From Clocks to Chaos

In Chapter 10 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss nonlinear dynamics and chaos. Leon Glass and Michael Mackey are pioneers in applying nonlinear dynamics to biomedical problems. Their book From Clocks to Chaos: The Rhythms of Life discusses the idea of dynamical diseases, which are “characterized by sudden changes in the qualitative dynamics of physiological processes, leading to abnormal dynamics and disease.” I started working at the National Institutes of Health the year their book was published (1988), and a group of us would meet periodically to discuss potential applications of nonlinear dynamics to important biomedical questions.

Figure 10.34 of Intermediate Physics for Medicine and Biology is reproduced from a highly cited paper Oscillation and Chaos in Physiological Control Systems published in Science (Volume 197, Pages 287-289) by Mackey and Glass (over 1500 citations). It shows how changing one parameter in the system (the delay time) causes a transition from regular to chaotic behavior. In the new homework problem below, you are asked to reproduce this figure. The problem is rather advanced, because it requires you to solve a differential equation numerically using a computer program. But for those readers who are comfortable with computer programming, it provides a nice exercise in numerical analysis of a delay differential equation. And some of you who are familiar with software such at MATLAB or Mathematica might be able to reproduce the figure without knowing anything about numerical methods using their built-in differential equation solver routines. (I don’t approve of this kind of thing, being rather old-school about writing your own computer programs.)

For more background, I recommend the Scholarpedia article on the Mackey-Glass equation. Also, for those needing help with numerical methods, I suggest one of the versions of Numerical Recipes. (The copy on my bookshelf is Numerical Recipes in Fortran 77, but you may have a different favorite computer language).

Section 10.11

Problem 42 1/2 Write a computer program to reproduce the numerical results in Fig. 10.34b and c. The calculation was originally performed by Glass and Mackey using the delay differential equation

where x is the white blood cell count (equal to P/θ in the figure), and β_o=0.2, γ=0.1, and n=10. The initial condition for x is 0.1. Figure 10.34b uses τ = 6, and Fig. 10.34c uses τ = 20. (See Sec. 6.14 for some guidance on how to solve differential equations numerically.)

For additional fun, plot x(t-τ) versus x(t) for each case (a phase plane plot). For τ = 20, this plot contains a strange attractor. I believe it is the illustration that is on the cover of From Clocks to Chaos (however, I long ago lost or lent out my copy, so I cannot verify this).

For more about this differential equation, click here, here and here.