Skip to main content
SpringerLink
Log in

Stochastic resonance in a fractional oscillator subjected to multiplicative trichotomous noise

  • Original Article
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, stochastic resonance in a fractional oscillator with a power-law friction kernel subject to random damping is investigated both theoretically and numerically. The influence of a fluctuating damping is modeled as multiplicative trichotomous noise. The exact expression of the first moment of the system\('\)s steady response has been calculated. It is shown that the interplay between multiplicative trichotomous noise and memory effect leads to stochastic resonance in the proposed system. The output amplitude gain (OAG) shows non-monotonic dependence on the driving frequency of the input signal and the characteristics of the noise. Furthermore, a multiresonance-like behavior of the OAG as function of the driving frequency and the inverse-stochastic resonance behavior of the OAG as function of the noise switching rate are observed, which is previously reported and believed to be absent in the case of the non-memory oscillator. Finally, some numerical simulations are performed to support the theoretical analyses.

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. Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys. A 14, L453457 (1981)

    Article  MathSciNet  Google Scholar 

  2. Gammaitoni, L., Hnggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70, 223287 (1998)

    Article  Google Scholar 

  3. Huelga, S.F., Plenio, M.B.: Stochastic resonance phenomena in quantum many-body systems. Phys. Rev. Lett. 98, 170601 (2007)

    Article  MATH  Google Scholar 

  4. Zhong, W.R., Shao, Y.Z., He, Z.H.: Pure multiplicative stochastic resonance of a theoretical anti-tumor model with seasonal modulability. Phys. Rev. E 73, 060902 (2006)

    Article  Google Scholar 

  5. McDonnell, M., Abbott, D.: What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology. PLos Comput. Biol. 5(5), 1000348 (2009)

    Article  MathSciNet  Google Scholar 

  6. Duan, F., Abbott, D.: Binary modulated signal detection in a bistable receiver with stochastic resonance. Phys. A 376, 173–190 (2013)

    Article  Google Scholar 

  7. Djurhuus, T., Krozer, V.: Numerical analysis of stochastic resonance in a bistable circuit. Int. J. Circ. Theor. Appl. Published online in Wiley (2016)

  8. Chang, C.H., Tian, Y.T.: Stochastic resonance in a biological motor under complex fluctuations. Phys. Rev. E 69, 021914 (2004)

  9. Takayasu, H., Sato, A.-H., Takayasu, M.: Stable infinite variance fluctuations in randomly amplified Langevin systems. Phys. Rev. Lett. 79, 966 (1997)

    Article  MATH  Google Scholar 

  10. Krawiecki, A., Hoyst, J.A.: Stochastic resonance as a model for financial market crashes and bubbles. Phys. A 317, 597–608 (2003)

    Article  MATH  Google Scholar 

  11. Gitterman, M.: Overdamped harmonic oscillator with multiplicative noise. Phys. A 352(24), 309–334 (2005)

    Article  MathSciNet  Google Scholar 

  12. Gitterman, M., Shapiro, I.: Stochastic resonance in a harmonic oscillator with random mass subject to asymmetric dichotomous noise. J. Stat. Phys. 144, 139149 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Goychuk, I.: Subdiffusive Brownian ratchets rocked by a periodic force. Chem. Phys. 375, 450457 (2010)

    Google Scholar 

  14. Mankin, R., Laas, K., Laas, T., Reiter, E.: Stochastic multiresonance and correlation-time-controlled stability for a harmonic oscillator with fluctuating frequency. Phys. Rev. E 78, 031120 (2008)

    Article  Google Scholar 

  15. Gitterman, M.: Harmonic oscillator with multiplicative noise: nonmonotonic dependence on the strength and the rate of dichotomous noise. Phys. Rev. E 67, 057103 (2003)

    Article  Google Scholar 

  16. Gitterman, M.: Stochastic oscillator with random mass: new type of Brownian motion. Phys. A 395, 11–21 (2014)

    Article  MathSciNet  Google Scholar 

  17. Mason, T.G., Weitz, D.A.: Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids. Phys. Rev. Lett 74, 1250–1253 (1995)

    Article  Google Scholar 

  18. Gtze, W., Sjgren, L.: Relaxation processes in supercooled liquids. Rep. Prog. Phys. 55, 241 (1992)

    Article  Google Scholar 

  19. Carlsson, T., Sjgren, L., Mamontov, E., PsiukMaksymowicz, K.: Irreducible memory function and slow dynamics in disordered systems. Phys. Rev. E 75, 031109 (2007)

    Article  Google Scholar 

  20. Gu, Q., Schiff, E.A., Grebner, S., Wang, F., Schwarz, R.: Non-Gaussian transport measurements and the Einstein relation in amorphous silicon. Phys. Rev. Lett. 76, 3196 (1996)

    Article  Google Scholar 

  21. Burov, S., Barkai, E.: Fractional Langevin equation: overdamped, underdamped, and critical behaviors. Phys. Rev. E 78, 031112 (2008)

