Friday, May 17, 2013

The Lorenz equations and chaos

Fifty years ago, Edward Lorenz (1917-2008) published an analysis of Rayleigh-Benard convection that began the study of a field of mathematics called chaos theory. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce chaos by analyzing the logistic map, which
"is an example of chaotic behavior or deterministic chaos. Deterministic chaos has four important characteristics: 1. The system is deterministic, governed by a set of equations that define the evolution of the system. 2. The behavior is bounded. It does not go off to infinity. 3. The behavior of the variables is aperiodic in the chaotic regime. The values never repeat. 4. The behavior depends very sensitively on the initial conditions."
The sensitivity to initial conditions is sometimes called the “butterfly effect,” a term coined by Lorenz. His model is a simplified description of the atmosphere, and has implications for weather prediction.

The mathematical model that Lorenz analyzed consists of three first-order coupled nonlinear ordinary differential equations. Because of their historical importance, I have written a new homework problem that introduces Lorenz’s equations. These particular equations don’t have any biological applications, but the general idea of chaos and nonlinear dynamics certainly does (see, for example, Glass and Mackey’s book From Clock’s to Chaos.
Section 10.7

Problem 33 1/2. Edward Lorenz (1963) published a simple, three-variable (x, y, z) model of Rayleigh-Benard convection
dx/dt = σ (y – x)
dy/dt = x (ρ – z) – y
dz/dt = xy – β z
where σ=10, ρ=28, and β=8/3.
(a) Which terms are nonlinear?
(b) Find the three equilibrium points for this system of equations.
(c) Write a simple program to solve these equations on the computer (see Sec. 6.14 for some guidance on how to solve differential equations numerically). Calculate and plot x(t) as a function of t for different initial conditions. Consider two initial equations that are very similar, and compute how the solutions diverge as time goes by.
(d) Plot z(t) versus x(t), with t acting as a parameter of the curve.

Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences 20: 130–141.

If you want to examine chaos in more detail, see Strogatz’s excellent book Nonlinear Dynamics and Chaos. He has an entire chapter (his Chapter 9) dedicated to the Lorenz equations.

The story of how Lorenz stumbled upon the sensitivity of initial conditions is a fascinating tale. Here is one version in a National Academy of Sciences Biographical Memoir about Lorenz written by Kerry Emanuel.
"At one point, in 1961, Ed had wanted to examine one of the solutions [to a preliminary version of his model that contained 12 equations] in greater detail, so he stopped the computer and typed in the 12 numbers from a row that the computer had printed earlier in the integration. He started the machine again and stepped out for a cup of coffee. When he returned about an hour later, he found that the new solution did not agree with the original one. At first he suspected trouble with the machine, a common occurrence, but on closer examination of the output, he noticed that the new solution was the same as the original for the first few time steps, but then gradually diverged until ultimately the two solutions differed by as much as any two randomly chosen states of the system. He saw that the divergence originated in the fact that he had printed the output to three decimal places, whereas the internal numbers were accurate to six decimal places. His typed-in new initial conditions were inaccurate to less than one part in a thousand.

‘At this point, I became rather excited,’ Ed relates. He realized at once that if the atmosphere behaved the same way, long-range weather prediction would be impossible owing to extreme sensitivity to initial conditions. During the following months, he persuaded himself that this sensitivity to initial conditions and the nonperiodic nature of the solutions were somehow related, and was eventually able to prove this under fairly general conditions. Thus was born the modern theory of chaos."
To learn more about the life of Edward Lorenz, see his obituary here and here. I have not read Chaos: Making a New Science by James Gleick, but I understand that he tells Lorenz’s story there. You can hear Lorenz lecture about chaos here.

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