Chaos: Making a New Science

by James Gleick

Now that science is looking, chaos seems to be everywhere. A rising column of cigarette smoke breaks into wild swirls. A flag snaps back and forth in the wind. A dripping faucet goes from a steady pattern to a random one. Chaos appears in the behavior of the weather, the behavior of an airplane in flight, the behavior of cars clustering on an expressway, the behavior of oil flowing in underground pipes. No matter what the medium, the behavior obeys the same newly discovered laws. That realization has begun to change the way business executives make decisions about insurance, the way astronomers look at the solar system, the way political theorists talk about the stresses leading to armed conflict.

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.” Of the three, the revolution in chaos applies to the universe we see and touch, to objects at human scale.

chaos

Traditionally, when physicists saw complex results, they looked for complex causes. When they saw a random relationship between what goes into a system and what comes out, they assumed that they would have to build randomness into any realistic theory, by artificially adding noise or error. The modern study of chaos began with the creeping realization in the 1960s that quite simple mathematical equations could model systems every bit as violent as a waterfall. Tiny differences in input could quickly become overwhelming differences in output—a phenomenon given the name “sensitive dependence on initial conditions.”

Newton’s banner actually waved another flag that said something like this: Given an approximate knowledge of a system’s initial conditions and an understanding of natural law, one can calculate the approximate behavior of the system. This assumption lay at the philosophical heart of science. As one theoretician liked to tell his students: “The basic idea of Western science is that you don’t have to take into account the falling of a leaf on some planet in another galaxy when you’re trying to account for the motion of a billiard ball on a pool table on earth. Very small influences can be neglected. There’s a convergence in the way things work, and arbitrarily small influences don’t blow up to have arbitrarily large effects.” Classically, the belief in approximation and convergence was well justified. It worked.

The European Centre’s assessments suggested that the world saved billions of dollars each year from predictions that were statistically better than nothing. But beyond two or three days the world’s best forecasts were speculative, and beyond six or seven they were worthless.

Lorenz saw it differently. Yes, you could change the weather. You could make it do something different from what it would otherwise have done. But if you did, then you would never know what it would otherwise have done. It would be like giving an extra shuffle to an already well-shuffled pack of cards. You know it will change your luck, but you don’t know whether for better or worse.

If the weather ever did reach a state exactly like one it had reached before, every gust and cloud the same, then presumably it would repeat itself forever after and the problem of forecasting would become trivial.

Lorenz saw that there must be a link between the unwillingness of the weather to repeat itself and the inability of forecasters to predict it—a link between aperiodicity and unpredictability.

Friction, for example. Without friction a simple linear equation expresses the amount of energy you need to accelerate a hockey puck. With friction the relationship gets complicated, because the amount of energy changes depending on how fast the puck is already moving. Nonlinearity means that the act of playing the game has a way of changing the rules. You cannot assign a constant importance to friction, because its importance depends on speed. Speed, in turn, depends on friction.

the earth’s magnetic field. The “geodynamo” is known to have flipped many times during the earth’s history, at intervals that seem erratic and inexplicable. Faced with such irregularity, theorists typically look for explanations outside the system, proposing such causes as meteorite strikes. Yet perhaps the geodynamo contains its own chaos.

At the time, though, few could see it. Lorenz described it to Willem Malkus, a professor of applied mathematics at M.I.T., a gentlemanly scientist with a grand capacity for appreciating the work of colleagues. Malkus laughed and said, “Ed, we know—we know very well—that fluid convection doesn’t do that at all.” The complexity would surely be damped out, Malkus told him, and the system would settle down to steady, regular motion. “Of course, we completely missed the point,” Malkus said a generation later—years after he had built a real Lorenzian waterwheel in his basement laboratory to show nonbelievers. “Ed wasn’t thinking in terms of our physics at all. He was thinking in terms of some sort of generalized or abstracted model which exhibited behavior that he intuitively felt was characteristic of some aspects of the external world. He couldn’t quite say that to us, though. It’s only after the fact that we perceived that he must have held those views.”

Few laymen realized how tightly compartmentalized the scientific community had become, a battleship with bulkheads sealed against leaks. Biologists had enough to read without keeping up with the mathematics literature—for that matter, molecular biologists had enough to read without keeping up with population biology. Physicists had better ways to spend their time than sifting through the meteorology journals. Some mathematicians would have been excited to see Lorenz’s discovery; within a decade, physicists, astronomers, and biologists were seeking something just like it, and sometimes rediscovering it for themselves. But Lorenz was a meteorologist, and no one thought to look for chaos on page 130 of volume 20 of the Journal of the Atmospheric Sciences.

Professional scientists, given brief, uncertain glimpses of nature’s workings, are no less vulnerable to anguish and confusion when they come face to face with incongruity. And incongruity, when it changes the way a scientist sees, makes possible the most important advances. So Kuhn argues, and so the story of chaos suggests.

