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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Delay Terms in the Slow Flow

Journal of Applied Nonlinear Dynamics 5(4) (2016) 471--484 | DOI:10.5890/JAND.2016.12.007

Si Mohamed Sah; Richard H. Rand

$^{1}$ Nanostructure Physics, KTH Royal Institute of Technology, Stockholm, Sweden

$^{2}$ Dept. of Mathematics, Dept. of Mechanical & Aerospace Engineering, Cornell University. Ithaca, NY 14853, USA.

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This work concerns the dynamics of nonlinear systems that are subjected to delayed self-feedback. Perturbation methods applied to such systems give rise to slow flows which characteristically contain delayed variables. We consider two approaches to analyzing Hopf bifurcations in such slow flows. In one approach, which we refer to as approach I, we follow many researchers in replacing the delayed variables in the slow flow with non-delayed variables, thereby reducing the DDE slow flow to an ODE. In a second approach, which we refer to as approach II, we keep the delayed variables in the slow flow. By comparing these two approaches we are able to assess the accuracy of making the simplifying assumption which replaces the DDE slow flow by an ODE. We apply this comparison to two examples, Duffing and van der Pol equations with self-feedback.


The author S.M. Sah gratefully acknowledges the financial support of the Ragnar Holm Fellowship at the Royal Institute of Technology (KTH).


  1. [1]  Ji, J.C. and Leung, A.Y.T. (2002), Resonances of a nonlinear SDOF system with two time-delays on linear feedback control, Journal of Sound and Vibration, 253, 985-1000.
  2. [2]  Maccari A. (2001), The resonances of a parametrically excited Van der Pol oscillator to a time delay state feedback, Nonlinear Dynamics, 26, 105-119.
  3. [3]  Hu, H., Dowell, E. H. and Virgin, L. N. (1998), Resonances of a harmonically forced duffing oscillator with time delay state feedback, Nonlinear Dynamics, 15, 311-327.
  4. [4]  Wahi, P. and Chatterjee, A. (2004), Averaging oscillations with small fractional damping and delayed terms. Nonlinear Dynamics. 38, 3-22.
  5. [5]  Atay, F.M. (1998), Van der Pol’s oscillator under delayed feedback. Journal of Sound and Vibration, 218(2), 333-339.
  6. [6]  Suchorsky, M.K., Sah, S.M. and Rand, R.H. (2010), Using delay to quench undesirable vibrations. Nonlinear Dynamics, 62, 107-116.
  7. [7]  Engelborghs, K., Luzyanina, T. and Roose, D. (2002), Numerical bifurcation analysis of delay differential equations using: DDE- BIFTOOL. ACM Transactions on Mathematical Software, 28(1), 1-21.
  8. [8]  Engelborghs, K., Luzyanina, T. and Samaey, G. (2001), DDE- BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations. Technical Report TW-330, Dept. Comp. Sci., K.U.Leuven, Leuven, Belgium
  9. [9]  Heckman, C.R. (2012), An introduction to DDE-BIFTOOL is available as Appendix B of the doctoral thesis of Christoffer Heckman: asymptotic and numerical analysis of delay- coupled microbubble oscillators (Doctoral Thesis). Cornell University
  10. [10]  Wirkus, S. and Rand, R.H. (2002), The dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dynamics, 30, 205-221.
  11. [11]  Morrison, T.M. and Rand, R.H. (2007), 2:1 Resonance in the delayed nonlinear Mathieu equation. Nonlinear Dynamics, 50, 341-352.
  12. [12]  Rand, R.H. (2012), Lecture notes in nonlinear vibrations published on-line by the Internet-First University Press http://
  13. [13]  Strogatz, S. H. (1994), Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, Massachusetts)
  14. [14]  Kalmar-Nagy, T. and Stepan, G. and Moon, F.C. (2001), Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, Nonlinear Dynamics, 26, 121-142.
  15. [15]  Verdugo, A. and Rand, R. (2008), Hopf Bifurcation in a DDE Model of Gene Expression, Communications in Nonlinear Science and Numerical Simulation, 13, 235-242.
  16. [16]  Erneux T. and Grasman J. (2008), Limit-cycle oscillators subject to a delayed feedback, Physical Review E, 78(2), 026209.