Advertisement

Indian Journal of Physics

, Volume 93, Issue 1, pp 61–66 | Cite as

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

  • Y. Y. Ma
  • L. J. NingEmail author
Original Paper
  • 112 Downloads

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.

Keywords

Stochastic bifurcation Van der Pol Fractional derivative Stochastic averaging method 

PACS No.

02.50.-r 05.45.-a 05.40.Ca 

Notes

Acknowledgements

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

References

  1. [1]
    S D Marinković, P M Rajković and M S Stanković Appl. Anal. Discrete Math. 1 311 (2007)MathSciNetCrossRefGoogle Scholar
  2. [2]
    S Luo and L Li Nonlinear Dyn. 73 339 (2013)MathSciNetCrossRefGoogle Scholar
  3. [3]
    A Schmidt and L Gaul Nonlinear Dyn. 29 37 (2002)CrossRefGoogle Scholar
  4. [4]
    R L Bagley and P J Torvik AIAA J. 23 918 (1985)ADSCrossRefGoogle Scholar
  5. [5]
    L C Chen and W Q Zhu Acta Mech. 207 109 (2009)CrossRefGoogle Scholar
  6. [6]
    L C Chen, M L Deng and W Q Zhu Acta Mech. 206 133 (2009)CrossRefGoogle Scholar
  7. [7]
    J A Rad, S Kazem, M Shaban, K Parand and A Yildirim Math. Methods Appl. Sci. 37 329 (2014)ADSCrossRefGoogle Scholar
  8. [8]
    F Hu, W Q Zhu and L C Chen Nonlinear Dyn. 70 1459 (2012)CrossRefGoogle Scholar
  9. [9]
    R C Koeller J. Appl. Mech. 51 299 (1984)ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    M Alvelid and M Enelund J. Sound. Vib. 300 662 (2007)ADSCrossRefGoogle Scholar
  11. [11]
    P J Torvik and R L Bagley J. Appl. Mech. 51 725 (1984)Google Scholar
  12. [12]
    R L Bagley and J Torvik AIAA J. 21 741 (2012)ADSCrossRefGoogle Scholar
  13. [13]
    R L Bagley and P J Torvik AIAA J. 23 918 (1985)ADSCrossRefGoogle Scholar
  14. [14]
    J A T Machado Math. Model. 46 560 (2012)Google Scholar
  15. [15]
    J A T Machado, A C Costa and M D Quelhas Commun. Nonlinear Sci. Numer. Simul. 16 2963 (2011)ADSCrossRefGoogle Scholar
  16. [16]
    F Mainardi Chaos Solitons Fractals 7 1461 (1996)ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    G Q Cai and Y K Lin Nonlinear Dyn. 24 3 (2001)MathSciNetCrossRefGoogle Scholar
  18. [18]
    Y F Jin and X Luo Nonlinear Dyn. 72 185 (2013)MathSciNetCrossRefGoogle Scholar
  19. [19]
    L C Chen, W Q Zhu Int. J. Nonlinear Mech. 46 1324 (2011)ADSCrossRefGoogle Scholar
  20. [20]
    Z L Huang, W Q Zhu, Y Q Ni and J M Ko J. Sound. Vib. 254 245 (2002)ADSCrossRefGoogle Scholar
  21. [21]
    Y Xiao, W Xu and L Wang Chaos 26 621 (2016)Google Scholar
  22. [22]
    Y G Yang, W Xu, Y H Sun, Y Xiao Commun. Nonlinear Sci. 42 62 (2017)CrossRefGoogle Scholar
  23. [23]
    J H Yang, M A F Sanjuán, H G Liu, G Litak and X Li Commun. Nonlinear Sci. Numer. Simul. 41 104 (2016)ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    S M Xiao and Y F Jin Nonlinear Dyn. 90 2069 (2017)CrossRefGoogle Scholar
  25. [25]
    S J Ma, W Xu, W Li and T Fang Chin. Phys. 15 1231 (2006)ADSCrossRefGoogle Scholar
  26. [26]
    L C Chen, Q Zhuang and W Q Zhu Acta Mech. 222 245 (2011)CrossRefGoogle Scholar
  27. [27]
    Y Xu, Y Li, D Liu, W Jia and H Huang Nonlinear Dyn. 74 745 (2013)CrossRefGoogle Scholar
  28. [28]
    Y Xu, Y G Li and D Liu J. Comput. Nonlinear Dyn. 9 031015 (2014)CrossRefGoogle Scholar
  29. [29]
    P D Spanos and B A Zeldin J. Eng. Mech. 123 290 (1997)CrossRefGoogle Scholar
  30. [30]
    O P Agrawal J. Vib. Acoust. 126 561 (2004)CrossRefGoogle Scholar
  31. [31]
    Y Jin Probabilistic Eng. Mech. 41 115 (2015)CrossRefGoogle Scholar
  32. [32]
    R S Barbosa, J A T Machado, B M Vinagre and A J Calderon J. Vib. Control 13 1291 (2007)CrossRefGoogle Scholar
  33. [33]
    M S Tavazoei, M Haeri, M Attari, S Bolouki and M Siami J. Vib. Control 15 803 (2009)MathSciNetCrossRefGoogle Scholar
  34. [34]
    W Eugene and Z Moshe Int. J. Eng. Sci. 3 213 (1965)CrossRefGoogle Scholar

Copyright information

© Indian Association for the Cultivation of Science 2018

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina

Personalised recommendations