Nonlinear Dynamics

, Volume 93, Issue 2, pp 767–778 | Cite as

On a “deterministic” explanation of the stochastic resonance phenomenon

  • I. I. Blekhman
  • V. S. SorokinEmail author
Original Paper


The present paper concerns the analysis of the stochastic resonance phenomenon that previously has been thoroughly studied and found numerous applications in physics, neuroscience, biology, medicine, mechanics, etc. A novel “deterministic” explanation of this phenomenon is proposed that allows broadening the range of dynamical systems for which the phenomenon can be predicted and analysed. Our results indicate that stochastic resonance, similarly to vibrational resonance, arises due to deterministic reasons: it occurs when a system is excited with two (or more) vastly different frequencies, one of which is much higher than another. The effective properties of the system, e.g. stiffness or mass, change under the action of the high-frequency excitation; and the low-frequency excitation acts on this “modified” system leading to low-frequency resonances. In the case of a broadband random excitation, the high-frequency part of the excitation spectrum affects the effective properties of the system. The low-frequency part of the spectrum acts on this modified system. Thus by varying the noise intensity one can change properties of the system and attain resonances. This explanation allows using “deterministic” approach, i.e. replacing noise by high-frequency excitation, when studying the stochastic resonance phenomenon. Employing this approach, we demonstrate that linear and nonlinear stochastic systems with varying parameters, i.e. parametrically excited systems, can exhibit the phenomenon and determine the corresponding resonance conditions.


Stochastic resonance Deterministic explanation High-frequency excitation Effective properties Oscillatory strobodynamics Parametric excitation 



