The European Physical Journal Special Topics

, Volume 216, Issue 1, pp 133–138 | Cite as

Stationary states in bistable system driven by Lévy noise

  • O.Yu. SliusarenkoEmail author
  • D.A. Surkov
  • V.Yu. Gonchar
  • A.V. Chechkin
Regular Article


We study the properties of the probability density function (PDF) of a bistable system driven by heavy tailed white symmetric Lévy noise. The shape of the stationary PDF is found analytically for the particular case of the Lévy index α = 1 (Cauchy noise). For an arbitrary Lévy index we employ numerical methods based on the solution of the stochastic Langevin equation and space fractional kinetic equation. In contrast to the bistable system driven by Gaussian noise, in the Lévy case, the positions of maxima of the stationary PDF do not coincide with the positions of minima of the bistable potential. We provide a detailed study of the distance between the maxima and the minima as a function of the depth of the potential and the Lévy noise parameters.


Probability Density Function European Physical Journal Special Topic Noise Intensity Bistable System Quartic Potential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    P. Lévy, Théorie de l’addition des variables aléatoires, vol. 1 (Gauthier-Villars, Paris, 1954)Google Scholar
  2. 2.
    B. Gnedenko, A. Kolmogorov, K. Chung, J. Doob, Limit distributions for sums of independent random variables, vol. 195 (Addison-Wesley Reading, MA, 1968)Google Scholar
  3. 3.
    A. Janicki, A. Weron, Simulation and chaotic behavior of α-stable stochastic processes, vol. 178 (CRC, 1994)Google Scholar
  4. 4.
    C. Nikias, M. Shao, Signal processing with alpha-stable distributions and applications (Wiley-Interscience, 1995)Google Scholar
  5. 5.
    A. Chechkin, R. Metzler, J. Klafter, V. Gonchar, Introduction to the Theory of Lévy Flights. Anomalous Transport: Foundations and Applications, edited by R. Klages, G. Radons, I.M. Sokolov (Wiley-VCH, 2008), p. 129Google Scholar
  6. 6.
    R. Metzler, A.V. Chechkin, J. Klafter, Encyclopedia of Complexity and System Science, edited by R.A. Mayers, Article 293 (Springer-Verlag, Berlin, 2009)Google Scholar
  7. 7.
    A. Chechkin, V. Gonchar, M. Szydłowski, Phys. Plasmas 9, 78 (2002)CrossRefADSGoogle Scholar
  8. 8.
    R. Jha, P. Kaw, D. Kulkarni, J. Parikh, A. Team, Phys. Plasmas 10, 699 (2003)CrossRefADSGoogle Scholar
  9. 9.
    V. Gonchar, et al., Plasma Phys. Rep. 29, 380 (2003)CrossRefADSGoogle Scholar
  10. 10.
    T. Mizuuchi, et al., J. Nuclear Materials 337, 332 (2005)CrossRefADSGoogle Scholar
  11. 11.
    P. Ditlevsen, H. Svensmark, S. Johnsen, Nature 379, 810 (1996)CrossRefADSGoogle Scholar
  12. 12.
    P. Ditlevsen, Phys. Rev. E 60, 172 (1999)CrossRefADSGoogle Scholar
  13. 13.
    P.D. Ditlevsen, Geophys. Res. Lett. 26, 1441 (1999)CrossRefADSGoogle Scholar
  14. 14.
    I. Eliazar, J. Klafter, J. Stat. Phys. 111, 739 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    B. West, V. Seshadri, Physica A: Stat. Theor. Phys. 113, 203 (1982)CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    F. Peseckis, Phys. Rev. A 36, 892 (1987)CrossRefADSGoogle Scholar
  17. 17.
    S. Jespersen, R. Metzler, H. Fogedby, Phys. Rev. E 59, 2736 (1999)CrossRefADSGoogle Scholar
  18. 18.
    A. Chechkin, V. Gonchar, J. Eksper. Theor. Phys. 91, 635 (2000)CrossRefADSGoogle Scholar
  19. 19.
    A. Dubkov, B. Spagnolo, V. Uchaikin, Int. J. Bifur. Chaos 18, 2649 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    B. Dybiec, E. Gudowska-Nowak, I.M. Sokolov, Phys. Rev. E 76, 041122 (2007)CrossRefADSGoogle Scholar
  21. 21.
    S. Denisov, W. Horsthemke, P. Hänggi, Phys. Rev. E 77, 061112 (2008)CrossRefADSGoogle Scholar
  22. 22.
    B. Dybiec, I. Sokolov, A. Chechkin, J. Stat. Mech.: Theor. Exper. 2010, P07008 (2010)Google Scholar
  23. 23.
    I. Pavlyukevich, B. Dybiec, A. Chechkin, I. Sokolov, Eur. Phys. J. Special Topics 191, 223 (2010)CrossRefADSGoogle Scholar
  24. 24.
    A. Dubkov, A. La Cognata, B. Spagnolo, J. Stat. Mech.: Theor. Exper., P01002 (2009)Google Scholar
  25. 25.
    A. La Cognata, D. Valenti, A. Dubkov, B. Spagnolo, Phys. Rev. E 82, 011121 (2010)CrossRefADSGoogle Scholar
  26. 26.
    A. Chechkin, V. Gonchar, J. Klafter, R. Metzler, L. Tanatarov, Chem. Phys. 284, 233 (2002)CrossRefADSGoogle Scholar
  27. 27.
    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and applications (Gordon and Breach, PA, 1993)Google Scholar
  28. 28.
    A. Chechkin, V. Gonchar, J. Klafter, R. Metzler, L. Tanatarov, J. Stat. Phys. 115, 1505 (2004)CrossRefzbMATHADSGoogle Scholar
  29. 29.
    I.I. Eliazar, M.H. Cohen, J. Phys. A: Math. Theor. 45, 332001 (2012)CrossRefMathSciNetGoogle Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  • O.Yu. Sliusarenko
    • 1
    Email author
  • D.A. Surkov
    • 2
  • V.Yu. Gonchar
    • 1
  • A.V. Chechkin
    • 1
    • 3
  1. 1.Akhiezer Institute for Theoretical Physics National Science Center “Kharkiv Institute of Physics and Technology”KharkivUkraine
  2. 2.Karazin National UniversityKharkivUkraine
  3. 3.Institute for Physics and Astronomy University of PotsdamPotsdam-GolmGermany

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