Skip to main content
SpringerLink
Log in

The Phase Transition in a Bistable Duffing System Driven by Lévy Noise

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In this paper, the phase transition and mean first passage time (MFPT) induced by Lévy noise is investigated in a bistable Duffing system numerically. First, we obtain the stationary probability density functions as the criterion of phase transition through the qualitative changes of the shapes of stationary probability distribution. Then, the influences of Lévy noise parameters are discussed. The results indicate that the stability index, noise intensity and skewness parameter can lead to phase transition behaviors in bistable Duffing system. In addition, the MFPT is calculated for various Lévy parameters. Different effects among stability index, noise intensity and the skewness parameter on the phase transition and MFPT are observed. Furthermore, distinctions between Lévy noise and Gaussian noise are provided.

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
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  2. Marathe, R., Bierbaum, V., Gomez, D., Klumpp, S.: Deterministic and stochastic descriptions of gene expression dynamics. J. Stat. Phys. 148, 608–627 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Zhang, X., Qian, H., Qian, M.: Stochastic theory of nonequilibrium steady states and its applications. Phys. Rep. 510, 1–86 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  4. Leimkuhler, B., Noorizadeh, E., Theil, F.: A gentle stochastic thermostat for molecular dynamics. J. Stat. Phys. 135, 261–277 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Ghosh, P.K., Misko, V.R., Marchesoni, F., Nori, F.: Self-propelled Janus particles in a ratchet: numerical simulations. Phys. Rev. L. 110, 268301 (2013)

    Article  ADS  Google Scholar 

  6. Kramer, B., Ohtsuki, T., Kettemann, S.: Random network models and quantum phase transitions in two dimensions. Phys. Rep. 417, 211–342 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  7. Xu, Y., Feng, J., Li, J.J., Zhang, H.Q.: Lévy noise induced switch in the gene transcriptional regulatory system. Chaos 23, 013110 (2013)

    Article  ADS  Google Scholar 

  8. Ghosh, P.K., Barik, D., Ray, D.S.: Noise-induced transition in a quantum system. Phys. Lett. A 342, 12–21 (2005)

    Article  ADS  Google Scholar 

  9. Cáceres, M.O.: Passage time statistics in a stochastic Verhulst model. J. Stat. Phys. 132, 487–500 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Ghosh, P.K., Bag, B.C., Ray, D.S.: Noised correlation-induced splitting of Kramers’ escape rate from a metastable state. J. Chem. Phys. 127, 044510 (2007)

    Article  ADS  Google Scholar 

  11. Ghosh, P.K., Bag, B.C., Ray, D.S.: Interference of stochastic resonances: splitting of Kramers’ rate. Phys. Rew. E 75, 032101 (2007)

    Article  ADS  Google Scholar 

  12. Nicolis, G., Nicolis, C., McKernan, D.: Stochastic resonance in chaotic dynamics. J. Stat. Phys. 70, 125–139 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70, 223–287 (1998)

    Article  ADS  Google Scholar 

  14. Xu, Y., Mahmoud, G.M., Xu, W., Lei, Y.M.: Suppressing chaos of a complex Duffing’s system using a random phase. Chaos Soliton. Fract. 23, 265–273 (2005)

    Article  ADS  MATH  Google Scholar 

  15. Ghosh, P.K., Marchesoni, F.: Brownian transport in narrow channels subject to transverse periodic force. Eur. Phys. J. Special Topics 187, 41–47 (2010)

    Article  ADS  Google Scholar 

  16. Bag, B.C.: Colored non-Gaussian noise driven systems: mean first passage time. Eur. Phys. J. B 34, 115–118 (2003)

    Article  ADS  Google Scholar 

  17. Dybiec, B., Gudowska-Nowak, E., Hänggi, P.: Escape driven by alpha-stable white noises. Phys. Rev. E 75, 021109 (2007)

    Article  ADS  Google Scholar 

  18. Chaudhuri, J.R., Chattopadhyay, S., Banik, S.K.: Multiplicative cross-correlated noise induced escape rate from a metastable state. J. Chem. Phys. 128, 154513 (2008)

    Article  ADS  Google Scholar 

  19. Øksendal, B.: Stochastic Differential Equations. Springer, New York (2003)

    Book  Google Scholar 

  20. Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  21. Comtet, A., Texier, C., Tourigny, Y.: Supersymmetric quantum mechanics with Lévy disorder in one dimension. J. Stat. Phys. 145, 1291–1323 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Bartumeus, F., Catalan, J., Fulco, U.L., Lyra, M.L., Viswanathan, G.M.: Optimizing the encounter rate in biological interactions: Lévy versus Brownian strategies. Phys. Rev. Lett. 88, 097901 (2002)

    Article  ADS  Google Scholar 

  23. West, B.J., Deering, W.: Fractal physiology for physicists: Lévy statistics. Phys. Rep. 246, 1–100 (1994)

    Article  ADS  Google Scholar 

  24. Shlesinger, M.F., Zaslavsky, G.M., Frisch, U.: Lévy Flights and Related Topics in Physics. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  25. Ditlevsen, P.D.: Observation of alpha-stable noise induced millennial climate changes from an ice-core record. Geophys. Res. Lett. 26, 1441–1444 (1999)

    Article  ADS  Google Scholar 

  26. Ai, B.Q., He, Y.F.: Directed transport driven by Lévy flights coexisting with sub-diffusion. J. Chem. Phys. 132, 094504 (2010)

    Article  ADS  Google Scholar 

  27. Imkeller, P., Pavlyukevich, I., Stauch, M.: First exit times of non-linear dynamical systems in R\(^{d}\) perturbed by multifractal Lévy noise. J. Stat. Phys. 141, 94–119 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Xu, Y., Feng, J., Li, J.J., Zhang, H.Q.: Stochastic bifurcation for a tumor-immune system with symmetric Lévy noise. Phys. A 392, 4739–4748 (2013)

    Article  MathSciNet  Google Scholar 

  29. Chechkin, A., Metzler, R., Gonchar, V., Klafter, J., Tanatarov, L.: First passage and arrival time densities for Lévy flights and the failure of the method of images. J. Phys. A 36, 537–544 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  30. Dybiec, B., Gudowska-Nowak, E.: Lévy stable noise-induced transitions: Stochastic resonance, resonant activation and dynamic hysteresis. J. Stat. Mech. 2009, P05004 (2009)

    Google Scholar 

  31. Chechkin, A., Gonchar, V., Klafter, J., Metzler, R., Tanatarov, L.: Stationary states of non-linear oscillators driven by Lévy noise. Chem. Phys. 284, 233–251 (2008)

    Article  ADS  Google Scholar 

  32. Chambers, J.M., Mallows, C.L., Stuck, B.W.: A method for simulating stable random variables. J. Am. Stat. Assoc. 71, 340–344 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  33. Weron, R.: On the Chambers–Mallows–Stuck method for simulating skewed stable random variables. Stat. Probab. Lett. 28, 165–171 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported by the NSF of China (11372247) and Shaanxi Province (2014JM1028), Program for NCET, the Fundamental Research Funds for the Central Universities (3102014JCQ01078) and NPU Foundation for Graduate Starting Seed.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China

    Yong Xu, Yongge Li, Juanjuan Li, Jing Feng & Huiqing Zhang

Authors
  1. Yong Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongge Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Juanjuan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jing Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Huiqing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, Y., Li, Y., Li, J. et al. The Phase Transition in a Bistable Duffing System Driven by Lévy Noise. J Stat Phys 158, 120–131 (2015). https://doi.org/10.1007/s10955-014-1129-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-014-1129-1

Keywords

Search

Navigation