Skip to main content
SpringerLink
Log in

Experimental Studies of Noise Effects in Nonlinear Oscillators

  • Chapter
  • First Online:
Nonlinear Dynamics and Complexity

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 8))

Abstract

In the paper the noisy behavior of nonlinear oscillators is explored experimentally. Two types of excitable stochastic oscillators are considered and compared, i.e., the FitzHugh–Nagumo system and the Van der Pol oscillator with a subcritical Andronov–Hopf bifurcation. In the presence of noise and at certain parameter values both systems can demonstrate the same type of stochastic behavior with effects of coherence resonance and stochastic synchronization. Thus, the excitable oscillators of both types can be classified as stochastic self-sustained oscillators. Besides, the noise influence on a supercritical Andronov–Hopf bifurcation is studied. Experimentally measured joint probability distributions enable to analyze the phenomenological stochastic bifurcations corresponding to the boundary of the noisy limit cycle regime. The experimental results are supported by numerical simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    All the terms in Eq. (15), regardless of the degree, are voltages taken off at different points of the scheme and measured in volts.

References

  1. Stratonovich RL (1961) Selected problems of the fluctuation theory in radiotechnique. Sov. Radio, Moscow (in Russian)

    Google Scholar 

  2. Freidlin MI, Wentzell AD (1984) Random perturbations in dynamical systems. Springer, New York

    Book  Google Scholar 

  3. Gardiner CW (1982) Handbook of stochastic methods. Springer series in synergetics, vol 13. Springer, Berlin

    Google Scholar 

  4. Risken H (1984) The Fokker–Planck equation. Springer series in synergetics, vol 18. Springer, Berlin

    Google Scholar 

  5. Van Kampen NG (1992) Stochastic processes in physics and chemistry, 2nd edn. North Holland, Amsterdam

    Google Scholar 

  6. Horsthemke W, Lefever R (1983) Noise induced transitions: theory and applications in physics, chemistry and biology. Springer series in synergetics, vol 15. Springer, Berlin

    Google Scholar 

  7. Graham R (1990) Macroscopic potentials, bifurcations and noise in dissipative systems. In: Moss F, McClintock PVE (eds) Theory of continuous Fokker–Planck systems. Noise in nonlinear dynamical systems, vol 1. Cambridge University Press, Cambridge

    Google Scholar 

  8. Arnold L (2003) Random dynamical systems. Springer, Berlin

    Google Scholar 

  9. Garcia-Ojalvo J, Sancho JM (1999) Noise in spatially extended systems. Springer, New York

    Book  MATH  Google Scholar 

  10. Benzi R, Sutera A, Vulpiani A (1981) J Phys Math Gen 14:L453

    Article  MathSciNet  Google Scholar 

  11. Gammaitoni L, Marchesoni F, Menichella-Saetta E, Santucci S (1989) Phys Rev Lett 62:349

    Article  Google Scholar 

  12. Anishchenko VS, Neiman AB, Moss F, Schimansky-Geier L (1999) Phys Usp 42:7

    Article  Google Scholar 

  13. Anishchenko V et al (2002) Nonlinear dynamics of chaotic and stochastic systems: Tutorial and modern development. Berlin, Springer

    Google Scholar 

  14. Pikovsky A, Kurths J (1997) Phys Rev Lett 78:775

    Article  MathSciNet  MATH  Google Scholar 

  15. Lindner B, Schimansky-Geier L (1999) Phys Rev E 60(6):7270

    Article  Google Scholar 

  16. Neiman AB (1994) Phys Rev E 49:3484–3488

    Article  Google Scholar 

  17. Shulgin B, Neiman A, Anishchenko V (1995) Phys Rev Lett 75(23):4157

    Article  Google Scholar 

  18. Anishchenko V, Neiman A (1997) Stochastic synchronization. In: Schimansky-Geier L, Pöschel T (eds) Stochastic dynamics. Springer, Berlin, p 155

