Skip to main content
SpringerLink
Log in

A perturbation-incremental scheme for studying Hopf bifurcation in delayed differential systems

  • Published:
Science in China Series E: Technological Sciences Aims and scope Submit manuscript

Abstract

A new method, called perturbation-incremental scheme (PIS), is presented to investigate the periodic solution derived from Hopf bifurcation due to time delay in a system of first-order delayed differential equations. The method is summarized as three steps, namely linear analysis at critical value, perturbation and increment for continuation. The PIS can bypass and avoid the tedious calculation of the center manifold reduction (CMR) and normal form. Meanwhile, the PIS not only inherits the advantages of the method of multiple scales (MMS) but also overcomes the disadvantages of the incremental harmonic balance (IHB) method. Three delayed systems are used as illustrative examples to demonstrate the validity of the present method. The periodic solution derived from the delay-induced Hopf bifurcation is obtained in a closed form by the PIS procedure. The validity of the results is shown by their consistency with the numerical simulation. Furthermore, an approximate solution can be calculated in any required accuracy.

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. Giannakopoulos F, Zapp A. Bifurcations in a planar system of differential delay equations modeling neural activity. Phys D, 2001, 1159(3–4): 215–232

    Article  MathSciNet  Google Scholar 

  2. Liao X F, Wong K W, Wu Z F. Bifurcation analysis on a two-neuron system with distributed delays. Phys D, 2001, 1149(1–2): 123–141

    Article  MathSciNet  Google Scholar 

  3. Krise S, Choudhury S R. Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations. Chaos Soliton Fract, 2003, 116(1): 59–77

    Article  MathSciNet  Google Scholar 

  4. MacDonald N. Time Lags in Biological Models. Berlin: Springer-Verlag, 1978

    MATH  Google Scholar 

  5. Nayfeh A H, Chin C M, Pratt J. Perturbation methods in nonlinear dynamics-applications to machining dynamics. J Manuf Sci E-T ASME, 1997, 1119(4A): 485–493

    Article  Google Scholar 

  6. Raghothama A, Narayanan S. Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dynam, 2002, 127(4): 341–365

    Article  MathSciNet  Google Scholar 

  7. Maccari A. The response of a parametrically excited van der Pol oscillator to a time delay state feedback. Nonlinear Dynam, 2001, 126(2): 105–119

    Article  MathSciNet  Google Scholar 

  8. Kalmár-Nagy T, Stépán G, Moon F C. Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dynam, 2001, 126(2): 121–142

    Article  Google Scholar 

  9. Pyragas K. Control of chaos via an unstable delayed feedback controller. Phys Rev Lett, 2001, 186(11): 2265–2268

    Article  Google Scholar 

  10. Pyragas K. Transmission of signals via synchronization of chaotic time-delay systems. Int J Bifurcat Chaos, 1998, 18(9): 1839–1842

    Google Scholar 

  11. Gregory D V, Rajarshi R. Chaotic communication using time-delayed optical systems. Int J Bifurcat Chaos, 1999, 19(11): 2129–2156

    Google Scholar 

  12. Reddy D V R, Sen A, Johnston G L. Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks. Phys D, 2000, 1144(3–4): 335–357

    Article  Google Scholar 

  13. Xu J, Chung K W. Effects of time delayed position feedback on a van der Pol-Duffing oscillator. Phys D, 2003, 1180(1–2): 17–39

    Article  MathSciNet  Google Scholar 

  14. Fridman E. Effects of small delays on stability of singularly perturbed systems. Automatica, 2002, 138(5): 897–902

    Article  MathSciNet  Google Scholar 

  15. Xu J, Lu Q S. Hopf bifurcation of time-delay Liénard equations. Int J Bifurcat Chaos, 1999, 19(5): 939–951

    MathSciNet  Google Scholar 

  16. Campbell S A, Bélair J, Ohira T, et al. Limit cycles, tori, and complex dynamics in a second-order differential equations with delayed negative feedback. J Dynam Diff Eq, 1995, 17(1): 213–236

    Article  Google Scholar 

  17. Fofana M S, Cabell B Q, Vawter N, et al. Illustrative applications of the theory of delay dynamical systems. Math Comp Model, 2003, 137(12–13): 1371–1382

    Article  MathSciNet  Google Scholar 

  18. Das S L, Chatterjee A. Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcation. Nonlinear Dynam, 2002, 130(2): 323–335

    Article  MathSciNet  Google Scholar 

  19. Moiola J L, Chiacchiarini H G. Bifurcation and Hopf degeneracies in nonlinear feedback systems with time delay. Int J Bifurcat Chaos, 1996, 16(4): 661–672

    Google Scholar 

  20. Chan H S Y, Chung K W, Xu Z. A perturbation-incremental method for strongly non-linear oscillators. Int J Nonlinear Mech, 1996, 131(1): 59–72

    Article  MathSciNet  Google Scholar 

  21. Chan H S Y, Chung K W, Qi D W. Some bifurcation diagrams for limit cycles of quadratic differential systems. Int J Bifurcat Chaos, 2001, 111(1): 197–206

    MathSciNet  Google Scholar 

  22. Olgac N, Sipahi R. An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems. IEEE T Automat Contr, 2002, 147(5): 793–797

    Article  MathSciNet  Google Scholar 

  23. Hale J K, Luncl S V. Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993

    MATH  Google Scholar 

  24. Diekmann O D. Delay Equations. New York: Springer-Verlag, 1995

    MATH  Google Scholar 

  25. Wu J, Faria T, Huang Y S. Synchronization and stable phase-locking in a network of neurons with memory. Math Comp Model, 1999, 130(1): 117–138

    Article  MathSciNet  Google Scholar 

  26. Xu J, Chung K W. An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks. SIAM J Appl Dyn Syst, 2007, 6(1): 29–60

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China

    Jian Xu

  2. Department of Mathematics, City University of Hong Kong Kowloon, Hong Kong, China

    Kwok Wai Chung

Authors
  1. Jian Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kwok Wai Chung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Xu.

Additional information

Supported by National Natural Science Funds for Distinguished Young Scholar (Grant No. 10625211), Key Program of National Natural Science Foundation of China (Grant No. 10532050), Program of Shanghai Subject Chief Scientist (Grant No. 08XD14044), and Hong Kong Research Grants Council under CERG (Grant No. CityU 1007/05E)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, J., Chung, K.W. A perturbation-incremental scheme for studying Hopf bifurcation in delayed differential systems. Sci. China Ser. E-Technol. Sci. 52, 698–708 (2009). https://doi.org/10.1007/s11431-009-0052-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11431-009-0052-1

Keyword

Search

Navigation