Skip to main content
SpringerLink
Log in

Bifurcation and stability analysis of commensurate fractional-order van der Pol oscillator with time-delayed feedback

  • Original Paper
  • Published:
Indian Journal of Physics Aims and scope Submit manuscript

Abstract

The stability and existence conditions of Hopf bifurcation of a commensurate fractional-order van der Pol oscillator with time-delayed feedback are studied. Firstly, the necessary and sufficient conditions for the asymptotic stability of the equilibrium point of fractional-order van der Pol oscillator with linear displacement feedback are obtained, and it is found that the conditions are not only related to the feedback gain, but also to the fractional order. Secondly, regarding time delay as a bifurcation parameter, the stability of the commensurate fractional-order van der Pol system with time-delayed feedback is investigated based on the characteristic equation. Under some conditions, the critical value of time delay is calculated. The equilibrium point is stable when the parameter is less than the critical value and will be unstable if the parameter is greater than it. Moreover, the conditions for the occurrence of Hopf bifurcation are obtained. Finally, choosing four typical system parameters, some numerical simulations are carried out to verify the correctness of the obtained theoretical results.

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

Similar content being viewed by others

References

  1. I Podlubny Fractional Differential Equations (San Diego: Academic Press) (1999)

    MATH  Google Scholar 

  2. Z H Wang and H Y Hu J. Sound Vib. 233 215 (2000)

    Article  ADS  Google Scholar 

  3. H Y Hu and Z H Wang Dynamics of Controlled Mechanical Systems with Delayed Feedback (New York: Springer) (2002)

    Book  Google Scholar 

  4. C P Li and G J Peng Chaos Solit. Fract. 22 443 (2004)

    Article  ADS  Google Scholar 

  5. E Ahmed, A M A El-Sayed and H A A El-Saka Phys. Lett. A 358 1 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  6. I Petráš Fractional-Order Nonlinear System (Beijing: Higher Education Press) (2011)

    Book  Google Scholar 

  7. C Donato and G Giuseppe Int. J. Bifurc. Chaos 18 1845 (2008)

    Article  Google Scholar 

  8. K H Sun, X Wang and J C Sprott Int. J. Bifurc. Chaos 20 1209 (2010)

    Article  Google Scholar 

  9. Y J Shen, S P Yang and H J Xing Acta Phys. Sinica 61 110505 (2012)

    Google Scholar 

  10. Y J Shen, S P Yang, H J Xing and H X Ma Commun. Nonlinear Sci. 17 3092 (2012)

    Article  Google Scholar 

  11. Y J Shen, S P Yang, H J Xing and H X Ma Int. J. Non-Linear Mech. 47 975 (2012)

    Article  ADS  Google Scholar 

  12. J J Sun and Q Ding Advances in Analysis and Control of Time-Delayed Dynamical Systems (Beijing: Higher Education Press) (2013)

    Book  Google Scholar 

  13. V E Tarasov Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Beijing: Higher Education Press) (2013)

  14. V V Uchaikin Fractional Derivatives for Physicists and Engineers (Beijing: Higher Education Press) (2013)

  15. C P Li and F H Zeng Int. J. Bifurc. Chaos 22 1230014 (2012)

    Article  Google Scholar 

  16. L C Chen, W H Wang, Z S Li and W Q Zhu Int. J. Non–Linear Mech. 48 44 (2013)

    Article  ADS  Google Scholar 

  17. Y J Shen, P Wei and S P Yang Nonlinear Dyn. 77 1629 (2014)

    Article  Google Scholar 

  18. L C Chen and W Q Zhu Theor. Appl. Mech. Lett. 4 013010 (2014)

    Article  Google Scholar 

  19. Y G Yang, W Xu, X D Gu and Y H Sun Chaos Solit. Fract. 77 190 (2015)

    Article  ADS  Google Scholar 

  20. Y J Shen, S F Wen, X H Li, S P Yang and H J Xing Nonlinear Dyn. 85 1457 (2016)

    Article  Google Scholar 

  21. Y Xu, Y G Li and D Liu Nonlinear Dyn. 83 2311 (2016)

    Article  Google Scholar 

  22. J Y Hou, X H Li and J F Chen J. Vibroeng. 18 4812 (2016)

    Article  Google Scholar 

  23. J F Chen, X H Li, J H Tang and Y F Liu Shock Vib. 5975329 (2017)

  24. Y Y Ma and L J Ning Indian J. Phys. 93 61 (2019)

    Article  ADS  Google Scholar 

  25. M S Tavazoei, M Haeri, M Attari, S Bolouki and M Siami J. Vib. Control 15 803 (2009)

    Article  MathSciNet  Google Scholar 

  26. M Attari, M Haeri and M S Tavazoei Nonlinear Dyn. 61 265(2010)

    Article  Google Scholar 

  27. Z J Guo, A Y T Leung and H X Yang Appl. Math. Model. 35 3918 (2011)

    Article  MathSciNet  Google Scholar 

  28. A Y T Leung, H X Yang and Z J Guo J. Sound Vib. 331 111 (2012)

    Article  Google Scholar 

  29. Y J Shen, S P Yang and C Y Sui Chaos Solit. Fract. 67 94 (2014)

    Article  ADS  Google Scholar 

  30. M Xiao, G P Jiang, W X Zheng, S L Yan, Y H Wan and C X Fan Asian J. Control 17 1756 (2015)

    Article  MathSciNet  Google Scholar 

  31. W H Deng, C P Li and J H Lü Nonlinear Dyn. 48 409 (2007)

    Article  Google Scholar 

  32. S F Wen, Y J Shen, S P Yang and J Wang Chaos Solit. Fract. 94 54 (2017)

    Article  ADS  Google Scholar 

  33. C D Huang, J D Cao, M Xiao, A Alsaedi and T Hayat Appl. Math. Comput. 292 210 (2017)

    MathSciNet  Google Scholar 

  34. G M Mahmoud, A A Arafa, T M Abed-Elhameed and E E Mahmoud Chaos Solit. Fract. 104 680 (2017)

    Article  ADS  Google Scholar 

  35. X L Li and J J Wei Chaos Solit. Fract. 26 519 (2005)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work reported in this paper was supported by the State Scholarship Fund from the China Scholarship Council (CSC), the National Natural Science Foundation of China (Nos. 11772206, 11672191, 11602152), the Natural Science Foundation of Hebei Province (E2018210056), Hundred Talents Innovation Program of Hebei Province (SLRC2017053), and the Training Program for Leading Talent in University Innovative Research Team in Hebei Province (LJRC018).

Author information

Authors and Affiliations

  1. Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, People’s Republic of China

    Jufeng Chen & Xianghong Li

  2. Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, 050043, People’s Republic of China

    Yongjun Shen & Shaopu Yang

  3. Transportation Institute, Shijiazhuang Tiedao University, Shijiazhuang, 050043, People’s Republic of China

    Shaofang Wen

Authors
  1. Jufeng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongjun Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xianghong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shaopu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Shaofang Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongjun Shen.

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, J., Shen, Y., Li, X. et al. Bifurcation and stability analysis of commensurate fractional-order van der Pol oscillator with time-delayed feedback. Indian J Phys 94, 1615–1624 (2020). https://doi.org/10.1007/s12648-019-01589-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12648-019-01589-2

Keywords

PACS Nos.

Search

Navigation