    Google Scholar 

  19. Anishchenko VS, Vadivasova TE, Strelkova GI (2010) Eur Phys J Spec Top 187:109–125

    Article  Google Scholar 

  20. Han SK, Yim TG, Postnov DE, Sosnovtseva OV (1999) Phys Rev Lett 83(9):1771

    Article  Google Scholar 

  21. Neiman A, Schimansky-Geier L, Cornell-Bell A, Moss F (1999) Phys Rev Lett 83(23):4896

    Article  Google Scholar 

  22. Hu B, Zhou Ch (2000) Phys Rev E 61(2):R1001

    Article  Google Scholar 

  23. Ushakov OV, Wùnsche H-J, Henneberger F, Khovanov IA, Schimansky-Geier L, Zaks MA (2005) Phys Rev Lett 95:123903(4)

    Article  Google Scholar 

  24. Zakharova A, Vadivasova T, Anishchenko V, Koseska A, Kurths J (2010) Phys Rev E 81:011106(1–6)

    Google Scholar 

  25. Lefever R, Turner J (1986) Phys Rev Lett 56:1631

    Article  MathSciNet  Google Scholar 

  26. Ebeling W, Herzel H, Richert W, Schimansky-Geier L (1986) Z Angew Math Mech 66:141

    Article  MathSciNet  MATH  Google Scholar 

  27. Schimansky-Geier L, Herzel H (1993) J Stat Phys 70:141

    Article  MATH  Google Scholar 

  28. Arnold L, Sri Namachshivaya N, Schenk-Yoppe JR (19996) Int J Bifurcat Chaos 6:1947

    Google Scholar 

  29. Olarrea J, de la Rubia FJ (1996) Phys Rev E 53(1):268

    Article  Google Scholar 

  30. Landa PS, Zaikin AA (1996) Phys Rev E 54(4):3535

    Article  Google Scholar 

  31. Crauel H, Flandol F (1998) J Dynam Differ Equat 10:259

    Article  MATH  Google Scholar 

  32. Bashkirtseva I, Ryashko L, Schurz H (2009) Chaos Solit Fract 39:7

    Article  MathSciNet  Google Scholar 

  33. Longtin A (1993) J Stat Phys 70:309

    Article  MATH  Google Scholar 

  34. Baltanas JP, Casado JM (1998) Phys D 122:231

    Article  Google Scholar 

  35. Izhikevich EM (2007) Dynamical systems in neuroscience: The geometry of excitability and bursting. Cambridge, MIT Press, Cambridge, MA

    Google Scholar 

  36. FitzHugh R (1955) Bull Math Biophys 17:257

    Article  Google Scholar 

  37. Scott AC (1975) Rev Mod Phys 47:487

    Article  Google Scholar 

  38. Anishchenko VS, Vadivasova TE, Feoktistov AV, Strelkova GI (2013) Stochastic oscillators. In: Rubio RG, Ryazantsev YS, Starov VM, Huang G-X, Chetverikov AP, Arena P, Nepomnyashchy AA, Ferrus A, Morozov EG (eds) Without bounds: A scientific canvas of nonlinearity and complex dynamics. Springer, Berlin, p 20

    Google Scholar 

  39. Makarov VA, del Rio E, Ebeling W, Velarde MG (2001) Phys Rev E 64:036601

    Article  Google Scholar 

  40. Feoktistov AV, Astakhov SV, Anishchenko VS (2010) Izv. VUZ. Appl Nonlinear Dynam 18:33 (in Russian)

    MATH  Google Scholar 

  41. Kuznetsov AP, Milovanov SV (2003) Izv. VUZ. Appl Nonlinear Dynam 11:16 (in Russian)

    MATH  Google Scholar 

  42. Franzoni L, Mannella R, McClintock PVE, Moss F (1987) Phys Rev E 36:834

    Article  Google Scholar 

  43. Anishchenko VS (2009) Complex oscillations in simple systems. URSS, Moscow (in Russian)

    Google Scholar 

Download references

Acknowledgements

This work is supported by the Russian Ministry of Education and Sciences in the framework of the state contract N 14.B37.21.0751.

Author information

Authors and Affiliations

  1. Saratov State University, Astrakhanskaya str., 83, Saratov, 410012, Russia

    Vadim S. Anishchenko, Tatjana E. Vadivasova, Alexey V. Feoktistov, Vladimir V. Semenov & Galina I. Strelkova

Authors
  1. Vadim S. Anishchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatjana E. Vadivasova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexey V. Feoktistov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vladimir V. Semenov
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Galina I. Strelkova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Galina I. Strelkova .

Editor information

Editors and Affiliations

  1. San Luis Potosi University, San Luis Potosi, Mexico

    Valentin Afraimovich

  2. School of Engineering Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, Illinois, USA

    Albert C. J. Luo

  3. Shandong Normal University, Ji’nan, People's Republic of China

    Xilin Fu

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Anishchenko, V.S., Vadivasova, T.E., Feoktistov, A.V., Semenov, V.V., Strelkova, G.I. (2014). Experimental Studies of Noise Effects in Nonlinear Oscillators. In: Afraimovich, V., Luo, A., Fu, X. (eds) Nonlinear Dynamics and Complexity. Nonlinear Systems and Complexity, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-02353-3_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-02353-3_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-02352-6

  • Online ISBN: 978-3-319-02353-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation