The European Physical Journal Special Topics

, Volume 222, Issue 10, pp 2481–2495 | Cite as

Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation

  • Anna ZakharovaEmail author
  • Alexey Feoktistov
  • Tatyana Vadivasova
  • Eckehard Schöll
Regular Article Nonlinear Dynamics of Stochastic Systems


We analyze noise-induced phenomena in nonlinear dynamical systems near a subcritical Hopf bifurcation. We investigate qualitative changes of probability distributions (stochastic bifurcations), coherence resonance, and stochastic synchronization. These effects are studied in dynamical systems for which a subcritical Hopf bifurcation occurs. We perform analytical calculations, numerical simulations and experiments on an electronic circuit. For the generalized Van der Pol model we uncover the similarities between the behavior of a self-sustained oscillator characterized by a subcritical Hopf bifurcation and an excitable system. The analogy is manifested through coherence resonance and stochastic synchronization. In particular, we show both experimentally and numerically that stochastic oscillations that appear due to noise in a system with hard excitation, can be partially synchronized even outside the oscillatory regime of the deterministic system.


Hopf Bifurcation European Physical Journal Special Topic Noise Intensity Deterministic System Unstable Limit Cycle 
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.
    R.L. Stratonovich, Topics in the Theory of Random Noise, Vol. 1 (Gordon and Breach, New York, 1963)Google Scholar
  2. 2.
    W. Horsthemke, R. Lefever, Noise-Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology (Springer Verlag, Berlin, 1984)Google Scholar
  3. 3.
    R. Graham, Macroscopic Potentials, Bifurcations and Noise in Dissipative Systems, Vol. 1, Noise in Nonlinear Dynamical Systems (Cambridge University Press, Cambridge, 1989)Google Scholar
  4. 4.
    H. Risken, The Fokker-Planck Equation, 2nd edn. (Springer, Berlin, 1996)Google Scholar
  5. 5.
    C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, Berlin, 2002)Google Scholar
  6. 6.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 2003)Google Scholar
  7. 7.
    L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics (Springer, Berlin, 2003)Google Scholar
  8. 8.
    V.S. Anishchenko, V. Astakhov, A. Neiman, T. Vadivasova, L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments (Springer, Berlin, 2007)Google Scholar
  9. 9.
    M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, Vol. 260, Series of Comprehensive Studies in Mathematics (Springer-Verlag, New York, 2012)Google Scholar
  10. 10.
    S.M. Soskin, R. Mannella, P.V.E. McClintock, Phys. Rep. 373, 247 (2003)MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    K. Wiesenfeld, J. Stat. Phys. 38, 1071 (1985)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    W. Ebeling, H. Herzel, W. Richert, L. Schimansky-Geier, J. Appl. Math. Mech. (ZAMM) 66, 141 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    R. Lefever, J.W. Turner, Phys. Rev. Lett. 56, 1631 (1986)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    L. Fronzoni, R. Mannella, P.V.E. McClintock, F. Moss, Phys. Rev. A 36, 834 (1987)ADSCrossRefGoogle Scholar
  15. 15.
    N.S. Namachchivaya, J. Appl. Math. Comput. 38, 101 (1990)CrossRefzbMATHGoogle Scholar
  16. 16.
    L. Arnold, N.S. Namachchivaya, K.R. Schenk-Hoppé, Int. J. Bifur. Chaos 6, 1947 (1996)CrossRefGoogle Scholar
  17. 17.
    J. Olarrea, F.J. de la Rubia, Phys. Rev. E 53, 268 (1996)ADSCrossRefGoogle Scholar
  18. 18.
    K.R. Schenk-Hoppé, Nonlinear Dyn. 11, 255 (1996)CrossRefGoogle Scholar
  19. 19.
    P.S. Landa, A.A. Zaikin, Phys. Rev. E 54, 3535 (1996)ADSCrossRefGoogle Scholar
  20. 20.
    H. Crauel, F. Flandoli, J. Dyn. Differ. Equ. 10, 259 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    N.B. Janson, A.G. Balanov, E. Schöll, Phys. Rev. Lett. 93, 010601 (2004)ADSCrossRefGoogle Scholar
  22. 22.
    I. Bashkirtseva, L. Ryashko, H. Schurz, Chaos, Solitons Fractals 39, 72 (2009)ADSCrossRefzbMATHGoogle Scholar
  23. 23.
