ratio close to 2. (In fact, some researchers in the 19th century, suspecting that rationality was associated with instability, speculated that a slight change in the distance between Saturn and Jupiter would be enough to send Saturn shooting out of our solar system.)

But the KAM theorem, combined with Birkhoff's earlier work, showed that there are little windows of stability within the ''unstable zones" associated with rationality; such a window of stability could account for the planets' apparent stability. (The theorem also answers questions about the orbit of Saturn's moon Hyperion, the gaps in Saturn's rings, and the distribution of asteroids; it is used extensively to understand the stability of particles in accelerators.)

In the years following the KAM theorem the emphasis in dynamical systems was on stability, for example in the work in the 1960s by topologist Stephen Smale and his group at the University of California, Berkeley. But more and more, researchers in the field have come to focus on instabilities as the key to understanding dynamical systems, and hence the world around us.

A key role in this shift was played by the computer, which has shown mathematicians how intermingled order and disorder are: systems assumed to be stable may actually be unstable, and apparently chaotic systems may have their roots in simple rules. Perhaps the first to see evidence of this, thanks to a computer, was not a mathematician but a meteorologist, Edward Lorenz, in 1961 (Lorenz, 1963). The story as told by James Gleick is that Lorenz was modeling Earth's atmosphere, using differential equations to estimate the impact of changes in temperature, wind, air pressure, and the like. One day he took what he thought was a harmless shortcut: he repeated a particular sequence but started halfway through, typing in the midpoint output from the previous printout—but only to the three decimal places displayed on his printout, not the six decimals calculated by the program. He then went for a cup of coffee and returned to find a totally new and dramatically altered outcome. The small change in initial conditions—figures to three decimal places, not six—produced an entirely different answer.

Sensitivity to Initial Conditions

Sensitivity to initial conditions is what chaos is all about: a chaotic system is one that is sensitive to initial conditions. In his talk McMullen described a mathematical example, "the simplest dynamical system in the quadratic family that one can study," iteration of the polynomial x2 + c when c = 0.

To iterate a polynomial or other function, one starts with a num-



The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement