Chaos Theory

In 1961 Edward Lorenz, a meteorologist at MIT, was running a small weather model on a Royal McBee computer. To repeat a run, he typed in a number from a printout, 0.506, instead of the 0.506127 stored in the machine. He went for coffee. When he came back, the new forecast had diverged completely from the old one. The rounding error, one part in several thousand, had grown until it dominated.

Lorenz realised this was not a fault of the computer but a property of the equations. His 1963 paper, a three-variable model of convection, showed motion that never repeats and never settles, winding forever around a butterfly-shaped set. If the atmosphere behaves like that, he concluded, detailed weather forecasts for weeks ahead are impossible. His 1972 talk, "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?", gave the idea its popular name.

In the 1970s chaos turned up in equations far simpler than the weather. Robert May showed in 1976 that the logistic map, a one-line population model, becomes chaotic as its growth rate rises. Tien-Yien Li and James Yorke proved that "period three implies chaos", naming the field, unaware that Oleksandr Sharkovsky had proved a sharper theorem in Kyiv eleven years earlier.

Then came a surprise in the other direction: order in chaos. At Los Alamos in 1975, Mitchell Feigenbaum studied how the logistic map's cycles double in length, 1, 2, 4, 8, and so on, on the way to chaos. The growth rates at which the doublings occur crowd together at a steady ratio of $4.6692\ldots$. He tried a different equation and found the same number. It is universal, a property of the route to chaos rather than of any particular system. Experiments on convecting liquid helium measured it in 1980, and Oscar Lanford proved it in 1982 with a computer-assisted proof.

Take the logistic map with growth rate 4, $x \mapsto 4x(1 - x)$, where $x$ is a population as a fraction of its maximum. Start two copies at $x = 0.3$ and $x = 0.3000001$, which differ by one part in three million, far below any real measurement error.

For about fifteen steps the copies agree to two decimal places. By step 25 they are unrelated. On average the difference doubles at every step. For this map the Lyapunov exponent is exactly $\ln 2$. An initial error of $10^{-7}$ therefore reaches size 1 after about $\log_2 10^7 \approx 23$ steps.

That sets the horizon of prediction, and improving the measurement buys surprisingly little. Measuring the starting value a million times more precisely, to $10^{-13}$, extends the forecast only by $\log_2 10^6 \approx 20$ steps, less than double. Weather behaves similarly: better observations and faster computers have pushed useful forecasts out by about a day per decade, but not to months.

The rule has no randomness at all. Any calculator will reproduce the table exactly. The unpredictability comes entirely from the amplification of small differences, which is the definition of chaos.

Chaos was first a science of computer experiments, and computers make rounding errors, the very thing chaos amplifies. Proofs lagged behind the pictures. Whether Lorenz's butterfly really exists as a strange attractor was the fourteenth of Smale's problems for the twenty-first century. In 2002 Warwick Tucker proved that it does, using a computer that tracked and bounded every rounding error. Many other pictures, including the chaotic sea of the standard map, are still unproved. The statistical laws that chaotic motion obeys belong to ergodic theory, and chaos in the complex plane produced the fractals of complex dynamics.