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Dynamics analysis in a non-smooth Lü system with two time scales

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Abstract

To explore the bursting behaviours in a dynamic system with non-smooth factor, this paper takes Lü system as an example, introduces a non-smooth term and a periodic external excitation, ensures that there exists an order gap between the natural frequency and the exited frequency, then a non-smooth dynamic system with two time scales is established. Through the stability analysis of the equilibrium point, the conditions of fold bifurcation and Hopf bifurcation are given. The numerical simulations show the bursting oscillations of the system under different parameter values and the dynamic behaviours of the trajectory at the non-smooth interface. In addition, combining with numerical calculation and related bifurcation theory, the bifurcation types of the system at the interface are determined. Finally, the mechanism of oscillations is revealed by the superposition of bifurcation curves and transformed phase portraits.

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References

  1. H N Li, S Y Xiao and S Y Wang, ASME PVP 2, 445 (2002)

    Google Scholar 

  2. W Ahmed, A Maqsood and R Riaz, Results Phys. 8 (2018)

  3. J Y Hou, X H Li, D W Zuo and Y N Li, Eur. Phys. J. Plus 6, 132 (2017)

    Google Scholar 

  4. Y Wang and J E Rubin, J. Comput. Neurosci. 3, 41 (2016)

    Google Scholar 

  5. D M Blitz, A E Pritchard, J K Latimer and A T Wakefield, J. Exp. Biol. 7, 220 (2017)

    Google Scholar 

  6. E Vitale, D Deneele, M Paris and G Russo, Appl. Clay Sci. 141, 36 (2017)

    Article  Google Scholar 

  7. L Cuevas, D Nei and C Manzie, Int. J. Robust Nonlinear Control 30, 14 (2020)

    Article  Google Scholar 

  8. M Peng, Z D Zhang, Z F Qu and Q S Bi, Pramana – J. Phys. 94, 14 (2020)

    Google Scholar 

  9. W H Mao, Z D Zhang and S Z Zhang, Chin. J. Theor. Appl. Mech. 53, 3 (2021)

    Google Scholar 

  10. A L Hodgkin and A F Huxley, The J. Physiol. 4, 117 (1952)

    Google Scholar 

  11. F Verhulst, Nonlinear Dyn. 4, 50 (2007)

    Google Scholar 

  12. A Haselbacher, F M Najjar, L Massa and R D Moser, J. Comput. Phys. 2, 229 (2010)

    Google Scholar 

  13. E M Izhikevich, Int. J. Bifurc. Chaos 6, 10 (2000)

    Google Scholar 

  14. J Rinzel, Bull. Math. Biol. 1, 52 (1990)

    Google Scholar 

  15. A Lerchner and J Rinzel, Neurocomputing 65, 777 (2005)

    Article  Google Scholar 

  16. M Brøns, M Desroches and M Krupa, Math. Popul. Studies 2, 22 (2015)

    Google Scholar 

  17. Z D Zhang, Y Y Li and Q S Bi, Phys. Lett. A 377, 975 (2013)

    Article  MathSciNet  Google Scholar 

  18. Y Q Xing, X K Chen, Z D Zhang and Q S Bi, Acta Phys. Sinica 9, 65 (2016)

    Google Scholar 

  19. H J Xu, Z D Zhang and M Peng, Nonlinear Dyn. 108, 2 (2022)

    Google Scholar 

  20. K Zimmermann, I Zeidis and V Lysenko, Appl. Math. Model. 89, 13 (2021)

    Article  Google Scholar 

  21. B Y Shen and Z D Zhang, Pramana – J. Phys. 95, 97 (2021)

    Google Scholar 

  22. Y F Lv, R Yuan and Y Z Pei, J. Comput. Appl. Math. 250 (2013)

  23. A A Transeth, R I Leine, C Glocker and K Y Pettersen, IEEE Trans. Robotics 2, 24 (2008)

    Google Scholar 

  24. F Angulo, G Olivar, G A Osorio, C M Escobar, J D Ferreira and J M Redondo, Commun. Nonlinear Sci. Numer. Simul. 12, 17 (2012)

    Google Scholar 

  25. J A Amador, G Olivar and F Angulo, Diff. Equ. Dyn. Syst. 21, 173 (2013)

    Article  Google Scholar 

  26. G R Chen and J H Lv, Lorenz systems family dynamic analysis of control and synchronization (Science Press, Beijing) (in Chinese)

  27. M Aqeel, A Azam and S Ahmad, Eur. Phys. J. Plus 10, 132 (2017)

    Google Scholar 

  28. L Qiang, N Benyamin and L Feng, Chaos Solitons Fractals 114, 230 (2018)

    Article  MathSciNet  Google Scholar 

  29. B Yan, S B He and S J Wang, Math. Prob. Eng.2468134, 1 (2020)

    Google Scholar 

  30. M L Ma , L W Xiong, Z Y Shu, Y J Fang and M J Wang, Int. J. Bifurc. Chaos 13, 31 (2021)

    Google Scholar 

  31. R Zhang, M Peng, Z D Zhang and Q S Bi, Chin. Phys. B 11, 27 (2018)

    Google Scholar 

  32. L Holden and T Erneux, SIAM J. Appl. Math. 4, 53 (1993)

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Acknowledgements

The author is thankful to editors and referees for the careful reading and valuable suggestions that improve the quality and description of this manuscript. This work was supported by National Natural Science Foundation of China (Grant No. 12102148) and Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No. 21KJB110010).

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Authors and Affiliations

  1. School of Mathematical Sciences, Jiangsu University, Zhenjiang , 212013, China

    Miao Peng

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  1. Miao Peng
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Correspondence to Miao Peng.

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Peng, M. Dynamics analysis in a non-smooth Lü system with two time scales. Pramana - J Phys 96, 227 (2022). https://doi.org/10.1007/s12043-022-02472-2

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  • DOI: https://doi.org/10.1007/s12043-022-02472-2

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