On the dynamics of the singularly perturbed Mackey-Glass equation

Abstract In this paper, we consider the singularly perturbed Mackey–Glass equation. By letting the perturbation parameter tends to zero, such an equation is formally reduced to a scalar difference equation. Local stability analysis of fixed points is investigated. The method of steps is employed to discretize the system. Moreover, numerical simulations including Lyapunov exponent, bifurcation diagrams and phase portraits are carried out to confirm the theoretical analysis obtained and to explore more complex dynamics of the system.

[1]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[2]  J. Dieudonne.  Foundations of Modern Analysis , 1969 .

[3]  U. an der Heiden.  Delays in physiological systems. , 1979, Journal of mathematical biology.

[4]  Hal L. Smith,et al.  An introduction to delay differential equations with applications to the life sciences / Hal Smith , 2010 .

[5]  S. Elaydi.  An introduction to difference equations , 1995 .

[6]  R. D. Driver,et al.  Ordinary and Delay Differential Equations , 1977 .

[7]  Michael C. Mackey,et al.  Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors , 1989 .

[8]  Peter W. Frank.  Foundations of Modern Analysis. J. Dieudonné. Academic Press, New York, 1960. xiv + 361 pp. Illus. $10 , 1960 .

[9]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[10]  L. Glass,et al.  PATHOLOGICAL CONDITIONS RESULTING FROM INSTABILITIES IN PHYSIOLOGICAL CONTROL SYSTEMS * , 1979, Annals of the New York Academy of Sciences.

[11]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[13]  S. P. Blythe,et al.  Nicholson's blowflies revisited , 1980, Nature.

[14]  N. D. Hayes.  Roots of the Transcendental Equation Associated with a Certain Difference‐Differential Equation , 1950 .

[15]  J. M. Aguirregabiria,et al.  Fractal basin boundaries of a delay-differential equation , 1987 .