zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] Marsden, J.E.; McCracken, M., The Hopf bifurcation and its applications, (1976), Spring-Verlag New York · Zbl 0346.58007
[2] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Spring-Verlag New York · Zbl 0515.34001
[3] Yuri, A.K., Elements of applied bifurcation theory, (1995), 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 · 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 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 London · Zbl 0752.34039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.