Skip Navigation Links
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


Distributed Position and Velocity Delay Effects in a Van der Pol System with Time-periodic Feedback

Journal of Vibration Testing and System Dynamics 8(2) (2024) 249--272 | DOI:10.5890/JVTSD.2024.06.007

Ryan Roopnarain, S. Roy Choudhury

Download Full Text PDF



The effects of a distributed delay on a parametrically forced Van der Pol limit cycle oscillator are considered. Delays modeling time lags due to a variety of factors in self-excited systems, have been considered earlier in the context of modification and control of limit cycle and quasiperiodic responses. Those studies are extended here to include the effects of periodically amplitude modulated {\it distributed} delays in both the position and velocity. A normal form or `slow flow' is employed to search for various bifurcations and transitions between regimes of different dynamics, including amplitude death and quasiperiodicity. The existence of quasiperiodic solutions then motivates the derivation of a second slow flow. A detailed comparison of the results and predictions from the second slow flow to numerical solutions is made. The second slow flow is also employed to approximate the amplitudes of the quasiperiodic solutions, yielding close agreement with the numerical results on the original system. Finally, the effect of varying the delay parameter is briefly considered, and the results and conclusions are summarized.


  1. [1]  Mitropolsky, Y.A. and Van Dao, N. (2013), Applied asymptotic methods in nonlinear oscillations, Vol. 55, Springer Science \& Business Media.
  2. [2]  Atay, F.M. (1998), Van der Pol's oscillator under delayed feedback, Journal of Sound and Vibration, 218(2), 333-339.
  3. [3]  Maccari, A. (2001), The response of a parametrically excited van der Pol oscillator to a time delay state feedback, Nonlinear Dynamics, 26(2), 105-119.
  4. [4]  Maccari, A. (2003), Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback. International Journal of Non-Linear Mechanics, 38(1), 123-131.
  5. [5]  Sah, S. and Belhaq, M. (2008), Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator, Chaos, Solitons $\&$ Fractals, 37(5), 1489-1496.
  6. [6]  Brockett, R. (1999), A stabilization problem. In Open problems in mathematical systems and control theory, 75-78, Springer, London.
  7. [7]  Kirrou, I. and Belhaq, M. (2015), Control of bistability in non-contact mode atomic force microscopy using modulated time delay, Nonlinear Dynamics, 81(1), 607-619.
  8. [8]  MacDonald, N. (1978), Time Lags in Biological Models, Vol. 27, Springer Science \& Business Media.
  9. [9]  Moreau, L. and Aeyels, D. (1999), Stabilization by means of periodic output feedback. In Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No. 99CH36304) (Vol. 1, pp. 108-109), IEEE.
  10. [10]  Moreau, L. and Aeyels, D. (2000), A note on stabilization by periodic output feedback for third-order systems, In Proceedings of the 14th International Symposium of Mathematical Theory of Networks and Systems (MTNS), Perpignan.
  11. [11]  Hamdi, M. and Belhaq, M. (2015), On the delayed van der Pol oscillator with time-varying feedback gain, Applied Mechanics and Materials, 706, 149-158.
  12. [12] Hamdi, M. and Belhaq, M. (2018), Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude. Journal of Vibration and Control, 24(24), 5726-5734.
  13. [13]  Roopnarain, R. and Choudhury, S.R. (2021), Distributed Delay Effects on Coupled van der Pol Oscillators, and a Chaotic van der Pol-Rayleigh System. Discontinuity, Nonlinearity, and Complexity, 10(01), 87-115.
  14. [14]  Roopnarain R. and Choudhury, S.R. (2021), Bifurcations and amplitude death from distributed delays in coupled Landau-Stuart oscillators and a chaotic parametrically forced Van der Pol-Rayleigh system. Far East Journal of Applied Mathematics, 109(2), 121-165.