Skip to main content
SpringerLink
Log in

An iteration method for calculating the periodic solution of time-delay systems after a Hopf bifurcation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A general description of the iteration method is presented in this paper for calculating the periodic solution resulted from a Hopf bifurcation of time-delay systems including the degenerated cases (the delays disappear): ordinary differential equations. Two algorithms are developed for scalar systems and for general systems, respectively. For scalar systems, the iteration method is straightforward, and for general systems, the method needs to solve two eigenvalue problems before the construction of the straightforward iteration scheme. As shown in the four illustrative examples, the iteration method works effectively. It involves easy computation only, the first iteration is usually enough for achieving the accurate bifurcation direction and an accurate estimation of the bifurcated periodic solution.

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. Kuznestov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1998)

    Google Scholar 

  2. Hassard, B., Kazarinoff, N., Wan, Y.-H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, London (1981)

    MATH  Google Scholar 

  3. Iooss, G., Joseph, D.D.: Elementary Stability and Bifurcation Theory. Springer, New York (1990)

    MATH  Google Scholar 

  4. Hale, J.K.: Averaging methods for differential equations with retarded arguments and a small parameter. J. Differ. Equ. 2, 57–73 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kalmár-Nagy, T., Stepán, G., Moon, F.C.: Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dyn. 26, 121–142 (2001)

    Article  MATH  Google Scholar 

  6. Xu, J., Chung, K.W.: Effects of time delayed position feedback on a van der Pol–Duffing oscillator. Phys. D 180, 17–39 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hale, J.K., Magalhães, L.T., Oliva, W.M.: Dynamics in Infinite Dimensions, 2nd edn. Springer, New York (2002)

    MATH  Google Scholar 

  8. Nayfeh, A.H., Chin, C.-M., Pratt, J.R.: Perturbation methods in nonlinear dynamics: applications to machining dynamics. ASME J. Manuf. Sci. Technol. 119, 485–493 (1996)

    Article  Google Scholar 

  9. Das, S.L., Chatterjee, A.: Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dyn. 30, 323–335 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gilsinn, D.E.: Estimating critical Hopf bifurcation parameters for a second-order delay differential equation with application to machine tool chatter. Nonlinear Dyn. 30, 103–154 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nayfeh, A.H.: Perturbation Methods. Shanghai Publishing House of Science and Technology, Shanghai (1984), in Chinese, translated by Wang, F.J., et al.

    Google Scholar 

  13. Casal, A., Freedman, M.: A Poincaré-Lindstedt approach to bifurcation problems for differential-delay equations. IEEE Trans. Autom. Control 25, 967–973 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rand, R., Verdugo, A.: Hopf bifurcation formula for first order differential-delay equations. Commun. Nonlinear Sci. Numer. Simul. 12, 859–864 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. MacDonald, N.: Harmonic balance in delay-differential equations. J. Sound Vib. 186, 649–656 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, Z.H., Hu, H.Y.: Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation. Int. J. Bifurc. Chaos 17(8) (2007, accepted for publication)

  17. Hoppensteadt, F.C.: Analysis and Simulation of Chaotic Systems, 2nd edn. Springer, New York (2000)

    MATH  Google Scholar 

  18. He, J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20, 1141–1199 (2006)

    Article  MATH  Google Scholar 

  19. He, J.H.: Periodic solutions and bifurcations of delay-differential equations. Phys. Lett. A 347, 228–230 (2005)

    Article  MathSciNet  Google Scholar 

  20. Schanz, M., Pelster, A.: Analytical and numerical investigations of the phase-locked loop with time delay. Phys. Rev. E 67, 056205 (2003)

    Article  Google Scholar 

  21. Pelster, A., Kleinert, H., Schanz, M.: High-order variational calculation for the frequency of time-periodic solutions. Phys. Rev. E 67, 016604 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 210016, Nanjing, China

    Z. H. Wang

Authors
  1. Z. H. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Z. H. Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, Z.H. An iteration method for calculating the periodic solution of time-delay systems after a Hopf bifurcation. Nonlinear Dyn 53, 1–11 (2008). https://doi.org/10.1007/s11071-007-9290-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-007-9290-6

Keywords

Search

Navigation