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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Giannakopoulos F, Zapp A. Bifurcations in a planar system of differential delay equations modeling neural activity. Phys D, 2001, 1159(3–4): 215–232
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
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
MacDonald N. Time Lags in Biological Models. Berlin: Springer-Verlag, 1978
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
Raghothama A, Narayanan S. Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dynam, 2002, 127(4): 341–365
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
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
Pyragas K. Control of chaos via an unstable delayed feedback controller. Phys Rev Lett, 2001, 186(11): 2265–2268
Pyragas K. Transmission of signals via synchronization of chaotic time-delay systems. Int J Bifurcat Chaos, 1998, 18(9): 1839–1842
Gregory D V, Rajarshi R. Chaotic communication using time-delayed optical systems. Int J Bifurcat Chaos, 1999, 19(11): 2129–2156
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
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
Fridman E. Effects of small delays on stability of singularly perturbed systems. Automatica, 2002, 138(5): 897–902
Xu J, Lu Q S. Hopf bifurcation of time-delay Liénard equations. Int J Bifurcat Chaos, 1999, 19(5): 939–951
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
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
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
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
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
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
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
Hale J K, Luncl S V. Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993
Diekmann O D. Delay Equations. New York: Springer-Verlag, 1995
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
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
Author information
Authors and Affiliations
Corresponding author
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
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11431-009-0052-1