Chaos: Making a New Science

by James Gleick

The most passionate advocates of the new science go so far as to say that twentieth-century science will be remembered for just three things: relativity, quantum mechanics, and chaos. Chaos, they contend, has become the century’s third great revolution in the physical sciences. Like the first two revolutions, chaos cuts away at the tenets of Newton’s physics. As one physicist put it: “Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and chaos eliminates the Laplacian fantasy of deterministic predictability.”

The Butterfly Effect acquired a technical name: sensitive dependence on initial conditions.

In science as in life, it is well known that a chain of events can have a point of crisis that could magnify small changes. But chaos meant that such points were everywhere. They were pervasive. In systems like the weather, sensitive dependence on initial conditions was an inescapable consequence of the way small scales intertwined with large.

Nonlinearity means that the act of playing the game has a way of changing the rules.

Shallow ideas can be assimilated; ideas that require people to reorganize their picture of the world provoke hostility.

Differential equations describe the way systems change continuously over time.

Chaos and instability, concepts only beginning to acquire formal definitions, were not the same at all. A chaotic system could be stable if its particular brand of irregularity persisted in the face of small disturbances.

The chaos Lorenz discovered, with all its unpredictability, was as stable as a marble in a bowl. You could add noise to this system, jiggle it, stir it up, interfere with its motion, and then when everything settled down, the transients dying away like echoes in a canyon, the system would return to the same peculiar pattern of irregularity as before. It was locally unpredictable, globally stable.

“When I started my professional work in mathematics in 1960, which is not so long ago, modern mathematics in its entirety—in its entirety—was rejected by physicists, including the most avant-garde mathematical physicists. So differentiable dynamics, global analysis, manifolds of mappings, differential geometry—everything just a year or two beyond what Einstein had used—was all rejected. The romance between mathematicians and physicists had ended in divorce in the 1930s. These people were no longer speaking. They simply despised each other. Mathematical physicists refused their graduate students permission to take math courses from mathematicians: Take mathematics from us. We will teach you what you need to know. The mathematicians are on some kind of terrible ego trip and they will destroy your mind. That was 1960. By 1968 this had completely turned around.” Eventually physicists, astronomers, and biologists all knew they had to have the news.

The spot is a self-organizing system, created and regulated by the same nonlinear twists that create the unpredictable turmoil around it. It is stable chaos.

This highly abstract description had practical weight for scientists trying to decide between different strategies of controlling error. In particular, it meant that, instead of trying to increase signal strength to drown out more and more noise, engineers should settle for a modest signal, accept the inevitability of errors and use a strategy of redundancy to catch and correct them. Mandelbrot also changed the way IBM’s engineers thought about the cause of noise. Bursts of errors had always sent the engineers looking for a man sticking a screwdriver somewhere. But Mandelbrot’s scaling patterns suggested that the noise would never be explained on the basis of specific local events.

Fractional dimension becomes a way of measuring qualities that otherwise have no clear definition: the degree of roughness or brokenness or irregularity in an object.

Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern.

In the end, the word fractal came to stand for a way of describing, calculating, and thinking about shapes that are irregular and fragmented, jagged and broken-up—shapes from the crystalline curves of snowflakes to the discontinuous dusts of galaxies. A fractal curve implies an organizing structure that lies hidden among the hideous complication of such shapes. High school students could understand fractals and play with them; they were as primary as the elements of Euclid. Simple computer programs to draw fractal pictures made the rounds of personal computer hobbyists.

This branch of dynamics concerned itself not with describing the final, stable behavior of a system but with the way a system chooses between competing options. A system like Lorenz’s now-classic model has just one attractor in it, one behavior that prevails when the system settles down, and it is a chaotic attractor. Other systems may end up with nonchaotic steady-state behavior—but with more than one possible steady state. The study of fractal basin boundaries was the study of systems that could reach one of several nonchaotic final states, raising the question of how to predict which. James Yorke, who pioneered the investigation of fractal basin boundaries a decade after giving chaos its name, proposed an imaginary pinball machine. Like most pinball machines it has a plunger with a spring. You pull back the plunger and release it to send the ball up into the playing area. The machine has the customary tilted landscape of rubber edges and electric bouncers that give the ball a kick of extra energy. The kick is important: it means that energy does not just decay smoothly. For simplicity’s sake this machine has no flippers at the bottom, just two exit ramps. The ball must leave by one ramp or the other. This is deterministic pinball—no shaking the machine. Only one parameter controls the ball’s destination, and that is the initial position of the plunger. Imagine that the machine is laid out so that a short pull of the plunger always means that the ball will end up rolling...

To researchers and engineers, there was a lesson in these pictures—a lesson and a warning. Too often, the potential range of behavior of complex systems had to be guessed from a small set of data. When a system worked normally, staying within a narrow range of parameters, engineers made their observations and hoped that they could extrapolate more or less linearly to less usual behavior. But scientists studying fractal basin boundaries showed that the border between calm and catastrophe could be far more complex than anyone had dreamed. “The whole electrical power grid of the East Coast is an oscillatory system, most of the time stable, and you’d like to know what happens when you perturb it,” Yorke said. “You need to know what the boundary is. The fact is, they have no idea what the boundary looks like.”

Barnsley’s central insight was this: Julia sets and other fractal shapes, though properly viewed as the outcome of a deterministic process, had a second, equally valid existence as the limit of a random process. By analogy, he suggested, one could imagine a map of Great Britain drawn in chalk on the floor of a room. A surveyor with standard tools would find it complicated to measure the area of these awkward shapes, with fractal coastlines, after all. But suppose you throw grains of rice into the air one by one, allowing them to fall randomly to the floor and counting the grains that land inside the map. As time goes on, the result begins to approach the area of the shapes—as the limit of a random process. In dynamical terms, Barnsley’s shapes proved to be attractors.