Dynamical Systems

In 1885 King Oscar II of Sweden offered a prize for progress on the stability of the solar system: will the planets continue in their orbits forever, or could one be flung out? Poincaré won in 1889 with a memoir on the three-body problem. While it was being printed in Acta Mathematica, Edvard Phragmén, who was editing it, asked about an unclear passage. Poincaré found that it hid a serious mistake.

Correcting it, he found the opposite of what he had claimed. Near certain unstable orbits, the paths of nearby motions cross each other infinitely often in a mesh so complicated, he later wrote, that he would not even attempt to draw it. Gösta Mittag-Leffler recalled the printed copies, and Poincaré paid for the reprinting, which cost more than the prize. The corrected memoir of 1890 contained the first description of chaos, though the word would not be used for more than eighty years.

Meanwhile others built tools. Aleksandr Lyapunov showed in 1892 how to prove stability with an energy-like function that never increases. George David Birkhoff proved Poincaré's last, unfinished theorem in 1913 and wrote the first general book on dynamical systems. The central question, whether regular motions survive small disturbances, was answered between 1954 and 1963 by Andrey Kolmogorov, Vladimir Arnold and Jürgen Moser. Most of them do survive, so the solar system is largely regular, but chaotic regions are threaded densely between them.

A pendulum's angle $\theta$ obeys $\theta'' = -\frac{g}{L} \sin\theta$. Because of the $\sin\theta$, there is no solution in elementary functions. But its phase space, with the angle along one axis and the angular velocity $v = \theta'$ along the other, shows everything.

Energy is conserved. Taking $g/L = 1$ for simplicity,

stays constant along every motion, so each motion traces a curve of constant $E$ in the phase plane.

• $E = -1$: the pendulum hangs at rest at the bottom, a single point.
• $-1 < E < 1$: the curves are closed loops around that point. The pendulum swings back and forth forever.
• $E > 1$: the curves run right across the plane without closing. The pendulum has enough energy to go over the top and keeps rotating in one direction.
• $E = 1$: the dividing curve, the separatrix, heads towards the upside-down position and takes infinitely long to get there.

That is a complete description of every possible motion, obtained without solving anything. It also shows stability: the bottom position is stable, since nearby curves stay nearby, while the top position is unstable. Near the separatrix, the time for one swing grows without limit. That is why a large swing takes longer than the small-swing period $2\pi\sqrt{L/g}$, about 2.0 seconds for a pendulum one metre long.

Poincaré's method was to draw such pictures for systems where they are far more complex. Add a small periodic push to the pendulum, and the separatrix breaks into exactly the tangle that Poincaré found in the three-body problem. Near it, the pendulum's motion becomes chaotic.

In 1960, working on a beach in Rio de Janeiro, Stephen Smale found the geometric mechanism behind Poincaré's tangle: the horseshoe. A map that stretches a square, folds it and lays it back across itself contains orbits that behave like sequences of coin tosses, and the behaviour survives any small change to the map. His 1967 programme made stretching and folding the core of the subject. Computer experiments were about to show the same behaviour in weather models and simple population equations, and chaos theory and ergodic theory grew from both sides.