Stochastic resonance in an overdamped system with a fractional power nonlinearity: Analytical and re-scaled analysis

  • Shuai Zhang
  • Yonglun Yao
  • Zhencai Zhu
  • Jianhua YangEmail author
  • Gang Shen
Regular Article


In our former work (Eur. Phys. J. Plus 132, 432 (2017)), we investigated the stochastic resonance phenomenon in the overdamped bistable system with a fractional power nonlinearity. However, the analytical explanations are missing and the former work only considers the low-frequency signal excitation case. In the present work, we give the analytical result based on the two-state theory. Moreover, through the general scale transformation method, we make the stochastic resonance occur in the system with an arbitrary high-frequency excitation. Further, as a new result, we find that the fractional-order value can also induce stochastic resonance. The meaning of the study lies in its application in engineering fields. By finding the optimal fractional-order value, we can obtain a much higher signal-to-noise ratio in the signal processing issues. Numerical simulations and experimental vibration signal simulations verify the analytical results.


  1. 1.
    R. Benzi, A. Sutera, A. Vulpiani, J. Phys. A 14, L453 (1981)CrossRefGoogle Scholar
  2. 2.
    S. Fauve, F. Heslot, Phys. Lett. A 97, 5 (1983)CrossRefGoogle Scholar
  3. 3.
    L. Gammaitoni, F. Marchesoni, E. Menichella-Saetta, S. Santucci, Phys. Rev. Lett. 62, 349 (1989)CrossRefGoogle Scholar
  4. 4.
    J.H. Yang, M.A.F. Sanjuán, H.G. Liu, G. Litak, X. Li, Commun. Nonlinear Sci. Numer. Simul. 41, 104 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Z.J. Qiao, Y.G. Lei, J. Lin, F. Jia, Mech. Syst. Signal Process. 84, 731 (2017)CrossRefGoogle Scholar
  6. 6.
    Y.F. Guo, J.G. Tan, Commun. Nonlinear Sci. Numer. Simul. 18, 2852 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    N.G. Stocks, Phys. Rev. Lett. 84, 2310 (2000)CrossRefGoogle Scholar
  8. 8.
    F. Duan, F. Chapeau-Blondeau, D. Abbott, Phys. Lett. A 380, 33 (2016)CrossRefGoogle Scholar
  9. 9.
    N.G. Stocks, N.D. Stein, P.V.E. McClintock, J. Phys. A 26, L385 (1993)CrossRefGoogle Scholar
  10. 10.
    N.V. Agudov, A.V. Krichigin, D. Valenti, B. Spagnolo, Phys. Rev. E 81, 051123 (2010)CrossRefGoogle Scholar
  11. 11.
    J.M. Li, M. Li, J.F. Zhang, J. Sound Vib. 40, 139 (2017)CrossRefGoogle Scholar
  12. 12.
    V.S. Anishchenko, A.B. Neiman, M.A. Safanova, J. Stat. Phys. 70, 183 (1993)CrossRefGoogle Scholar
  13. 13.
    A. Crisanti, M. Falcioni, G. Paladin, A. Vulpiani, J. Phys. A 27, L597 (1994)CrossRefGoogle Scholar
  14. 14.
    J.F. Lindner, B.K. Meadows, W.L. Ditto, M.E. Inchiosa, A.R. Bulsara, Phys. Rev. Lett. 75, 3 (1995)CrossRefGoogle Scholar
  15. 15.
    S. Kim, S.H. Park, H.B. Pyo, Phys. Rev. Lett. 82, 1620 (1999)CrossRefGoogle Scholar
  16. 16.
