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Slow–fast dynamics in a perturbation model of double pendulum system with singularity of triple zero eigenvalues

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Abstract

Slow–fast dynamics such as bursting behaviors are common in many physical and engineering systems. In the previous study, some focused on the bursting behavior caused by the codimension-1 bifurcations, and others focus on the bursting behavior due to the particular structures. However, systems under critical conditions may exhibit many complicated dynamics due to the high co-dimensional bifurcations. Our research aims to investigate the bursting oscillations near a triple zero eigenvalues singularity. A perturbation model of the double pendulum system with external excitation is taken as an example and investigated the dynamical mechanism of bursting behaviors. Because of the order gap between the exciting frequency and natural frequency, the perturbation model with the external excitation can be regarded as a generalized autonomous system. By overlapping the transformed phase portrait and the equilibrium branch, four types of bursting oscillations are determined: fold/fold type, zero-Hopf/zero-Hopf type, symmetrical zero-Hopf/sup-Hopf/fold-cycle type, and symmetrical zero-Hopf/sup-Hopf/fold-cycle/sub-Hopf type. Primarily, we find that due to the singularity of compound fold conditions, many bifurcations of the limit cycle occur, which cause many complex dynamics such as 2-D tori bursting, breaking of symmetric structure, and chaotic bursting. These results play an essential role in understanding the stability of a system with high co-dimensional bifurcation conditions. They can be expected to provide a theoretical basis for formulating a control strategy.

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Some or all data, models, or code generated or used during the study are available from the corresponding author by request.

References

  1. Shafiei, M., Jafari, S., Parastesh, F., Ozer, M., Kapitaniak, T., Perc, M.: Time delayed chemical synapses and synchronization in multilayer neuronal networks with ephaptic inter-layer coupling. Commun. Nonlinear Sci. Numer. Simul. 84, 105175 (2020)

    Article  MATH  Google Scholar 

  2. Bashkirtseva, I., Ryashko, L.: Slow–fast oscillatory dynamics and phantom attractors in stochastic modeling of biochemical reactions, Chaos: An Interdisciplin J Nonlinear Sci 32 (3) (2022) 033126

  3. Brzeski, P., Chong, A., Wiercigroch, M., Perlikowski, P.: Impact adding bifurcation in an autonomous hybrid dynamical model of church bell. Mech. Syst. Signal Process. 104, 716–724 (2018)

    Article  Google Scholar 

  4. Tarasova, V.V., Tarasov, V.E.: Logistic map with memory from economic model. Chaos, Solitons & Fractals 95, 84–91 (2017)

    Article  MATH  Google Scholar 

  5. Chowdhury, P.R., Petrovskii, S., Banerjee, M.: Effect of slow-fast time scale on transient dynamics in a realistic prey-predator system. Mathematics 10(5), 699 (2022)

    Article  Google Scholar 

  6. Rinzel, J., Huguet, G.: Nonlinear dynamics of neuronal excitability, oscillations, and coincidence detection. Commun. Pure Appl. Math. 66(9), 1464–1494 (2013)

    Article  MATH  Google Scholar 

  7. Izhikevich, E.M.: Neural excitability, spiking and bursting. Int J Bifurcat Chaos 10(06), 1171–1266 (2000)

    Article  MATH  Google Scholar 

  8. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500–544 (1952)

    Article  Google Scholar 

  9. Fujimoto, K., Kaneko, K.: How fast elements can affect slow dynamics. Physica D 180(1–2), 1–16 (2003)

    Article  MATH  Google Scholar 

  10. Rinzel, J.: A formal classification of bursting mechanisms in excitable systems, in: Mathematical topics in population biology, morphogenesis and neurosciences, Springer, 1987, pp. 267–281

  11. Izhikevich, E.M.: Dynamical systems in neuroscience, MIT press, (2007)

  12. Saha, T., Pal, P.J., Banerjee, M.: Relaxation oscillation and canard explosion in a slow-fast predator-prey model with beddington-deangelis functional response. Nonlinear Dyn. 103(1), 1195–1217 (2021)

    Article  Google Scholar 

  13. Wen, Q., Liu, S., Lu, B.: Firing patterns and bifurcation analysis of neurons under electromagnetic induction. Electron. Res. Archive 29(5), 3205 (2021)

    Article  MATH  Google Scholar 

  14. Izhikevich, E.M., Hoppensteadt, F.: Classification of bursting mappings. Int. J. Bifurcat. Chaos 14(11), 3847–3854 (2004)

    Article  MATH  Google Scholar 

  15. Shan, C.: Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Dis. Continuous Dynam. Syst.-B 27(3), 1447 (2022)

    Article  MATH  Google Scholar 

  16. Chen, C., Chen, X.: Rich sliding motion and dynamics in a filippov plant-disease system. Int. J. Bifurcat. Chaos 28(01), 1850012 (2018)

    Article  MATH  Google Scholar 

  17. Krishnapriya, P., Pitchaimani, M.: Analysis of time delay in viral infection model with immune impairment. J. Appl. Math. Comput. 55(1), 421–453 (2017)

    Article  MATH  Google Scholar 

  18. Kuznetsov, Y.A.: Practical computation of normal forms on center manifolds at degenerate bogdanov-takens bifurcations. Int. J. Bifurcat. Chaos 15(11), 3535–3546 (2005)

    Article  MATH  Google Scholar 

  19. Bi, Q., Yu, P.: Computation of normal forms of differential equations associated with non-semisimple zero eigenvalues. Int. J. Bifurcat. Chaos 8(12), 2279–2319 (1998)

