Stability and Neimark-Sacker bifurcation of numerical discretization of delay differential equations. (English) Zbl 1198.65107

He, Zhimin; Lai, Xin; Hou, Aiyu

Chaos Solitons Fractals 41, No. 4, 2010-2017 (2009).

Summary: A kind of a discrete delay model obtained by Euler method is investigated. Firstly, the linear stability of the model is studied. It is found that there exist Neimark-Sacker bifurcations when the delay passes a sequence of critical values. Then the explicit algorithm for determining the direction and stability of the Neimark-Sacker bifurcations are derived by using the normal form theory and center manifold theorem. Finally, computer simulations are provided to illustrate the analytical results found.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

65L03	Numerical methods for functional-differential equations
65P30	Numerical bifurcation problems
37G99	Local and nonlocal bifurcation theory for dynamical systems
34K18	Bifurcation theory of functional-differential equations
34K20	Stability theory of functional-differential equations

References:

[1]	Marsden, J. E.; McCracken, M., The Hopf bifurcation and its applications (1976), Spring-Verlag: Spring-Verlag New York · Zbl 0346.58007
[2]	Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1983), Spring-Verlag: Spring-Verlag New York · Zbl 0515.34001
[3]	Yuri, A. K., Elements of applied bifurcation theory (1995), Spring-Verlag: Spring-Verlag New York · Zbl 0829.58029
[4]	Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and applications of Hopf bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002
[5]	Koto, T., Neimark-Sacker bifurcations in the Euler method for a delay differential equations, BIT, 39, 110-115 (1999) · Zbl 0918.65054
[6]	Neville, J. F.; Wulf, V., The use of boundary locus points in the identification of bifurcation point in numerical approximation of delay differential equations, J Comput Appl Math, 111, 153-162 (1999) · Zbl 0941.65132
[7]	Neville, J. F.; Wulf, V., Numerical Hopf bifurcation for a class of delay differential equations, J Comput Appl Math, 115, 601-616 (2000) · Zbl 0946.65065
[8]	Peng, M.; Ugar, A., The use of the Euler method in identification of multiple bifurcations and chaotic behavior in numerical approximation of delay differential equations, Chaos, Solitons & Fractals, 21, 883-891 (2004) · Zbl 1054.65126
[9]	Peng, M., Bifurcation and chaotic behavior in the Euler method for a Kaplan-Yorke prototype delay model, Chaos, Solitons & Fractals, 20, 489-496 (2004) · Zbl 1048.37030
[10]	Zhang, C.; Liu, M.; Zheng, B., Hopf bifurcation in numerical approximation of a class delay differential equations, Appl Math Comput, 146, 335-349 (2003) · Zbl 1037.34068
[11]	Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1027.37002
[12]	Yuan, Z.; Hu, D.; Huang, L., Stability and bifurcation analysis on a discrete-time system of two neurons, Appl Math Lett, 17, 1239-1245 (2004) · Zbl 1058.92012
[13]	Yuan, Z.; Hu, D.; Huang, L., Stability and bifurcation analysis on a discrete-time neural network, J Comput Appl Math, 177, 89-100 (2005) · Zbl 1063.93030
[14]	Zhang, C.; Zu, Y.; Zheng, B., Stability and bifurcation of a discrete red blood cell survival model, Chaos, Solitons & Fractals, 28, 386-394 (2006) · Zbl 1079.92027
[15]	Ruan, S.; Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynam Continuous Discrete Impuls Syst Ser A: Math Anal, 10, 863-874 (2003) · Zbl 1068.34072
[16]	Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics (1992), Kluwer Academic Publishers: Kluwer Academic Publishers London · Zbl 0752.34039