Nonlinear Dynamics

, Volume 73, Issue 4, pp 2241–2260 | Cite as

Applications of the integral equation method to delay differential equations

  • Yueli Chen
  • Jian Xu
Original Paper


In this paper, we compare two approaches for determining the amplitude equations; namely, the integral equation method and the method of multiple scales. To describe and compare the methods, we consider three examples: the parametric resonance of a Van der Pol oscillator under state feedback control with a time delay, the primary resonance of a harmonically forced Duffing oscillator under state feedback control with a time delay, and the primary resonance together with 1:1 internal resonance of a two degree-of-freedom model. Using the integral equation method and the method of multiple scales, the amplitude equations are obtained. The stability of the periodic solution is examined by using the Floquet theorem together with the Routh–Hurwitz criterion (without time delay) and the Nyquist criterion (with time delay). By comparison with the solution obtained by the numerical integration, we find that the accuracy of the integral equation method is much better.


Integral equation method Method of multiple scales Parametric resonance Primary resonance Internal resonance 



This research is supported by the State Key Program of National Natural Science Foundation of China under Grant No. 11032009 and National Natural Science Foundation of China under Grant No. 11272236.


  1. 1.
    Nayfeh, A.H., Mook, D.T.: Non-linear Oscillations. Wiley, New York (1979) Google Scholar
  2. 2.
    Nayfeh, A.H.: Perturbation Techniques. Wiley, New York (1979) Google Scholar
  3. 3.
    Maccari, A.: The response of a parametrically excited van der pol oscillator to a time delay state feedback. Nonlinear Dyn. 26, 105–119 (2001) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Maccari, A.: Delayed feedback control for a parametrically excited van der pol oscillator. Phys. Scr. 76, 526–532 (2007) CrossRefGoogle Scholar
  5. 5.
    Maccari, A.: Vibration amplitude control for a van der pol-Duffing oscillator with time delay. J. Sound Vib. 317, 20–29 (2008) CrossRefGoogle Scholar
  6. 6.
    Maccari, A.: Vibration control for an externally excited nonlinear system. Phys. Scr. 70, 79–85 (2004) MATHCrossRefGoogle Scholar
  7. 7.
    Hu, H., Dowell, E.H., Virgin, L.N.: Resonances of a harmonically forced Duffing oscillator with time delay state feedback. Nonlinear Dyn. 15, 311–327 (1998) MATHCrossRefGoogle Scholar
  8. 8.
    El-Bassiouny, A.F.: Resonances of a nonlinear SDOF system with time-delay in linear feedback control. Phys. Scr. 81, 015007 (2010) (12 pp.) CrossRefGoogle Scholar
  9. 9.
    El-Bassiouny, A.F.: Nonlinear analysis for a ship with a general roll-damping model. Phys. Scr. 75, 691–701 (2007) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Zhao, Y.Y., Xu, J.: Mechanism analysis of delayed nonlinear vibration absorber. Chin. J. Theor. Appl. Mech. 40(1), 98–106 (2008) MathSciNetGoogle Scholar
  11. 11.
    Schmidt, G., Tondl, A.: Nonlinear Vibrations. Cambridge University Press, Cambridge (1986) MATHGoogle Scholar
  12. 12.
    Hu, H.Y., Wang, Z.H.: Calculation of the rightmost characteristic root of retarded time-delay systems via Lambdert W function. J. Sound Vib. 315, 757–767 (2008) Google Scholar
  13. 13.
    Fu, M.Y., Olbrot, A.W., Polis, M.P.: Robust stability for time-delay systems: the edge theorem and graphical tests. IEEE Trans. Autom. Control 34, 813–820 (1989) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Wahi, P., Chatterjee, A.: Garlerkin projections for delay differential equations. ASME J. Dyn. Syst. Meas. Control 127, 80–87 (2005) CrossRefGoogle Scholar
  15. 15.
    Das, S.L., Chatterjee, A.: Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dyn. 30, 323–335 (2002) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Hu, H.Y., Wang, Z.H.: Singular perturbation methods for nonlinear dynamic systems with time delays. Chaos Solitons Fractals 40, 13–27 (2009) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Wang, H.L., Hu, H.Y.: Remarks on the perturbation methods in solving the second-order delay differential equations. Nonlinear Dyn. 33, 379–398 (2003) MATHCrossRefGoogle Scholar
  18. 18.
    Das, S.L., Chatterjee, A.: Second order multiple scales for oscillators with large delay. Nonlinear Dyn. 39, 375–394 (2005) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiP.R. China

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