    Article  MathSciNet  Google Scholar 

  22. Kou, S.C., Xie, X.S.: Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. Phys. Rev. Lett. 93, 180603 (2004)

    Article  Google Scholar 

  23. Zhong, S.C., Wei, K., Gao, S.L., Ma, H.: Stochastic resonance in a linear fractional Langevin equation. J. Stat. Phys. 150(5), 867–880 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Arqub, O.A., El-Ajou, A., Momani, S.: Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. J. Comput. Phys. 293, 385–399 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Arqub, O.A., Maayah, B.: Solutions of Bagley-Torvik and Painlev equations of fractional order using iterative reproducing kernel algorithm. Neural Comput. Appl. (2016). doi:10.1007/s00521-016-2484-4

    Google Scholar 

  26. Zhong, S.C., Ma, H., Peng, H., Zhang, L.: Stochastic resonance in a harmonic oscillator with fractional-order external and intrinsic dampings. Nonlinear Dyn. 82, 535545 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mankin, R., Rekker, A.: Memory-enhanced energetic stability for a fractional oscillator with fluctuating frequency. Phys. Rev. E 81, 041122 (2010)

    Article  Google Scholar 

  28. Litak, G., Borowiec, M.: On simulation of a bistable system with fractional damping in the presence of stochastic coherence resonance. Nonlinear Dyn. 77, 681–686 (2014)

    Article  MathSciNet  Google Scholar 

  29. Bier, M.: Reversals of noise induced flow. Phys. Lett. A 211, 12–18 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Berghaus, C., Kahlert, U., Schnakenberg, J.: Current reversal induced by a cyclic stochastic process. Phys. Lett. A 224, 243248 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lin, L.F., Chen, C., Wang, H.Q.: Trichotomous noise induced stochastic resonance in a fractional oscillator with random damping and random frequency. J. Stat. Mech. 023201 (2016)

  32. Zheng, L., Li, J.: Stochastic resonance of an under-damped linear system driven by trichotomous noise. In: International Conference on Vulnerability and Risk Analysis and Management, pp. 1933–1940 (1940)

  33. Lang, R.L., Yang, L., Qin, H.L., Di, G.H.: Trichotomous noise induced stochastic resonance in a linear system. Nonlinear Dyn. 69, 1423–1427 (2012)

    Article  MathSciNet  Google Scholar 

  34. Mankin, R., Laas, K., Laas, T., Reiter, E.: Stochastic multiresonance and correlation-time-controlled stability for a armonic oscillator with fluctuating frequency. Phys. Rev. E 78, 031120 (2008)

    Article  Google Scholar 

  35. Doering, C.R., Horsthemke, W., Riordan, J.: Nonequilibrium fluctuation-induced transport. Phys. Rev. Lett. 72, 2984 (1994)

    Article  Google Scholar 

  36. Elston, T.C., Doering, C.R.: Numerical and analytical studies of nonequilibrium fluctuation-induced transport processes. J. Stat. Phys. 83, 359 (1996)

    Article  Google Scholar 

  37. Soika, E., Mankin, R., Ainsaar, A.: Resonant behavior of a fractional oscillator with fluctuating frequency. Phys. Rev. E 81, 011141 (2010)

    Article  Google Scholar 

  38. Soika, E., Mankin, R., Lumi, N.: Parametric resonance of a harmonic oscillator with fluctuating mass. In: AIP Conference Proceedings, p. 233 AIP, New York (2012)

  39. Zhang, W., Di, G.: Stochastic resonance in a harmonic oscillator with damping trichotomous noise. Nonlinear Dyn. 77, 1589 (2014)

    Article  MathSciNet  Google Scholar 

  40. Jiang, S., Guo, F., Zhou, Y.R., Gu, T.X.: Stochastic resonance in a harmonic oscillator with randomizing damping by asymmetric dichotomous noise. In: International Conference on Communication Circuits and Systems, p. 1113, Fukuoka, Japan (2007)

  41. Gitterman, M.: Harmonic oscillator with fluctuating damping parameter. Phys. Rev. E 69, 041101 (2004)

    Article  Google Scholar 

  42. West, B., Seshadri, V.: Model of gravity wave growth due to fluctuations in the air-sea coupling parameter. J. Geophys. Res. 86, 4293–4298 (1981)

    Article  Google Scholar 

  43. Gitterman, M., Shapiro, B.Y., Shapiro, I.: Phase transitions in vortex matter driven by bias current. Phys. Rev. B 65, 174510 (2002)

  44. Kubo, R.: The fluctuation dissipation theorem. Rep. Prog. Phys. 29, 255284 (1966)

    Article  Google Scholar 

  45. Shapiro, V.E., Loginov, V.M.: Formulae of differentiation and their use for solving stochastic equations. Phys. A 91, 563 (1978)

    Article  MathSciNet  Google Scholar 

  46. Deng, W.H., Li, C.: Numerical Modelling. In: Tech Press, Rijeka 355374 (2012)

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No.11301361)

Author information

Authors and Affiliations

  1. College of Mathematics, Sichuan University, Chengdu, 610065, China

    Ruibin Ren, Maokang Luo & Ke Deng

Authors
  1. Ruibin Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maokang Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ke Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ke Deng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ren, R., Luo, M. & Deng, K. Stochastic resonance in a fractional oscillator subjected to multiplicative trichotomous noise. Nonlinear Dyn 90, 379–390 (2017). https://doi.org/10.1007/s11071-017-3669-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3669-9

Keywords

Search

Navigation