The work is carried out with financial support from the Russian Science Foundation, Grant 17-79-30056 (Project “REC Mekhanobr-Tekhnika”).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Wiesenfeld, K., Moss, F.: Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs. Nature 373(6509), 33–36 (1995)CrossRefGoogle Scholar
  2. 2.
    Bulsara, A., Gammaitoni, L.: Tuning in to noise. Phys. Today 49(3), 39–45 (1996)CrossRefGoogle Scholar
  3. 3.
    Gammaitoni, L., Hanggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70(1), 223–287 (1998)CrossRefGoogle Scholar
  4. 4.
    Chapeau-Blondeau, F., Rousseau, D.: Noise improvements in stochastic resonance: from signal amplification to optimal detection. Fluct. Noise Lett. 2, 221–233 (2002)CrossRefGoogle Scholar
  5. 5.
    Comte, J., et al.: Stochastic resonance: another way to retrieve subthreshold digital data. Phys. Lett. A 309(1), 39–43 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Moss, F., Ward, L., Sannita, W.: Stochastic resonance and sensory information processing: a tutorial and review of application. Clin. Neurophysiol. 115(2), 267–281 (2004)CrossRefGoogle Scholar
  7. 7.
    Priplata, A., Patritti, B., Niemi, J., et al.: Noise-enhanced balance control in patients with diabetes and patients with stroke. Ann. Neurol. 59(1), 4–12 (2006)CrossRefGoogle Scholar
  8. 8.
    McDonnell, M., Abbot, D.: What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology. Comput. Biol. 5(5), e1000348 (2009)MathSciNetGoogle Scholar
  9. 9.
    Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F.: Stochastic resonance: a remarkable idea that changed our perception of noise. Eur. Phys. J. B 69(1), 1–3 (2009)CrossRefGoogle Scholar
  10. 10.
    Chapeau-Blondeau, F., Rousseau, D.: Raising the noise to improve performance in optimal processing. J. Stat. Mech. Theory Exp. (2009). Google Scholar
  11. 11.
    Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys. A 14(11), L 453 (1981)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Benzi, R., Parisi, G., Sutera, A., Vulpiani, A.: Stochastic resonance in climatic change. Tellus 34(1), 10–15 (1982)CrossRefzbMATHGoogle Scholar
  13. 13.
    Benzi, R., Parisi, G., Sutera, A., Vulpiani, A.: A theory of stochastic resonance in climatic change. SIAM J. Appl. Math. 43(3), 565–578 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Longtin, A.: Stochastic resonance in neuron models. J. Stat. Phys. 70(1), 309–327 (1993)CrossRefzbMATHGoogle Scholar
  15. 15.
    Jung, P.: Threshold devices: fractal noise and neural talk. Phys. Rev. E 50, 2513–2522 (1994)CrossRefGoogle Scholar
  16. 16.
    Wiesenfeld, K., Pierson, D., Pantazelou, E., Dames, C., Moss, F.: Stochastic resonance on a circle. Phys. Rev. Lett. 72(14), 2125–2129 (1994)CrossRefGoogle Scholar
  17. 17.
    Gingl, Z., Kiss, L., Moss, F.: Non-dynamical stochastic resonance: theory and experiments with white and arbitrarily coloured noise. Europhys. Lett. 29(3), 191–196 (1995)CrossRefGoogle Scholar
  18. 18.
    Gammaitoni, L.: Stochastic resonance and the dithering effect in threshold physical systems. Phys. Rev. E 52, 4691–4698 (1995)CrossRefGoogle Scholar
  19. 19.
    McNamara, B., Wiesenfeld, K.: Theory of stochastic resonance. Phys. Rev. A 39(9), 4854–4869 (1989)CrossRefGoogle Scholar
  20. 20.
    Hanggi, P., Talkner, P., Borkovec, M.: Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62(2), 251–341 (1990)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Cottone, F., Vocca, H., Gammaitoni, L.: Nonlinear energy harvesting. Phys. Rev. Lett. 102(8), 080601 (2009)CrossRefGoogle Scholar
  22. 22.
    Stephenson, A.: On induced stability. Philos. Mag. 6(15), 233–236 (1908)CrossRefzbMATHGoogle Scholar
  23. 23.
    Kapitza, P.L.: Pendulum with a vibrating suspension. Usp. Fiz. Nauk 44, 7–15 (1951)CrossRefGoogle Scholar
  24. 24.
    Bleich, H.: Effect of vibrations on the motion of small gas bubbles in a liquid. J. Am. Rocket Soc. 26, 11, 978, 958–964 (1956)Google Scholar
  25. 25.
    Sorokin, V.S., Blekhman, I.I., Vasilkov, V.B.: Motion of a gas bubble in fluid under vibration. Nonlinear Dyn. 67(1), 147–158 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Blekhman, I.I.: Vibrational Mechanics. Nonlinear Dynamic Effects, General Approach, Applications, p. 509. World Scientific, Singapore (2000)CrossRefGoogle Scholar
  27. 27.
    Blekhman, I.I. (ed.): Selected Topics in Vibrational Mechanics, p. 409. World Scientific, Hackensack (2002)zbMATHGoogle Scholar
  28. 28.
    Thomsen, J.: Vibrations and Stability: Advanced Theory, Analysis and Tools, p. 404. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  29. 29.
