Chaos at fifty
In 1963 an MIT meteorologist revealed deterministic predictability to be an illusion and gave birth to a field that still thrives.
In classical physics, one is taught that given the initial state of a system, all of its future states can be calculated. In the celebrated words of Pierre Simon Laplace, “An intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it—an intelligence sufficiently vast to submit these data to analysis . . . for it, nothing would be uncertain and the future, as the past, would be present to its eyes.”1 Or, put another way, the clockwork universe holds true.
Herein lies the rub: Exact knowledge of a real-world initial state is never possible—the adviser can always demand a few more digits of experimental precision from the student, but the result will never be exact. Still, until the 19th century, the tacit assumption had always been that approximate knowledge of the initial state implies approximate knowledge of the final state. Given their success describing the motion of the planets, comets, and stars and the dynamics of countless other systems, physicists had little reason to assume otherwise.Starting in the 19th century, however, and culminating with a 1963 paper by MIT meteorologist Edward Lorenz, pictured in figure 1a, a series of developments revealed that the notion of deterministic predictability, although appealingly intuitive, is in practice false for most systems. Small uncertainties in an initial state can indeed become large errors in a final one. Even simple systems for which all forces are known can behave unpredictably. Determinism, surprisingly enough, does not preclude chaos.A gallery of monsters
Chaos theory, as we know it today,2 took shape mostly during the last quarter of the 20th century. But researchers had experienced close encounters with the phenomenon as early as the late 1880s, beginning with Henri Poincaré’s studies of the three-body problem in celestial mechanics. Poincaré observed that in such systems “it may happen that small differences in the initial conditions produce very great ones in the final phenomena. . . . Prediction becomes impossible.”3Dynamical systems like the three-body system studied by Poincaré are best described in phase space, in which dimensions correspond to the dynamical variables, such as position and momentum, that allow the system to be described by a set of first-order ordinary differential equations. The prevailing view had long been that, left alone, a conventional classical system will eventually settle toward either a steady state, described by a point in phase space; a periodic state, described by a closed loop; or a quasi-periodic state, which exhibits n > 1 incommensurable periodic modes and is described by an n-dimensional torus in phase space....MORE
HT that it was the anniversary, MoneyBeat: