Chaos: Making a New Science

by James Gleick

Every scientist who turned to chaos early had a story to tell of discouragement or open hostility. Graduate students were warned that their careers could be jeopardized if they wrote theses in an untested discipline, in which their advisors had no expertise. A particle physicist, hearing about this new mathematics, might begin playing with it on his own, thinking it was a beautiful thing, both beautiful and hard—but would feel that he could never tell his colleagues about it.

Those who recognized chaos in the early days agonized over how to shape their thoughts and findings into publishable form. Work fell between disciplines—for example, too abstract for physicists yet too experimental for mathematicians.

To chaos researchers, mathematics has become an experimental science, with the computer replacing laboratories full of test tubes and microscopes. Graphic images are the key.

As Kuhn notes, established sciences take for granted a body of knowledge that serves as a communal starting point for investigation. To avoid boring their colleagues, scientists routinely begin and end their papers with esoterica. By contrast, articles on chaos from the late 1970s onward sounded evangelical, from their preambles to their perorations.

Every clock and every wristwatch (until the era of vibrating quartz) relied on a pendulum of some size or shape. (For that matter, the oscillation of quartz is not so different.)

Basic electronic circuits are described by equations exactly the same as those describing a swinging bob. The electronic oscillations are millions of times faster, but the physics is the same.

Physical motion, for Aristotle, was not a quantity or a force but rather a kind of change, just as a person’s growth is a kind of change.

Galileo’s advantage over the ancient Greeks was not that he had better data. On the contrary, his idea of timing a pendulum precisely was to get some friends together to count the oscillations over a twenty-four–hour period—a labor-intensive experiment.

In fact, so powerful was his theory that he saw a regularity that did not exist. He contended that a pendulum of a given length not only keeps precise time but keeps the same time no matter how wide or narrow the angle of its swing. A wide-swinging pendulum has farther to travel, but it happens to travel just that much faster. In other words, its period remains independent of its amplitude.

The changing angle of the bob’s motion creates a slight nonlinearity in the equations. At low amplitudes, the error is almost nonexistent. But it is there, and it is measurable even in an experiment as crude as the one Galileo describes.

If a chemist finds two substances in a constant proportion of 2.001 one day, and 2.003 the next day, and 1.998 the day after, he would be a fool not to look for a theory that would explain a perfect two-to–one ratio.

Air resistance is a notorious experimental nuisance, a complication that had to be stripped away to reach the essence of the new science of mechanics.

To separate the effects of gravity on a given mass from the effects of air resistance was a brilliant intellectual achievement. It allowed Galileo to close in on the essence of inertia and momentum. Still, in the real world, pendulums eventually do exactly what Aristotle’s quaint paradigm predicted. They stop.

A physicist could not truly understand turbulence or complexity unless he understood pendulums—and understood them in a way that was impossible in the first half of the twentieth century. As chaos began to unite the study of different systems, pendulum dynamics broadened to cover high technologies from lasers to superconducting Josephson junctions.

The surprising, erratic behavior comes from a nonlinear twist in the flow of energy in and out of this simple oscillator. The swing is damped and it is driven: damped because friction is trying to bring it to a halt, driven because it is getting a periodic push. Even when a damped, driven system is at equilibrium, it is not at equilibrium, and the world is full of such systems, beginning with the weather, damped by the friction of moving air and water and by the dissipation of heat to outer space, and driven by the constant push of the sun’s energy.

Traditionally, a dynamicist would believe that to write down a system’s equations is to understand the system. How better to capture the essential features? For a playground swing or a toy, the equations tie together the pendulum’s angle, its velocity, its friction, and the force driving it. But because of the little bits of nonlinearity in these equations, a dynamicist would find himself helpless to answer the easiest practical questions about the future of the system.

In some sense, they realized, physics understood perfectly the fundamental mechanisms of pendulum motion but could not extend that understanding to the long term.

SMALE MADE A BAD CONJECTURE. In the most rigorous mathematical terms, he proposed that practically all dynamical systems tended to settle, most of the time, into behavior that was not too strange. As he soon learned, things were not so simple.

Topology studies the properties that remain unchanged when shapes are deformed by twisting or stretching or squeezing. Whether a shape is square or round, large or small, is irrelevant in topology, because stretching can change those properties.

Both subjects, topology and dynamical systems, went back to Henri Poincaré, who saw them as two sides of one coin. Poincaré, at the turn of the century, had been the last great mathematician to bring a geometric imagination to bear on the laws of motion in the physical world. He was the first to understand the possibility of chaos; his writings hinted at a sort of unpredictability almost as severe as the sort Lorenz discovered.

Differential equations describe the way systems change continuously over time. The tradition was to look at such things locally, meaning that engineers or physicists would consider one set of possibilities at a time. Like Poincaré, Smale wanted to understand them globally, meaning that he wanted to understand the entire realm of possibilities at once.

As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system’s parameters, making a fluid more viscous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system.

The equations governing a pencil standing on its point have a good mathematical solution with the center of gravity directly above the point—but you cannot stand a pencil on its point because the solution is unstable.

Physicists assumed that any behavior they could actually observe regularly would have to be stable, since in real systems tiny disturbances and uncertainties are unavoidable. You never know the parameters exactly. If you want a model that will be both physically realistic and robust in the face of small perturbations, physicists reasoned that you must surely want a stable model.

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.

A modern physics student would explore the behavior of such an oscillator by looking at the line traced on the screen of an oscilloscope. Van der Pol did not have an oscilloscope, so he had to monitor his circuit by listening to changing tones in a telephone handset. He was pleased to discover regularities in the behavior as he changed the current that fed it. The tone would leap from frequency to frequency as if climbing a staircase, leaving one frequency and then locking solidly onto the next. Yet once in a while van der Pol noted something strange. The behavior sounded irregular, in a way that he could not explain.

