Numerical bifurcation control of Mackey-Glass system. (English) Zbl 1221.93235

Summary: A mathematical physiological model, the Mackey – Glass system of a delay differential equation, is considered. With a greater delay, a periodic solution arises, which characterizes the disease of chronic granulocytic leukemia (CGL). To treat such disease, a blood transfusion feedback control is considered, from the point of view of mathematical control. By using a nonstandard finite-difference (NSFD) scheme to the control system, we obtain a numerical discrete system and analyze its Neimark – Sacker and fold bifurcations. The results imply that the condition of the illness could be relieved by transfusing blood to the patient, if the control is a delay control. Finally, the effectiveness of the control are illustrated by several numerical simulations.


93D15 Stabilization of systems by feedback
92C30 Physiology (general)
34K18 Bifurcation theory of functional-differential equations
37N25 Dynamical systems in biology
93B52 Feedback control
92-08 Computational methods for problems pertaining to biology
Full Text: DOI


[1] Mackey, M.C.; Glass, L., Oscillations and chaos in physiological control systems, Science, 197, 287-289, (1977) · Zbl 1383.92036
[2] Wei, J., Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 20, 2483-2498, (2007) · Zbl 1141.34045
[3] Wulf, V.; Ford, N.J., Numerical Hopf bifurcation for a class of delay differential equation, J. comput. appl. math., 115, 601-616, (2000) · Zbl 0946.65065
[4] V. Wulf, numerical analysis of delay differential equations undergoing a Hopf bifurcation, Ph.D. diss., University of Liverpool, (1999).
[5] Ford, N.J.; Wulf, V., The use of boundary locus plots in the identification of bifurcation points in numerical approximation of delay differential equations, J. comput. appl. math., 111, 153-162, (1999) · Zbl 0941.65132
[6] Su, H.; Ding, X.H., Dynamics of a nonstandard finite-difference scheme for mackey – glass system, J. math. anal. appl., 344, 932-941, (2008) · Zbl 1153.65074
[7] Mickens, R.E., A nonstandard finite-difference scheme for the lotka – volterra system, Appl. numer. math., 45, 309-314, (2003) · Zbl 1025.65047
[8] Patidar, K.C., On the use of nonstandard finite difference methods, J. differ. equ. appl., 11, 8, 735-758, (2005) · Zbl 1073.65545
[9] Hassard, B.D.; Kazarinoff, N.D.; Wa, Y.H., Theory and applications of Hopf bifurcation, (1981), Cambridge University · Zbl 0474.34002
[10] Yuri, A.K., Elements of applied bifurcation theory, (1995), Spring-Verlag New York · Zbl 0829.58029
[11] Wiggins, S., Introduction to applied nonliear dynamical systems and chaos, (1990), Springer-Verlag New York
[12] Cheng, Z.; Cao, J., Hopf bifurcation control for delayed complex networks, J. franklin I., 344, 846-857, (2007) · Zbl 1126.37033
[13] Chen, G.; Moiola, J.L.; Wang, H.O., Bifurcation control: theories, method, and applications, Int. J. bifurcat. chaos, 10, 3, 511-548, (2000) · Zbl 1090.37552
[14] Kramer, M.A.; Lopour, B.A.; Kirsch, H.E.; Szeri, A.J., Bifurcation control of a seizing human cortex, Phys. rev. E., 73, 041928, (2006)
[15] Engelborghs, K.; Lemaire, V.; Belair, J.; Roose, D., Numerical bifurcation analysis of delay differential equations arising from physical modeling, J. math. biol., 42, 361-385, (2001) · Zbl 0983.65136
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.