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concepts as unpredictability. The Lyapunov exponents in a system provided a way of measuring the conflicting effects of stretching, contracting, and folding in the phase space of an attractor. They gave a picture of all the properties of a system that lead to stability or instability. An exponent greater than zero meant stretching—nearby points would separate. An exponent smaller than zero meant contraction. For a fixed-point attractor, all the Lyapunov exponents were negative, since the direction of pull was inward toward the final steady state. An attractor in the form of a periodic orbit had one exponent of exactly zero and other exponents that were negative. A strange attractor, it turned out, had to have at least one positive Lyapunov exponent.

To their chagrin, the Santa Cruz students did not invent this idea, but they developed it in the most practical ways possible, learning how to measure Lyapunov exponents and relate them to other important properties. They used computer animation to make movies illustrating the beating together of order and chaos in dynamical systems. Their analysis showed vividly how some systems could create disorder in one direction while remaining trim and methodical in another. One movie showed what happened to a tiny cluster of nearby points—representing initial conditions—on a strange attractor as the system evolved in time. The cluster began to spread out and lose focus. It turned into a dot and then a blob. For certain kinds of attractors, the blob would quickly spread all over. Such attractors were efficient at mixing. For other attractors, though, the spreading would only occur in certain directions. The blob would become a band, chaotic along one axis and orderly along another. It was as if the system had an orderly impulse and a disorderly one together, and they were decoupling. As one impulse led to random unpredictability, the other kept time like a precise clock. Both impulses could be defined and measured.

CHAOTIC MIXING. One blob mixes rapidly; another blob, just a bit closer to the center, barely mixes at all. In experiments by Julio M. Ottino and others with real fluids, the process of mixing—ubiquitous in nature and industry, yet still poorly understood—proved intimately bound up with the mathematics of chaos. The patterns revealed a stretching and folding that led back to the horseshoe map of Smale.

THE MOST CHARACTERISTICALLY Santa Cruzian imprint on chaos research involved a piece of mathematics cum philosophy known as information theory, invented in the late 1940s by a researcher at the Bell Telephone Laboratories, Claude Shannon. Shannon called his work “The Mathematical Theory of Communication,” but it concerned a rather special quantity called information, and the name information theory stuck. The theory was a product of the electronic age. Communication lines and radio transmissions were carrying a certain thing, and computers would soon be storing this same thing on punch cards or magnetic cylinders, and the thing was neither knowledge nor meaning. Its basic units were not ideas or concepts or even, necessarily, words or numbers. This thing could be sense or nonsense—but the engineers and mathematicians could measure it, transmit it, and test the transmission for accuracy. Information proved as good a word as any, but people had to remember that they were using a specialized value-free term without the usual connotations of facts, learning, wisdom, understanding, enlightenment.

Hardware determined the shape of the theory. Because information was stored in binary on-off switches newly designated as bits, bits became the basic measure of information. From a technical point of view, information theory became a handle for grasping how noise in the form of random errors interfered with the flow of bits. It gave a way of predicting the necessary carrying capacity of communication lines or compact disks or any technology that encoded language, sounds, or images. It offered a theoretical means of reckoning the effectiveness of different schemes for correcting errors