adding and broken Farey tree sequence of bifurcations for mixed-mode oscillations and chaos in the simplest three-variable nonlinear system. J. Chem. Phys. 112(14), 6122–6130 (2000)

    Google Scholar 

  49. Channell, P., Cymbalyuk, G., Shilnikov, A.L.: Applications of the Poincaré mapping technique to analysis of neuronal dynamics. Neurocomputing 70(10–12), 2107–2111 (2007)

    Google Scholar 

  50. Shilnikov, A.L., Cymbalyuk, G.: Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe. Phys. Rev. Lett. 94(4), 048101 (2005)

    Google Scholar 

  51. Neiman, A.B., Dierkes, K., Lindner, B., Han, L., Shilnikov, A.L.: Spontaneous voltage oscillations and response dynamics of a Hodgkin–Huxley type model of sensory hair cells. J. Math. Neurosci. 1(1), 11 (2011)

    MathSciNet  MATH  Google Scholar 

  52. Freire, E., Rodríguez-Luis, A.J., Gamero, E., Ponce, E.: A case study for homoclinic chaos in an autonomous electronic circuit: a trip from Takens–Bogdanov to Hopf–Shilnikov. Physica D 62(1–4), 230–253 (1993)

    MathSciNet  MATH  Google Scholar 

  53. Ermentrout, B.: Simulating, Analyzing and Animating Dynamical System: A Guide to Xppaut for Researchers and Students, pp. 161–169. SIAM, Philadelphia (2002)

    MATH  Google Scholar 

  54. Grebogi, C., Ott, E., Yorke, J.A.: Crisis, sudden change in chaotic attractor and transient chaos. Physica D 7(1–3), 181–200 (1983)

    MathSciNet  MATH  Google Scholar 

  55. Grebogi, C., Ott, E., Yorke, J.A.: Chaos strange attractor, and fractal basin boundaries in nonlinear dynamics. Science 238(4827), 632–638 (1987)

    MathSciNet  MATH  Google Scholar 

  56. Arnold, V.I., Afraimovich, V.S., Ilyashenko, Y.S., Shilnikov, L.P.: Dynamical Systems V: Bifurcation Theory and Catastrophe Theory, pp. 148–149. Springer, New York (1993)

    Google Scholar 

  57. Grebogi, C., Ott, E., Romeiras, F., Yorke, J.A.: Critical exponents for crisis induced intermittency. Phys. Rev. A 36(11), 5365 (1988)

    MathSciNet  Google Scholar 

  58. Baer, S.M., Erneux, T., Rinzel, J.: The slow passage through a Hopf bifurcation: delay, memory effects, and resonance. SIAM J. Appl. Dyn. Syst. 49(1), 55–71 (1989)

    MathSciNet  MATH  Google Scholar 

  59. Baesens, C.: Slow sweep through a period-doubling cascade: delayed bifurcations and renormalisation. Physica D 53(2–4), 319–375 (1991)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11872201, 11572148).

Author information

Authors and Affiliations

  1. Department of Mechanics, Nanjing University of Aeronautics and Astronautics, Nanjing, 200016, People’s Republic of China

    Zhenyang Chen

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, People’s Republic of China

    Fangqi Chen

  3. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Fangqi Chen

Authors
  1. Zhenyang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fangqi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhenyang Chen.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Z., Chen, F. Complex aperiodic mixed mode oscillations induced by crisis and transient chaos in a nonlinear system with slow parametric excitation. Nonlinear Dyn 100, 659–677 (2020). https://doi.org/10.1007/s11071-020-05500-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05500-1

Keywords

Search

Navigation