    Article  MATH  Google Scholar 

  20. Harlim, J., Langford, W.F.: The cusp-hopf bifurcation. Int. J. Bifurcat. Chaos 17(08), 2547–2570 (2007)

    Article  MATH  Google Scholar 

  21. Golubitsky, M., Josic, K., Kaper, T. J.: An unfolding theory approach to bursting in fast-slow systems, Global Analysis of Dynamical Systems (2001) 277–308

  22. Saggio, M.L., Spiegler, A., Bernard, C., Jirsa, V.K.: Fast-slow bursters in the unfolding of a high codimension singularity and the ultra-slow transitions of classes. J. Math. Neurosci. 7(1), 1–47 (2017)

    Article  MATH  Google Scholar 

  23. Zaihua, W., Haiyan, H.: Stability and bifurcation of delayed dynamic systems: from theory to application. Adv. Mech. 43(1), 3–20 (2013)

    Google Scholar 

  24. Dudkowski, D., Wojewoda, J., Czołczyński, K., Kapitaniak, T.: Is it really chaos? the complexity of transient dynamics of double pendula. Nonlinear Dyn. 102, 759–770 (2020)

    Article  Google Scholar 

  25. Brzeski, P., Wojewoda, J., Kapitaniak, T., Kurths, J., Perlikowski, P.: Sample-based approach can outperform the classical dynamical analysis-experimental confirmation of the basin stability method. Sci. Rep. 7(1), 1–10 (2017)

    Google Scholar 

  26. Mandadi, V., Huseyin, K.: Non-linear bifurcation analysis of non-gradient systems. Int. J. Non-Linear Mech. 15(3), 159–172 (1980)

    Article  MATH  Google Scholar 

  27. Yu, P., Bi, Q.: Analysis of non-linear dynamics and bifurcations of a double pendulum. J. Sound Vib. 217(4), 691–736 (1998)

    Article  MATH  Google Scholar 

  28. Huang, L., Wu, G., Zhang, Z., Bi, Q.: Fast-slow dynamics and bifurcation mechanism in a novel chaotic system. Int. J. Bifurcat. Chaos 29(10), 1930028 (2019)

    Article  MATH  Google Scholar 

  29. Zhang, Z., Chen, Z., Bi, Q.: Modified slow-fast analysis method for slow-fast dynamical systems with two scales in frequency domain. Theor. Appl. Mech. Lett. 9(6), 358–362 (2019)

    Article  Google Scholar 

  30. Zhang, X., Zhang, B., Han, X., Bi, Q.: On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified van der pol–duffing system with slow-varying periodic excitation, Nonlinear Dynamics (2022) 1–18

  31. Yu, P., Leung, A.: A perturbation method for computing the simplest normal forms of dynamical systems. J. Sound Vib. 261(1), 123–151 (2003)

    Article  MATH  Google Scholar 

  32. Wiggins, S., Golubitsky, M.: Introduction to applied nonlinear dynamical systems and chaos, Vol. 2, Springer, (1990)

  33. Han, X., Liu, Y., Bi, Q., Kurths, J.: Frequency-truncation fast-slow analysis for parametrically and externally excited systems with two slow incommensurate excitation frequencies. Commun. Nonlinear Sci. Numer. Simul. 72, 16–25 (2019)

    Article  MATH  Google Scholar 

  34. Zhang, M., Bi, Q.: On occurrence of bursting oscillations in a dynamical system with a double hopf bifurcation and slow-varying parametric excitations. Int. J. Non-Linear Mech. 128, 103629 (2021)

    Article  Google Scholar 

  35. Lü, X.-Y., Jing, H., Ma, J.-Y., Wu, Y.: P t-symmetry-breaking chaos in optomechanics. Phys. Rev. Lett. 114(25), 253601 (2015)

    Article  Google Scholar 

  36. Olson, C.L., Olsson, M.: Dynamical symmetry breaking and chaos in duffing’s equation. Am. J. Phys. 59(10), 907–911 (1991)

    Article  Google Scholar 

  37. Xu, J., Jiang, S.: Delay-induced bogdanov-takens bifurcation and dynamical classifications in a slow-fast flexible joint system. Int. J. Bifurcat. Chaos 25(09), 1550121 (2015)

    Article  MATH  Google Scholar 

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Acknowledgements

The first author, Weipeng Lyu, is a doctoral candidate in the Faculty of Civil Engineering and Mechanics at Jiangsu University, China.

Funding

The work is supported by the National Natural Science Foundation of China (Grant Nos. 12002299, 11972173, 11872188 and 11632008) and Natural Science Foundation for colleges and universities in Jiangsu Province (Grant Nos. 20KJB110010).

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Authors and Affiliations

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

    Weipeng Lyu, Liping Zhang & Qinsheng Bi

  2. Faculty of Mathematics and Statistics, Yancheng Normal University, Yancheng, 224005, Jiangsu, People’s Republic of China

    Weipeng Lyu, Liping Zhang & Haibo Jiang

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  1. Weipeng Lyu
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  2. Liping Zhang
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Correspondence to Qinsheng Bi.

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Lyu, W., Zhang, L., Jiang, H. et al. Slow–fast dynamics in a perturbation model of double pendulum system with singularity of triple zero eigenvalues. Nonlinear Dyn 111, 3239–3252 (2023). https://doi.org/10.1007/s11071-022-08020-2

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