    A. Zakharova, T. Vadivasova, V. Anishchenko, A. Koseska, J. Kurths, Phys. Rev. E 81, 011106 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    Y. Xu, R. Gu, H. Zhang, W. Xu, J. Duan, Phys. Rev. E 83, 056215 (2011)ADSCrossRefGoogle Scholar
  25. 25.
    R. Benzi, A. Sutera, A. Vulpiani, J. Phys. A 14, L453 (1981)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998)ADSCrossRefGoogle Scholar
  27. 27.
    V.S. Anishchenko, A.B. Neiman, F. Moss, L. Schimansky-Geier, Phys. Usp. 42, 7 (1999)ADSCrossRefGoogle Scholar
  28. 28.
    G. Hu, T. Ditzinger, C.Z. Ning, H. Haken, Phys. Rev. Lett. 71, 807 (1993)ADSCrossRefGoogle Scholar
  29. 29.
    A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78, 775 (1997)MathSciNetADSCrossRefzbMATHGoogle Scholar
  30. 30.
    R.E. Lee DeVille, E. Vanden-Eijnden, C.B. Muratov, Phys. Rev. E 73, 031105 (2005)ADSCrossRefGoogle Scholar
  31. 31.
    R. Aust, P. Hövel, J. Hizanidis, E. Schöll, Eur. Phys. J. Special Topics 187, 77 (2010)ADSCrossRefGoogle Scholar
  32. 32.
    A.G. Balanov, N.B. Janson, E. Schöll, Physica D 199, 1 (2004)ADSCrossRefzbMATHGoogle Scholar
  33. 33.
    J.L.A. Dubbeldam, B. Krauskopf, D. Lenstra, Phys. Rev. E 60, 6580 (1999)ADSCrossRefGoogle Scholar
  34. 34.
    G. Giacomelli, M. Giudici, S. Balle, J.R. Tredicce, Phys. Rev. Lett. 84, 3298 (2000)ADSCrossRefGoogle Scholar
  35. 35.
    J.F.M. Avila, H.L.D. de, S. Cavalcante, J.R.R. Leite, Phys. Rev. Lett. 93, 144101 (2004)ADSCrossRefGoogle Scholar
  36. 36.
    D. Ziemann, R. Aust, B. Lingnau, E. Schöll, K. Lüdge, Europhys. Lett. 103, 14002 (2013)ADSCrossRefGoogle Scholar
  37. 37.
    J. Hizanidis, A.G. Balanov, A. Amann, E. Schöll, Phys. Rev. Lett. 96, 244104 (2006)ADSCrossRefGoogle Scholar
  38. 38.
    J. Hizanidis, E. Schöll, Phys. Rev. E 78, 066205 (2008)ADSCrossRefGoogle Scholar
  39. 39.
    B. Lindner, J. García-Ojalvo, A. Neiman, L. Schimansky-Geier, Phys. Rep. 392, 321 (2004)ADSCrossRefGoogle Scholar
  40. 40.
    A. Neiman, Phys. Rev. E 49, 3484 (1994)ADSCrossRefGoogle Scholar
  41. 41.
    B. Shulgin, A. Neiman, V. Anishchenko, Phys. Rev. Lett. 75, 4157 (1995)ADSCrossRefGoogle Scholar
  42. 42.
    S.K. Han, T.G. Yim, D.E. Postnov, O.V. Sosnovtseva, Phys. Rev. Lett. 83, 1771 (1999)ADSCrossRefGoogle Scholar
  43. 43.
    V.S. Anishchenko, T. Vadivasova, G. Strelkova, Eur. Phys. J. Special Topics 187, 109 (2010)ADSCrossRefGoogle Scholar
  44. 44.
    O.V. Ushakov, H.J. Wünsche, F. Henneberger, I.A. Khovanov, L. Schimansky-Geier, M.A. Zaks, Phys. Rev. Lett. 95, 123903 (2005)ADSCrossRefGoogle Scholar
  45. 45.
    A. Zakharova, J. Kurths, T. Vadivasova, A. Koseska, PLoS ONE 6, e19696 (2011)ADSCrossRefGoogle Scholar
  46. 46.
    I.A. Khovanov, D.G. Luchinsky, R. Mannella, P.V.E. McClintock, Fluctuational Escape from a Quasiattractor, Lecture Notes in Physics (Springer-Verlag, Berlin Heidelberg, 2000)Google Scholar
  47. 47.
    Note that a very broad, smeared-out minimum may still be observed outside, but close to the bimodal regimeGoogle Scholar
  48. 48.
    E. Schöll, A.G. Balanov, N.B. Janson, A. Neiman, Stoch. Dyn. 5, 281 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    J. Pomplun, A. Amann, E. Schöll, Europhys. Lett. 71, 366 (2005)MathSciNetADSCrossRefGoogle Scholar
  50. 50.
    A. Feoktistov, V. Anishchenko, Rus. J. Nonlin. Dyn. 8, 897 (2012)Google Scholar
  51. 51.
    A. Feoktistov, S. Astakhov, V. Anishchenko, Izvestiya VUZ. Appl. Nonlinear Dyn. 18, 33 (2010)zbMATHGoogle Scholar
  52. 52.
    S. Wieczorek, B. Krauskopf, T. Simpson, D. Lenstra, Phys. Rep. 416, 1 (2005)ADSCrossRefGoogle Scholar
  53. 53.
    J. Pausch, C. Otto, E. Tylaite, N. Majer, E. Schöll, K. Lüdge, New J. Phys. 14, 053018 (2012)ADSCrossRefGoogle Scholar
  54. 54.
    K. Lüdge, B. Lingnau, C. Otto, E. Schöll, Nonlinear Phenom. Complex Syst. 15, 350 (2012)Google Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  • Anna Zakharova
    • 1
    Email author
  • Alexey Feoktistov
    • 2
  • Tatyana Vadivasova
    • 2
  • Eckehard Schöll
    • 1
  1. 1.Institut für Theoretische Physik, Technische Universität BerlinBerlinGermany
  2. 2.Saratov State UniversitySaratovRussia

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