Skip to main content
SpringerLink
Log in

Stochastic P-bifurcation of fractional derivative Van der Pol system excited by Gaussian white noise

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

Abstract

This paper aimed to investigate the stochastic P-bifurcation of Van der Pol oscillator with a fractional derivative damping term driven by Gaussian white noise excitation. Firstly, based on the method of stochastic averaging method and Stratonovich–Khasminskii theorem, the corresponding Fokker–Plank–Kolmogorov (FPK) equation is deduced. To describe the P-bifurcation of system, the stationary probability densities of amplitude can be obtained by solving the FPK equation. Then, the effects of the fractional order, the fractional coefficient, and the intensity of Gaussian white noise on the fractional systems are discussed in detail. The results show that increasing order α will change obviously the number and the height of peaks under certain parameter conditions. Finally, comparing the analytical and numerical results, a very satisfactory agreement can be found.

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

Similar content being viewed by others

References

  1. S D Marinković, P M Rajković and M S Stanković Appl. Anal. Discrete Math. 1 311 (2007)

    Article  MathSciNet  Google Scholar 

  2. S Luo and L Li Nonlinear Dyn. 73 339 (2013)

    Article  MathSciNet  Google Scholar 

  3. A Schmidt and L Gaul Nonlinear Dyn. 29 37 (2002)

    Article  Google Scholar 

  4. R L Bagley and P J Torvik AIAA J. 23 918 (1985)

    Article  ADS  Google Scholar 

  5. L C Chen and W Q Zhu Acta Mech. 207 109 (2009)

    Article  Google Scholar 

  6. L C Chen, M L Deng and W Q Zhu Acta Mech. 206 133 (2009)

    Article  Google Scholar 

  7. J A Rad, S Kazem, M Shaban, K Parand and A Yildirim Math. Methods Appl. Sci. 37 329 (2014)

    Article  ADS  Google Scholar 

  8. F Hu, W Q Zhu and L C Chen Nonlinear Dyn. 70 1459 (2012)

    Article  Google Scholar 

  9. R C Koeller J. Appl. Mech. 51 299 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  10. M Alvelid and M Enelund J. Sound. Vib. 300 662 (2007)

    Article  ADS  Google Scholar 

  11. P J Torvik and R L Bagley J. Appl. Mech. 51 725 (1984)

    Google Scholar 

  12. R L Bagley and J Torvik AIAA J. 21 741 (2012)

    Article  ADS  Google Scholar 

  13. R L Bagley and P J Torvik AIAA J. 23 918 (1985)

    Article  ADS  Google Scholar 

  14. J A T Machado Math. Model. 46 560 (2012)

    Google Scholar 

  15. J A T Machado, A C Costa and M D Quelhas Commun. Nonlinear Sci. Numer. Simul. 16 2963 (2011)

    Article  ADS  Google Scholar 

  16. F Mainardi Chaos Solitons Fractals 7 1461 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  17. G Q Cai and Y K Lin Nonlinear Dyn. 24 3 (2001)

    Article  MathSciNet  Google Scholar 

  18. Y F Jin and X Luo Nonlinear Dyn. 72 185 (2013)

    Article  MathSciNet  Google Scholar 

  19. L C Chen, W Q Zhu Int. J. Nonlinear Mech. 46 1324 (2011)

    Article  ADS  Google Scholar 

  20. Z L Huang, W Q Zhu, Y Q Ni and J M Ko J. Sound. Vib. 254 245 (2002)

    Article  ADS  Google Scholar 

  21. Y Xiao, W Xu and L Wang Chaos 26 621 (2016)

    Google Scholar 

  22. Y G Yang, W Xu, Y H Sun, Y Xiao Commun. Nonlinear Sci. 42 62 (2017)

    Article  Google Scholar 

  23. J H Yang, M A F Sanjuán, H G Liu, G Litak and X Li Commun. Nonlinear Sci. Numer. Simul. 41 104 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  24. S M Xiao and Y F Jin Nonlinear Dyn. 90 2069 (2017)

    Article  Google Scholar 

  25. S J Ma, W Xu, W Li and T Fang Chin. Phys. 15 1231 (2006)

    Article  ADS  Google Scholar 

  26. L C Chen, Q Zhuang and W Q Zhu Acta Mech. 222 245 (2011)

    Article  Google Scholar 

  27. Y Xu, Y Li, D Liu, W Jia and H Huang Nonlinear Dyn. 74 745 (2013)

    Article  Google Scholar 

  28. Y Xu, Y G Li and D Liu J. Comput. Nonlinear Dyn. 9 031015 (2014)

    Article  Google Scholar 

  29. P D Spanos and B A Zeldin J. Eng. Mech. 123 290 (1997)

    Article  Google Scholar 

  30. O P Agrawal J. Vib. Acoust. 126 561 (2004)

    Article  Google Scholar 

  31. Y Jin Probabilistic Eng. Mech. 41 115 (2015)

    Article  Google Scholar 

  32. R S Barbosa, J A T Machado, B M Vinagre and A J Calderon J. Vib. Control 13 1291 (2007)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. W Eugene and Z Moshe Int. J. Eng. Sci. 3 213 (1965)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities under Nos. GK201502007 and GK201701001.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, China

    Y. Y. Ma & L. J. Ning

Authors
  1. Y. Y. Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. J. Ning
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. J. Ning.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, Y.Y., Ning, L.J. Stochastic P-bifurcation of fractional derivative Van der Pol system excited by Gaussian white noise. Indian J Phys 93, 61–66 (2019). https://doi.org/10.1007/s12648-018-1231-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12648-018-1231-3

Keywords

PACS No.

Search

Navigation