Nonlinear Dynamics

, Volume 45, Issue 3–4, pp 305–317 | Cite as

Noise-Induced Chaos in Duffing Oscillator with Double Wells

  • Chunbiao Gan


Stochastic Melnikov method is employed to predict noise-induced chaotic response in the Duffing oscillator with double wells. The safe basin is simulated to show the noise-induced fractal boundary. Three cases are considered: harmonic excitation, both harmonic and Gaussian white noise excitations, and Gaussian white noise excitation. The leading Lyapunov exponent estimated by Rosenstein's algorithm is shown to quantify the chaotic nature of the sample time series of the system. The results show that the boundary of the safe basin can be fractal even if the system is excited only by external Gaussian white noise.


Duffing oscillator Gaussian white noise fractal basin boundary leading Lyapunov exponent noise-induced chaos 


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  1. 1.
    Bulsara, A. R., Schieve, W. C., and Jacobs, E.W., ‘Homoclinic chaos in systems perturbed by weak Langevin noise’, Physical Review A 41(2), 1990, 668–681.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Frey, M. and Simiu, E., ‘Equivalence between motions with noise-induced jumps and chaos with Smale horseshoes’, in Proceedings of the 9th Engineering Mechanics Conference, ASCE, 1992, 660–663.Google Scholar
  3. 3.
    Frey, M. and Simiu, E., ‘Noise-induced chaos and phase space flux’, Physica D 63, 1993, 321–340.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Xie, W. C., ‘Effect of noise on chaotic motion of buckled column under periodic excitation’, Nonlinear and Stochastic Dynamics, AMD Vol. 192/DE Vol. 73, 1994, 215–225.Google Scholar
  5. 5.
    Simiu, E. and Frey, M., ‘Melnikov processes and noise-induced exits from a well’, Journal of Engineering Mechanics 122(3), 1996, 263–270.CrossRefGoogle Scholar
  6. 6.
    Lin, H. and Yim, S. C. S., ‘Analysis of a nonlinear system exhibiting chaotic, noisy chaotic, and random behaviors’, ASME Journal of Applied Mechanics 63, 1996, 509–516.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Wiggins, S., Global Bifurcations and Chaos: Analytical Methods, Springer, New York, 1988.zbMATHGoogle Scholar
  8. 8.
    Soliman, M. S. and Thompson, J. M. T., ‘Integrity measures quantifying the erosion of smooth and fractal basins of attraction’, Journal of Sound and Vibration 35, 1989, 453–475.MathSciNetCrossRefGoogle Scholar
  9. 9.
    McDonald, S. W., Grebogi, C., Ott, E., and Yorke, J. A., ‘Fractal basin boundaries’, Physica D 17, 1985, 125–153.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Moon, F. C. and Li, G. X., ‘Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential’, Physical Review Letter 55, 1985, 1439–1442.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Soliman, M. S., ‘Fractal erosion of basins of attraction in coupled nonlinear systems’, Journal of Sound and Vibration 182, 1995, 727–740.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Senjanovic, I., Parunov, J., and Cipric, G., ‘Safety analysis of ship rolling in rough sea’, Chaos, Solitons and Fractals 4, 1997, 659–680.zbMATHCrossRefGoogle Scholar
  13. 13.
    Freitas, M. S. T., Viana, R. L., and Grebogi, C., ‘Erosion of the safe basin for the transversal oscillations of a suspension bridge’, Chaos, Solitons and Fractals 18, 2003, 829–841.zbMATHCrossRefGoogle Scholar
  14. 14.
    Xu, J., Lu, Q. S., and Huang, K. L., ‘Controlling erosion of safe basin in nonlinear parametrically excited systems’, ACTA Mechanica Sinica 12, 1996, 281–288.zbMATHGoogle Scholar
  15. 15.
    Gan, C. B., Lu, Q. S., and Huang, K. L., ‘Non-stationary effects on safe basins of a softening Duffing oscillator’, ACTA Mechanica Solida Sinica 11(3), 1998, 253–260.Google Scholar
  16. 16.
    Gan, C., ‘Noise-Induced chaos and basin erosion in softening Duffing oscillator’, Chaos, Solitons & Fractals 25, 2005, 1069–1081.zbMATHCrossRefGoogle Scholar
  17. 17.
    Wolf, A., Swift, J. R., Swinney, H. L., and Vastano, J. A., ‘Determining Lyapunov exponents from a time series’, Physica D 16, 1985, 285–317.zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kantz, H. and Schreiber, T., Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, England, 1997.zbMATHGoogle Scholar
  19. 19.
    Sano, M. and Sawada, Y., ‘Measurement of the Lyapunov spectrum from a chaotic time series’, Physical Review Letter 55, 1985, 1082–1085.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Eckmann, J. P., Kamphorts, S. O., Ruelle, D., and Ciliberto, S., ‘Lyapunov exponents from a time series’, Physical Review A 34, 1986, 4971–4979.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rosenstein, M. T., Collins, J. J., and Luca, C. J., ‘A practical method for calculating leading Lyapunov exponents from small data sets’, Physica D 65, 1993, 117–134.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Shinozuka, M., ‘Digital simulation of random processes and its applications’, Journal of Sound and Vibration 25(1), 1972, 111–128.CrossRefGoogle Scholar
  23. 23.
    Takens, F., ‘Detecting strange attractors in turbulence’, in Lecture Notes in Mathematics. D. A. Rand and L. S. Young (eds.), Vol. 898, Springer, New York, 1981.Google Scholar

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© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mechanics, CMEEZhejiang UniversityHangzhouChina

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