Skip to main content
SpringerLink
Log in

Investigating the stabilizing effect of stochastic excitation on a chaotic dynamical system

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The response of an interactive Mathieu–Duffing system in R 4, subjected to a harmonic excitation is investigated. For a deterministic circular frequency, chaotic behavior is observed. Subsequently it is shown that when the excitation becomes stochastic, chaos is subsided and trajectories tend to a diffused attracting set. The stabilizing effect of stochastic excitation is verified by finding the largest Lyapunov exponent for the two cases.

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

Similar content being viewed by others

References

  1. Wang, R., Yasuda, K., Zhang, Z.: A generalized analysis technique of the stationary FPK equation in nonlinear systems under Gaussian white noise excitations. Int. J. Eng. Sci. 38, 1315–1330 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Polidori, D.C., Beck, J.L., Papadimitriou, C.: A new stationary PDF approximation for non-linear oscillators. Int. J. Non-Linear Mech. 35, 657–673 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wang, R., Yasuda, K.: Exact stationary response of SDOF nonlinear stochastic oscillators, Acoustique, ondes, vibrations/Acoustics, waves, vibrations. C. R. Acad. Sci. Paris, Sér. IIb 328, 349–357 (2000)

    MATH  Google Scholar 

  4. Hilton, H.H., Sri Namachchivaya, N.: Effects of noise in nonlinear system exhibiting simple bifurcations. Struct. Saf. 6, 211–221 (1989)

    Article  Google Scholar 

  5. Ma, S., Xu, W.: Period-doubling bifurcation in an extended van der Pol system with bounded random parameter. Commun. Nonlinear Sci. Numer. Simul. 13(10), 2256–2265 (2008)

    Article  Google Scholar 

  6. Hea, Q., Xua, W., Rong, H., Fanga, T.: Stochastic bifurcation in Duffing–Van der Pol oscillators. Physica A 338, 319–334 (2004)

    Article  MathSciNet  Google Scholar 

  7. Schenk, K.R.: Stochastic Hopf bifurcation: an example. Int. J. Non-Linear Mech. 31(5), 685–692 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Xua, W., Hea, Q., Fanga, T., Rong, H.: Stochastic bifurcation in Duffing system subject to harmonic excitation and in presence of random noise. Int. J. Non-Linear Mech. 39, 1473–1479 (2004)

    Article  Google Scholar 

  9. Lepik, U., Hein, H.: On response of nonlinear oscillators with random frequency of excitation. J. Sound Vib. 288, 275–292 (2005)

    Article  Google Scholar 

  10. Hein, H., Lepik, U.: Response of nonlinear oscillators with random frequency of excitation, revisited. J. Sound Vib. 301, 1040–1049 (2007)

    Article  Google Scholar 

  11. Huang, Z.L., Zhu, W.Q., Ni, Y.Q., Ko, J.M.: Stochastic averaging of strongly non-linear oscillators under bounded noise excitation. J. Sound Vib. 254(2), 245–267 (2002)

    Article  MathSciNet  Google Scholar 

  12. Li, J., Xua, W., Rena, Z., Leia, Y.: Maximal Lyapunov exponent and almost-sure stability for Stochastic Mathieu–Duffing Systems. J. Sound Vib. 286, 395–402 (2005)

    Article  Google Scholar 

  13. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

  14. Shin, K., Hammond, J.K.: The instantaneous Lyapunov exponent and its application to chaotic dynamical systems. J. Sound Vib. 218(3), 389–403 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Shiraz University, Fars, Iran

    H. Emadi & M. Mahzoon

Authors
  1. H. Emadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Mahzoon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Emadi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Emadi, H., Mahzoon, M. Investigating the stabilizing effect of stochastic excitation on a chaotic dynamical system. Nonlinear Dyn 67, 505–515 (2012). https://doi.org/10.1007/s11071-011-9999-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-9999-0

Keywords

Search

Navigation