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Stochastic P-bifurcation in a self-sustained tristable oscillator under random excitations

  • Regular Article - Statistical and Nonlinear Physics
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Abstract

Stochastic P-bifurcation in a self-sustained tristable oscillator subjected to two random excitations is explored. The self-sustained oscillator here has two stable limit cycles and a stable state, which is known for modeling the flutter of airfoils with large span in low-speed wind tunnels. By means of the stochastic averaging method, the stationary probability density function of the system amplitude for characterizing stochastic P-bifurcation is derived. The stationary probability density function can be easily transferred between the unimodal, bimodal and trimodal structure according to the critical parameters displayed in the bifurcation diagram. The effects of two noises and time delay on stochastic P-bifurcation are analyzed theoretically, whose correctness is verified numerically. The involvement of two noises and time delay not only expands the range of bifurcation parameters, but also enriches the bifurcation phenomenon.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: During the writing of the paper, the parameters of theoretical analysis and numerical simulation are clearly described in the paper. Therefore, the data will not be deposited.]

References

  1. W. Zhu, ASME 59, 18 (2006)

    Google Scholar 

  2. W. Zhu, Z. Ying, Adv. Mech. 43, 6 (2013)

    Google Scholar 

  3. K. Yuan, Z. Qiu, Sci. China. Phys. Mech. 53, 8 (2010)

    Google Scholar 

  4. S.D. Djeundam, R. Yamapi, T. Kofane, M.A. Aziz-Alaoui, Chaos 23, 033125 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  5. H. Zang, T. Zhang, Y. Zhang, Appl. Math. Comput. 260, 22 (2015)

    Google Scholar 

  6. A.R. Guido, G. Adiletta, Nonlinear Dyn. 19, 26 (1999)

    Google Scholar 

  7. Z. Wu, Y. Hao, Acta. Phys. Sin. 64, 060501 (2015)

    Article  Google Scholar 

  8. F. Chen, L. Zhou, Y. Chen, Sci. China. Technol. Sci. 54, 11 (2011)

    Article  ADS  Google Scholar 

  9. L. Ning, Nonlinear Dyn. 102, 12 (2020)

    Article  Google Scholar 

  10. Y. Li, Z. Wu, J. Vibroeng. 21, 12 (2019)

    Google Scholar 

  11. Z. Sun, J. Fu, Y. Xiao, W. Xu, Chaos 25, 083102 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  12. A.A. Andronov, A.A. Vitt, S.E. Khaikin, The Theory of Oscillations (Pergamon, Oxford, 1966)

    MATH  Google Scholar 

  13. G. Pushpita, S. Shrabani, S.S. Riaz, R. Deb Shankar, Phys. Rev. E. 83, 036205 (2011)

    Article  MathSciNet  Google Scholar 

  14. D. Biswas, T. Banerjee, J. Kurths, Chaos 27, 063110 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  15. A.C. Chamgoué, R. Yamapi, P. Woafo, Nonlinear Dyn. 73, 16 (2013)

    Google Scholar 

  16. H. Fröhlich, F. Kremer, Coherent Excitations in Biological Systems (Springer, New York, 1983)

    Book  Google Scholar 

  17. S. Kar, D.S. Ray, Front. Phys. 67, 137 (2004)

    Google Scholar 

  18. O. Decroly, A. Goldbeter, Proc. Natl. Acad. Sci. 79, 6917 (1982)

    Article  ADS  Google Scholar 

  19. J.C. Leloupf, A. Goldbeter, J. Theor. Biol. 198, 14 (1999)

    Google Scholar 

  20. M. Stich, M. Ipsen, A.S. Mikhailov, Phys. Rev. Lett. 86, 4406–9 (2001)

    Article  ADS  Google Scholar 

  21. I.M. De la Fuente, Biosystems 50, 14 (1999)

    Google Scholar 

  22. J. Dunlap, Endeavour 20, 1 (1996)

    Article  Google Scholar 

  23. M. Alamgir, I.R. Epstein, J. Am. Chem. Soc. 105, 2 (1983)

    Google Scholar 

  24. J. Yan, A. Goldbeter, J. R. Soc. Interface 16, 12 (2019)

    Google Scholar 

  25. E. Brun, B. Derighetti, D. Meier, R. Holzner, M. Ravani, J. Opt. Soc. 2, 156 (1985)

    Article  ADS  Google Scholar 

  26. W. Wischert, A. Wunderlin, A. Pelster, M. Olivier, J. Groslambert, Phys. Rev. E. 49, 16 (1994)

    Article  Google Scholar 

  27. L. Ning, Z. Ma, Int. J. Bifurc. Chaos 28, 1850127 (2018)

    Article  Google Scholar 

  28. J. Yang, M.A. Sanjuán, H. Liu, Commun. Nonlinear Sci. Numer. Simul. 22, 10 (2015)

    Google Scholar 

  29. J. Yang, X. Liu, Chaos 20, 033124 (2010)

    Article  ADS  Google Scholar 

  30. R. Yamapi, G. Filatrella, M.A. Aziz-Alaoui, Chaos 20, 013114 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  31. Y. Xu, J. Feng, J. Li, H. Zhang, Phys. A 392, 4739 (2013)

    Article  MathSciNet  Google Scholar 

  32. A. Zakharova, N. Semenova, V. Anishchenko, E. Schöll, Chaos 27, 4320 (2017)

    Article  Google Scholar 

  33. Z. Wu, Y. Hao, Phys. Mech. Astron. 43, 5 (2013)

    Google Scholar 

  34. M. Gaudreault, F. Drolet, J. Viñals, Phys. Rev. E. 85, 056214 (2012)

    Article  ADS  Google Scholar 

  35. Z. Sun, J. Zhang, X. Yang, W. Xu, Chaos 27, 3102 (2017)

    Google Scholar 

  36. Z. Huang, W. Zhu, Y. Suzuki, J. Sound. Vib. 238, 22 (2000)

    Google Scholar 

  37. J.B. Roberts, P.D. Spanos, Int. J. Nonlinear Mech. 21, 23 (1986)

    Article  Google Scholar 

  38. W. Zhu, Z. Huang, Y. Suzuki, Int. J. Nonlinear Mech. 36, 15 (2001)

    Google Scholar 

  39. L. Liu, W. Xu, X. Yue, W. Jia, Probab. Eng. Mech. 61, 103085 (2020)

    Article  Google Scholar 

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Acknowledgements

This work was partially funded by the National Natural Science Foundation of China under Grant no. 11202120. The authors also would like to express their appreciation to the reviewers for their insightful reading and helpful comments.

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Authors and Affiliations

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

    Yingying Wang & Lijuan Ning

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  1. Yingying Wang
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  2. Lijuan Ning
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Contributions

Lijuan Ning and Yingying Wang designed the research; Yingying Wang performed the research; Lijuan Ning and Yingying Wang wrote the manuscript and revised it.

Corresponding author

Correspondence to Lijuan Ning.

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Wang, Y., Ning, L. Stochastic P-bifurcation in a self-sustained tristable oscillator under random excitations. Eur. Phys. J. B 95, 34 (2022). https://doi.org/10.1140/epjb/s10051-022-00286-0

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  • DOI: https://doi.org/10.1140/epjb/s10051-022-00286-0

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