Skip to main content
SpringerLink
Log in

Effect of External Excitations on a Nonlinear System with Time Delay

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The trivial equilibrium of a two-degree-of-freedom autonomous system may become unstable via a Hopf bifurcation of multiplicity two and give rise to oscillatory bifurcating solutions, due to presence of a time delay in the linear and nonlinear terms. The effect of external excitations on the dynamic behaviour of the corresponding non-autonomous system, after the Hopf bifurcation, is investigated based on the behaviour of solutions to the four-dimensional system of ordinary differential equations. The interaction between the Hopf bifurcating solutions and the high level excitations may induce a non-resonant or secondary resonance response, depending on the ratio of the frequency of bifurcating periodic motion to the frequency of external excitation. The first-order approximate periodic solutions for the non-resonant and super-harmonic resonance response are found to be in good agreement with those obtained by direct numerical integration of the delay differential equation. It is found that the non-resonant response may be either periodic or quasi-periodic. It is shown that the super-harmonic resonance response may exhibit periodic and quasi-periodic motions as well as a co-existence of two or three stable motions.

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. Gilsinn, D. E., ‘Estimating critical Hopf bifurcation parameters for a second-order delay differential equation with application to machine tool chatter’, Nonlinear Dynamics 30, 2002, 103–154.

    Article  Google Scholar 

  2. Kalmar-Nagy, T., Stepan, G., and Moon, F. C., ‘Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations’, Nonlinear Dynamics 26, 2001, 121–142.

    Article  Google Scholar 

  3. Raghothama, A. and Narayanan, S., ‘Periodic response and chaos in nonlinear systems with parametric excitation and time delay’, Nonlinear Dynamics 27, 2002, 341–365.

    Article  Google Scholar 

  4. Ji, J. C. and Leung, A. Y. T., ‘Resonances of a non-linear SDOF system with two time-delays in linear feedback control’, Journal of Sound and Vibration 253, 2002, 985–1000.

    Article  MathSciNet  Google Scholar 

  5. Maccari, A., ‘The resonances of a parametrically excited van der Pol oscillator to a time delay state feedback’, Nonlinear Dynamics 26, 2001, 105–119.

    Article  Google Scholar 

  6. Hu, H. Y., Dowell, E. H., and Virgin, L. N., ‘Resonances of a harmonically forced Duffing oscillator with time delay state feedback’, Nonlinear Dynamics 15, 1998, 311–327.

    Article  Google Scholar 

  7. Plaut, R. H. and Hsieh, J. C., ‘Non-linear structural vibrations involving a time delay in damping’, Journal of Sound and Vibration 117, 1987, 497–510.

    Google Scholar 

  8. Wirkus, S. and Rand, R., ‘The dynamics of two coupled van der Pol oscillators with delay coupling’, Nonlinear Dynamics 30, 2002, 205–221.

    Article  Google Scholar 

  9. Ji, J. C. and Hansen, C. H., ‘Hopf bifurcation of a magnetic bearing system with time delay’, ASME Journal of Vibration and Acoustics, in press.

  10. Hale, J., Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.

    Google Scholar 

  11. Hale, J. K. and Verduyn Lunel, S. M., Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.

    Google Scholar 

  12. Halanay, A., Differential Equations, Stability, Oscillations, Time Lags, Academic Press, New York, 1966.

    Google Scholar 

  13. Yu, P., ‘Computation of normal forms via a perturbation technique’, Journal of Sound and Vibration 211, 1998, 19–38.

    Article  Google Scholar 

  14. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley-Interscience, New York, 1979.

    Google Scholar 

  15. Yu, P., Taneri, U., and Huseyin, K., ‘Forced oscillations, bifurcations and stability of a molecular system (part I): Non-resonance’, International Journal of Systems Science 27, 1996, 1339–1350.

    Google Scholar 

  16. Yu, P., Taneri, U., and Huseyin, K., ‘Forced oscillations, bifurcations and stability of a molecular system (part II): Resonances’, International Journal of Systems Science 27, 1996, 1351–1361.

    Google Scholar 

  17. Hairer, E., Norsett, S. P., and Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd revised edn., Springer-Verlag, Berlin, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia

    J. C. Ji & Colin H. Hansen

  2. School of Mechanical Engineering, Hebei University of Technology, Tianjin, 300130, P.R. China

    Xinye Li

Authors
  1. J. C. Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Colin H. Hansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xinye Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. C. Ji.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ji, J.C., Hansen, C.H. & Li, X. Effect of External Excitations on a Nonlinear System with Time Delay. Nonlinear Dyn 41, 385–402 (2005). https://doi.org/10.1007/s11071-005-0418-2

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-005-0418-2

Key words

Search

Navigation