Complexity_ A Guided Tour - Melanie Mitchell [12]
Possibly the first clear example of a chaotic system was given in the late nineteenth century by the French mathematician Henri Poincaré. Poincaré was the founder of and probably the most influential contributor to the modern field of dynamical systems theory, which is a major outgrowth of Newton’s science of dynamics. Poincaré discovered sensitive dependence on initial conditions when attempting to solve a much simpler problem than predicting the motion of a hurricane. He more modestly tried to tackle the so-called three-body problem: to determine, using Newton’s laws, the long-term motions of three masses exerting gravitational forces on one another. Newton solved the two-body problem, but the three-body problem turned out to be much harder. Poincaré tackled it in 1887 as part of a mathematics contest held in honor of the king of Sweden. The contest offered a prize of 2,500 Swedish crowns for a solution to the “many body” problem: predicting the future positions of arbitrarily many masses attracting one another under Newton’s laws. This problem was inspired by the question of whether or not the solar system is stable: will the planets remain in their current orbits, or will they wander from them? Poincaré started off by seeing whether he could solve it for merely three bodies.
He did not completely succeed—the problem was too hard. But his attempt was so impressive that he was awarded the prize anyway. Like Newton with calculus, Poincaré had to invent a new branch of mathematics, algebraic topology, to even tackle the problem. Topology is an extended form of geometry, and it was in looking at the geometric consequences of the three-body problem that he discovered the possibility of sensitive dependence on initial conditions. He summed up his discovery as follows:
If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon has been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomenon. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible….
In other words, even if we know the laws of motion perfectly, two different sets of initial conditions (here, initial positions, masses, and velocities for objects), even if they differ in a minuscule way, can sometimes produce greatly different results in the subsequent motion of the system. Poincaré found an example of this in the three-body problem.
Henri Poincaré, 1854–1912 (AIP Emilio Segre Visual Archives)
It was not until the invention of the electronic computer that the scientific world began to see this phenomenon as significant. Poincaré, way ahead of his time, had guessed that sensitive dependence on initial conditions would stymie attempts at long-term weather prediction. His early hunch gained some evidence when, in 1963, the meteorologist Edward Lorenz found that even simple computer models of weather phenomena were subject to sensitive dependence on initial conditions. Even with today’s modern, highly complex meteorological computer models, weather predictions are at best reasonably accurate only to about one week in the