Chaos: Making a New Science

In Chapter 10 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I include a homework problem about the Lorenz model.

Problem 36. Edward Lorenz (1963) published a simple, three-variable (x, y, z) model of Rayleigh–Bénard convection:

dx/dt = σ (y − x)

dy/dt = x (ρ − z) − y

dz/dt = x y − 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 Sect. 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 conditions that are very similar, and compute how the solutions diverge as time goes by.

(d) Plot z(t) vs. x(t), with t acting as a parameter of the curve.

This model played a critical role in the history of nonlinear dynamics; it resulted in Lorenz discovering chaos. More specifically, his calculation revealed one of the hallmarks of chaotic behavior: sensitivity to initial conditions, also known as the butterfly effect.

Chaos: Making a New Science,
by James Gleick.

I will let James Gleick, author of Chaos: Making a New Science, tell the story. Lorenz’s equations are a model for the weather, which he solved using one of the first computers.

One day in the winter of 1961, wanting to examine one sequence at greater length, Lorenz took a shortcut. Instead of starting the whole run over, he started midway through. To give the machine its initial conditions, he typed the numbers straight from the earlier printout. Then he walked down the hall to get away from the noise and drink a cup of coffee. When he returned an hour later, he saw something unexpected, something that planted the seed for a new science.

This new run should have exactly duplicated the old. Lorenz had copied the numbers into the machine himself. The program had not changed. Yet as he stared at the new printout, Lorenz saw his weather diverging so rapidly from the pattern of the last run that, within just a few months, all resemblance had disappeared. He looked at one set of numbers, then back at the other. He might as well have chosen two random weathers out of a hat. His first thought was that another vacuum tube had gone bad.

Suddenly he realized the truth. There had been no malfunction. The problem lay in the numbers he had typed. In the computer’s memory, six decimal places were stored: .506127. On the printout, to save space, just three appeared: .506. Lorenz had entered the shorter, rounded-off numbers, assuming that the difference—one part in a thousand—was inconsequential…

He decided to look more closely at the way two nearly identical runs of weather flowed apart. He copied one of the wavy lines of output onto a transparency and laid it over the other, to inspect the way it diverged. First, two humps matched detail for detail. Then one line began to lag a hairsbreadth behind. By the time the two runs reached the next hump, they were distinctly out of phase. By the third or fourth hump, all similarity had vanished.

It was only a wobble from a clumsy computer. Lorenz could have assumed something was wrong with his particular machine or his particular model—probably should have assumed. It was not as though he had mixed sodium and chlorine and got gold. But for reasons of mathematical intuition that his colleagues would begin to understand only later, Lorenz felt a jolt: something was philosophically out of joint. The practical import could be staggering. Although his equations were gross parodies of the earth’s weather, he had a faith that they captured the essence of the real atmosphere. That first day, he decided that long-range weather forecasting must be doomed.