    H.T. Li, W.Y. Qin, W.Z. Deng, R.L. Tian, Eur. Phys. J. Plus 131, 60 (2016)CrossRefGoogle Scholar
  17. 17.
    Y. Hirano, Y. Segawa, T. Kawai, T. Matsumoto, J. Phys. Chem. C 117, 140 (2012)CrossRefGoogle Scholar
  18. 18.
    D. Alcor, J.F. Allemand, E. Cogne-Laage, V. Croquette, F. Ferrage, L. Jullien, A. Kononov, A. Lemarchand, J. Phys. Chem. B 109, 1318 (2005)CrossRefGoogle Scholar
  19. 19.
    G. Rodrigo, N.G. Stocks, Trends Biochem. Sci. 43, 483 (2018)CrossRefGoogle Scholar
  20. 20.
    D.Q. Guo, M. Perc, Y.S. Zhang, P. Xu, D.Z. Yao, Phys. Rev. E 96, 022415 (2017)CrossRefGoogle Scholar
  21. 21.
    Q.W. Li, Z. Li, IEEE Trans. Veh. Technol. 63, 1717 (2014)CrossRefGoogle Scholar
  22. 22.
    P. Pfeffer, F. Hartmann, S. Höfling, M. Kamp, L. Worschech, Phys. Rev. Appl. 4, 014011 (2015)CrossRefGoogle Scholar
  23. 23.
    Y.G. Leng, T.Y. Wang, Y. Guo, Y.G. Xu, S.B. Fan, Mech. Syst. Signal Process. 21, 138 (2007)CrossRefGoogle Scholar
  24. 24.
    Y. Qin, Y. Tao, Y. He, B.P. Tang, J. Sound Vib. 333, 7386 (2014)CrossRefGoogle Scholar
  25. 25.
    J.M. Li, X.F. Chen, Z.H. Du, Z.W. Fang, Renew. Energy 60, 7 (2013)CrossRefGoogle Scholar
  26. 26.
    Z.H. Lai, Y.G. Leng, Mech. Syst. Signal Process. 81, 60 (2016)CrossRefGoogle Scholar
  27. 27.
    J.H. Yang, M.A.F. Sanjuán, P.P. Chen, H.G. Liu, Eur. Phys. J. Plus 132, 432 (2017)CrossRefGoogle Scholar
  28. 28.
    B. McNamara, K. Wiesenfeld, Phys. Rev. A 39, 4854 (1989)CrossRefGoogle Scholar
  29. 29.
    A. Ichiki, Y. Tadokoro, Phys. Lett. A 377, 185 (2013)CrossRefGoogle Scholar
  30. 30.
    J.M.G. Vilar, J.M. Rubi, Phys. Rev. Lett. 77, 2863 (1996)CrossRefGoogle Scholar
  31. 31.
    V.N. Chizhevsky, G. Giacomelli, Phys. Rev. A 71, 011801 (2005)CrossRefGoogle Scholar
  32. 32.
    Z. Gingl, R. Vajtai, L.B. Kiss, Chaos Solitons Fractals 11, 1929 (2000)CrossRefGoogle Scholar
  33. 33.
    X.F. Zhang, N.Q. Hu, Z. Cheng, L. Hu, Chin. J. Mech. Eng. 25, 1287 (2012)CrossRefGoogle Scholar
  34. 34.
    Y.G. Leng, T.Y. Wang, Acta Phys. Sin. 52, 2432 (2003)Google Scholar
  35. 35.
    J.R. Yang, C.J. Wu, J.H. Yang, H.G. Liu, J. Comput. Nonlinear. Dyn. 13, 031009 (2018)CrossRefGoogle Scholar
  36. 36.
    J.Y. Tang, X.F. Chen, J.Y. Wang, H.X. Chen, H.R. Cao, Y.Y. Zi, Z.J. He, Mech. Syst. Signal Process. 23, 811 (2009)CrossRefGoogle Scholar
  37. 37.
    Y.G. Leng, Y.S. Leng, T.Y. Wang, Y. Guo, J. Sound Vib. 292, 788 (2006)CrossRefGoogle Scholar
  38. 38.
    D.W. Huang, J.H. Yang, J.L. Zhang, H.G. Liu, Proc. Inst. Mech. Eng. C 13, 2352 (2018)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mechatronic EngineeringChina University of Mining and TechnologyXuzhouChina
  2. 2.Jiangsu Key Laboratory of Mine Mechanical and Electrical EquipmentChina University of Mining and TechnologyXuzhouChina

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