For a simple system like a pendulum, the phase space might just be a rectangle: the pendulum’s angle at a given instant would determine the east-west position of a point and the pendulum’s speed would determine the north-south position. For a pendulum swinging regularly back and forth, the trajectory through phase space would be a loop, around and around as the system lived through the same sequence of positions over and over again.

His tools were topological transformations of shapes in phase space—transformations like stretching and squeezing. Sometimes these transformations had clear physical meaning. Dissipation in a system, the loss of energy to friction, meant that the system’s shape in phase space would contract like a balloon losing air—finally shrinking to a point at the moment the system comes to a complete halt.

To make a simple version of Smale’s horseshoe, you take a rectangle and squeeze it top and bottom into a horizontal bar. Take one end of the bar and fold it and stretch it around the other, making a C-shape, like a horseshoe. Then imagine the horseshoe embedded in a new rectangle and repeat the same transformation, shrinking and folding and stretching.

Smale put his horseshoe through an assortment of topological paces, and, the mathematics aside, the horseshoe provided a neat visual analogue of the sensitive dependence on initial conditions that Lorenz would discover in the atmosphere a few years later.

Originally, Smale had hoped to explain all dynamical systems in terms of stretching and squeezing—with no folding, at least no folding that would drastically undermine a system’s stability. But folding turned out to be necessary, and folding allowed sharp changes in dynamical behavior.

For three centuries it had been a case of the more you know, the less you know. Astronomers noticed a blemish on the great planet not long after Galileo first pointed his telescopes at Jupiter. Robert Hooke saw it in the 1600s. Donati Creti painted it in the Vatican’s picture gallery. As a piece of coloration, the spot called for little explaining. But telescopes got better, and knowledge bred ignorance. The last century produced a steady march of theories, one on the heels of another. For example:

In spectacular detail, astronomers saw the spot itself as a hurricane-like system of swirling flow, shoving aside the clouds, embedded in zones of east-west wind that made horizontal stripes around the planet. Hurricane was the best description anyone could think of, but for several reasons it was inadequate.

Hurricanes rotate in a cyclonic direction, counterclockwise above the Equator and clockwise below, like all earthly storms; the Red Spot’s rotation is anticyclonic.

Also, as astronomers studied the Voyager pictures, they realized that the planet was virtually all fluid in motion. They had been conditioned to look for a solid planet surrounded by a paper-thin atmosphere like earth’s, but if Jupiter had a solid core anywhere, it was far from the surface. The planet suddenly looked like one big fluid dynamics experiment, and there sat the Red Spot, turning steadily around and around, thoroughly unperturbed by the chaos around it.

Voyager had made the mystery doubly maddening by showing small-scale features of the flow, too small to be seen by the most powerful earthbound telescopes. The small scales displayed rapid disorganization, eddies appearing and disappearing within a day or less. Yet the spot was immune.

Freed from the ersatz hurricane theory, they found more appropriate analogues elsewhere. The Gulf Stream, for example, winding through the western Atlantic Ocean, twists and branches in subtly reminiscent ways. It develops little waves, which turn into kinks, which turn into rings and spin off from the main current—forming slow, long-lasting, anticyclonic vortices.

Since Newton’s laws apply everywhere, Marcus programmed a computer with a system of fluid equations. To capture Jovian weather meant writing rules for a mass of dense hydrogen and helium, resembling an unlit star. The planet spins fast, each day flashing by in ten earth hours. The spin produces a strong Coriolis force, the sidelong force that shoves against a person walking across a merry-go–round, and the Coriolis force drives the spot.

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.

But unlike most physicists, Marcus eventually learned Lorenz’s lesson, that a deterministic system can produce much more than just periodic behavior. He knew to look for wild disorder, and he knew that islands of structure could appear within the disorder.

In the emergence of chaos as a new science in the 1970s, ecologists were destined to play a special role. They used mathematical models, but they always knew that the models were thin approximations of the seething real world.

If regular equations could produce irregular behavior—to an ecologist, that rang certain bells.

Biologists’ mathematical models tended to be caricatures of reality, as did the models of economists, demographers, psychologists, and urban planners, when those soft sciences tried to bring rigor to their study of systems changing over time. The standards were different.

A physicist, looking at a particular system (say, two pendulums coupled by a spring), begins by choosing the appropriate equations. Preferably, he looks them up in a handbook; failing that, he finds the right equations from first principles. He knows how pendulums work, and he knows about springs. Then he solves the equations, if he can. A biologist, by contrast, could never simply deduce the proper equations by just thinking about a particular animal population. He would have to gather data and try to find equations that produced similar output.

Population biology learned quite a bit about the history of life, how predators interact with their prey, how a change in a country’s population density affects the spread of disease. If a certain mathematical model surged ahead, or reached equilibrium, or died out, ecologists could guess something about the circumstances in which a real population or epidemic would do the same.

Changes year to year are often more important than changes on a continuum. Unlike people, many insects, for example, stick to a single breeding season, so their generations do not overlap.

A year-by–year facsimile produces no more than a shadow of a system’s intricacies, but in many real applications the shadow gives all the information a scientist needs.

Or feedback can produce stability, as a thermostat does in regulating the temperature of a house: any temperature above a fixed point leads to cooling, and any temperature below it leads to heating.

A naïve approach to population biology might suggest a function that increases the population by a certain percentage each year. That would be a linear function—xnext = rx—and it would be the classic Malthusian scheme for population growth, unlimited by food supply or moral restraint.

An ecologist imagining real fish in a real pond had to find a function that matched the crude realities of life—for example, the reality of hunger, or competition.