    Blekhman, I.I.: Theory of Vibrational Processes and Devices: Vibrational Mechanics and Vibrational Rheology, p. 640. Ruda I Metalli, St. Petersburg (2013). (in Russian) Google Scholar
  30. 30.
    Landa, P.S., McClintock, P.: Vibrational resonance. J. Phys. A Math. Gen. 33, L433–L438 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Landa, P.S.: Regular and Chaotic Oscillations, p. 397. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  32. 32.
    Baltanas, J., et al.: Experimental evidence, numerics, and theory of vibrational resonance in bistable systems. Phys. Rev. E 67, 066119 (2003)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Blekhman, I.I., Landa, P.S.: Conjugate resonances and bifurcations in nonlinear systems under biharmonical excitation. Int. J. Non Linear Mech. 39, 421–426 (2004)CrossRefzbMATHGoogle Scholar
  34. 34.
    Gandhimathi, V.M., et al.: Vibrational and stochastic resonances in two coupled overdamped anharmonic oscillators. Phys. Lett. A 360, 279–286 (2006)CrossRefzbMATHGoogle Scholar
  35. 35.
    Yang, J.H., Zhu, H.: Vibrational resonance in Duffing systems with fractional-order damping. Chaos 22, 013112 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Rajasekar, S., Sanjuan, M.: Nonlinear Resonances, p. 409. Springer, Heidelberg (2016)CrossRefGoogle Scholar
  37. 37.
    Berdichevsky, V., Gitterman, M.: Stochastic resonance in linear systems subject to multiplicative and additive noise. Phys. Rev. E 60(2), 1494–1499 (1999)CrossRefGoogle Scholar
  38. 38.
    Gitterman, M.: Harmonic oscillator with multiplicative noise: nonmonotonic dependence on the strength and the rate of dichotomous noise. Phys. Rev. E 67, 057103 (2003)CrossRefGoogle Scholar
  39. 39.
    Gitterman, M.: Harmonic oscillator with fluctuating damping parameter. Phys. Rev. E 69, 041101 (2004)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Guo, F., Li, H., Liu, J.: Stochastic resonance in a linear system with random damping parameter driven by trichotomous noise. Physica A 409, 1–7 (2014)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Seshia, A., et al.: A vacuum packaged surface micromachined resonant accelerometer. J. Microelectromech. Syst. 11(6), 784–793 (2002)CrossRefGoogle Scholar
  42. 42.
    Krylov, S., Harari, I., Cohen, Y.: Stabilization of electrostatically actuated microstructures using parametric excitation. J. Micromech. Microeng. 15(6), 1188–1204 (2005)CrossRefGoogle Scholar
  43. 43.
    Rhoads, J., Shaw, S., Turner, K.: Nonlinear dynamics and its applications in micro- and nanoresonators. J. Dyn. Syst. Meas. Control 132(3), 034001 (2010)CrossRefGoogle Scholar
  44. 44.
    Zaitsev, S., et al.: Nonlinear damping in a micromechanical oscillator. Nonlinear Dyn. 67, 859–883 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Floris, C.: Stochastic stability of damped Mathieu oscillator parametrically excited by a Gaussian noise. Math. Probl. Eng. 2012, 375913 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Nayfeh, A., Mook, D.: Nonlinear Oscillations, p. 720. Wiley-Interscience, New York (1979)zbMATHGoogle Scholar
  47. 47.
    Bogoliubov, N., Mitropolskii, J.: Asymptotic Methods in the Theory of Non-linear Oscillations, p. 537. Gordon and Breach, New York (1961)Google Scholar
  48. 48.
    Sanders, J., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems, p. 249. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  49. 49.
    Blekhman, I.I.: Oscillatory strobodynamics—a new area in nonlinear oscillations theory, nonlinear dynamics and cybernetical physics. Cybern. Phys. 1, 5–10 (2012)Google Scholar
  50. 50.
    Blekhman, I.I., Sorokin, V.S.: Effects produced by oscillations applied to nonlinear dynamic systems: a general approach and examples. Nonlinear Dyn. 83, 2125–2141 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Guo, F.: Multiplicative noise-induced vibrational resonance in a monostable system with one high-frequency and two low-frequency forces. Phys. Scr. 83, 025008 (2011)CrossRefzbMATHGoogle Scholar
  52. 52.
    Blekhman, I.I., Sorokin, V.S.: On the separation of fast and slow motions in mechanical systems with high-frequency modulation of the dissipation coefficient. J. Sound Vib. 329(23), 4936–4949 (2010)CrossRefGoogle Scholar
  53. 53.
    Stocks, N.G., et al.: Stochastic resonance in monostable systems. J. Phys. A Math. Gen. 26, L385 (1993)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mekhanobr-Tekhnika Research and Engineering Corp.St. PetersburgRussia
  2. 2.Institute of Problems in Mechanical Engineering RASSt. PetersburgRussia
  3. 3.Department of Mechanical EngineeringThe University of AucklandAucklandNew Zealand

Personalised recommendations