The Mackey-Glass Equation
by Mike Martin with Mike Mackey and Leon Glass

On this page, plots of solutions for the Mackey-Glass delay-differential equation are generated. The equation is:

τ represents a non-negative time delay in evaluating a signal.   n is a non-negative shape parameter that helps to describe the time-delayed feedback/response.   β and γ are the two remaining, non-negative parameters in the equation.   Since this is a delay differential equation, the initial conditions must be specified on an interval and that makes this an infinite-dimensional system. The initial conditions for this problem is set to a value of 0.5 on the interval [ -τ, 0], but it can be set to other values or even functions in the entry field below. In the original work by Mackey and Glass, x (t) represents a concentration of circulating white blood cells and τ is the time lag in the physiological control system used to determine the subsequent production of new circulating blood cells.   The first plot generated below is a time-delay embedding of x (t - τ) versus x (t). The second plot is a plot of x versus t on the interval [Tmin, Tmax] and is intended to give a view of long-term, steady-state behavior.   The next graph is a plot of x versus t on the interval [-τ, Tmax] and is intended to exhibit the full behavior of the signal, including the initial conditions.   The last graph is that of the power spectrum for the signal using 2048 points on the interval [Tmin, Tmax].   The user inputs values of the parameters β, γ, τ, n. Also time bounds, grid bounds (the maximum value of the power spectrum), and the initial conditions must be specified.



Enter in the value of the parameters and select the plotting bounds, then press the "Plot/Evaluate" button.

β =  

γ =  

τ =  

n =  


x (t) =     for   t ≤ 0


0   ≤   Tmin =     ≤   t   ≤     = Tmax


Max Power Spectrum =  






References:

ScholarPedia's Mackey-Glass Equation

Wolfram's Mackey-Glass Demonstration Project

M.C. Mackey & L. Glass. "Oscillation and chaos in physiological control systems", Science (1977) 197, 287- 289.