this problem, which was awarded to French mathematician and physicist Henri Poincaré (1854–1912), a brilliant scientist and graceful writer with an instinct for delving beneath the surface of scientific problems.

In fact, Poincaré did not solve the three-body problem, and some of his ideas were later found to be wrong. But in grappling with it, he conceived many major insights into dynamical systems. Among these was the crucial realization that the three-body problem was in an essential way unsolvable, that it was far more complex and tangled than anyone had imagined. He also recognized the fruitfulness of studying complex dynamics qualitatively, using geometry, not just formulas.

The significance of Poincaré's ideas was not fully appreciated at the time, and it is said that even Poincaré himself came to doubt their implications. But a number of mathematicians followed up the many leads he developed. In France immediately after World War I, Pierre Fatou (1878–1929) and Gaston Julia (1893–1978) discovered and explored what is now known as Julia sets, seen today as characteristic examples of chaotic behavior. In the United States George Birkhoff developed many fundamental techniques; in 1913 he gave the first correct proof of one of Poincaré's major conjectures.

Although dynamical systems was not in the mainstream of mathematical research in the United States until relatively recently, work continued in the Soviet Union. A major step was taken in the 1950s, by the Russian Andrei Kolmogorov, followed by his compatriot Vladimir Arnold and German mathematician Jürgen Moser, who proved what is known as the KAM (Kolmogorov-Arnold-Moser) theorem.

They found that systems of bodies orbiting in space are often astonishingly stable and ordered. In a book on the topic, Moser wrote that Poincaré and Birkhoff had already found that there are "solutions which do not experience collisions and do not escape," even given infinite time (Moser, 1973, p. 4). "But are these solutions exceptional?" he asked. The KAM theorem says that they are not.

In answering that question, the subtle and complicated KAM theorem has a great deal to say about when such systems are stable and when they are not. Roughly, when the periods, or years, of two bodies orbiting a sun can be expressed as a simple ratio of whole numbers, such as 2/5 or 3—or by a number close to such a number—then they will be potentially unstable: resonance can upset the balance of gravitational forces. When the periods cannot be expressed as such a ratio, the system will be stable.

This is far from the whole story, however. The periods of Saturn and Jupiter have the ratio 2/5; those of Uranus and Neptune have a