In Kuhn’s scheme, normal science consists largely of mopping up operations. Experimentalists carry out modified versions of experiments that have been carried out many times before. Theorists add a brick here, reshape a cornice there, in a wall of theory. It could hardly be otherwise. If all scientists had to begin from the beginning, questioning fundamental assumptions, they would be hard pressed to reach the level of technical sophistication necessary to do useful work. In Benjamin Franklin’s time, the handful of scientists trying to understand electricity could choose their own first principles—indeed, had to. One researcher might consider attraction to be the most important electrical effect, thinking of electricity as a sort of “effluvium” emanating from substances. Another might think of electricity as a fluid, conveyed by conducting material. These scientists could speak almost as easily to laymen as to each other, because they had not yet reached a stage where they could take for granted a common, specialized language for the phenomena they were studying.

Shallow ideas can be assimilated; ideas that require people to reorganize their picture of the world provoke hostility. A physicist at the Georgia Institute of Technology, Joseph Ford, started quoting Tolstoy: “I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives.”

By the middle of the eighties a process of academic diffusion had brought chaos specialists into influential positions within university bureaucracies. Centers and institutes were founded to specialize in “nonlinear dynamics” and “complex systems.”

Those studying chaotic dynamics discovered that the disorderly behavior of simple systems acted as a creative process. It generated complexity: richly organized patterns, sometimes stable and sometimes unstable, sometimes finite and sometimes infinite, but always with the fascination of living things. That was why scientists played with toys.

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.

The result of a mathematical development should be continuously checked against one’s own intuition about what constitutes reasonable biological behavior. When such a check reveals disagreement, then the following possibilities must be considered: A mistake has been made in the formal mathematical development; The starting assumptions are incorrect and/or constitute a too drastic oversimplification; One’s own intuition about the biological field is inadequately developed; A penetrating new principle has been discovered. —HARVEY J. GOLD, Mathematical Modeling of Biological Systems

Mathematicians proved theorems by ratiocination; physicists’ proofs used heavier equipment. The objects that made up their worlds were different. Their examples were different.

Yorke had offered more than a mathematical result. He had sent a message to physicists: Chaos is ubiquitous; it is stable; it is structured. He also gave reason to believe that complicated systems, traditionally modeled by hard continuous differential equations, could be understood in terms of easy discrete maps.

Biologists had overlooked bifurcations on the way to chaos because they lacked mathematical sophistication and because they lacked the motivation to explore disorderly behavior. Mathematicians had seen bifurcations but had moved on. May, a man with one foot in each world, understood that he was entering a domain that was astonishing and profound.

In epidemiology, for example, it was well known that epidemics tend to come in cycles, regular or irregular. Measles, polio, rubella—all rise and fall in frequency. May realized that the oscillations could be reproduced by a nonlinear model and he wondered what would happen if such a system received a sudden kick—a perturbation of the kind that might correspond to a program of inoculation. Naïve intuition suggests that the system will change smoothly in the desired direction. But actually, May found, huge oscillations are likely to begin. Even if the long-term trend was turned solidly downward, the path to a new equilibrium would be interrupted by surprising peaks. In fact, in data from real programs, such as a campaign to wipe out rubella in Britain, doctors had seen oscillations just like those predicted by May’s model. Yet any health official, seeing a sharp short-term rise in rubella or gonorrhea, would assume that the inoculation program had failed.

And yet relation appears, A small relation expanding like the shade Of a cloud on sand, a shape on the side of a hill. —WALLACE STEVENS “Connoisseur of Chaos”

Mathematicians’ subjects became self-contained; their method became formally axiomatic. A mathematician could take pride in saying that his work explained nothing in the world or in science. Much good came of this attitude, and mathematicians treasured it. Steve Smale, even while he was working to reunite mathematics and natural science, believed, as deeply as he believed anything, that mathematics should be something all by itself. With self-containment came clarity. And clarity, too, went hand in hand with the rigor of the axiomatic method.

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.

Discontinuity, bursts of noise, Cantor dusts—phenomena like these had no place in the geometries of the past two thousand years. The shapes of classical geometry are lines and planes, circles and spheres, triangles and cones. They represent a powerful abstraction of reality, and they inspired a powerful philosophy of Platonic harmony. Euclid made of them a geometry that lasted two millennia, the only geometry still that most people ever learn. Artists found an ideal beauty in them, Ptolemaic astronomers built a theory of the universe out of them. But for understanding complexity, they turn out to be the wrong kind of abstraction.

A simple, Euclidean, one-dimensional line fills no space at all. But the outline of the Koch curve, with infinite length crowding into finite area, does fill space. It is more than a line, yet less than a plane. It is greater than one-dimensional, yet less than a two-dimensional form. Using techniques originated by mathematicians early in the century and then all but forgotten, Mandelbrot could characterize the fractional dimension precisely. For the Koch curve, the infinitely extended multiplication by four-thirds gives a dimension of 1.2618.

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