Chaos: Making a New Science

by James Gleick

When Harry Swinney studied the liquid-vapor critical point in carbon dioxide, he did so with Landau’s conviction that his findings would carry over to the liquid-vapor critical point in xenon—and indeed they did.

Swinney and Gollub prepared to combat the messiness of moving fluids with an arsenal of neat experimental techniques built up over years of studying phase transitions in the most delicate of circumstances. They had laboratory styles and measuring equipment that a fluid dynamicist would never have imagined. To probe the rolling currents, they used laser light. A beam shining through the water would produce a deflection, or scattering, that could be measured in a technique called laser doppler interferometry.

When they began reporting results, Swinney and Gollub confronted a sociological boundary in science, between the domain of physics and the domain of fluid dynamics. The boundary had certain vivid characteristics. In particular, it determined which bureaucracy within the National Science Foundation controlled their financing.

But the experiment had never stopped. “There was the transition, very well defined,” Swinney said. “So that was great. Then we went on, to look for the next one.” There the expected Landau sequence broke down. Experiment failed to confirm theory. At the next transition the flow jumped all the way to a confused state with no distinguishable cycles at all.

Ruelle had heard talks by Steve Smale about the horseshoe map and the chaotic possibilities of dynamical systems. He had also thought about fluid turbulence and the classic Landau picture. He suspected that these ideas were related—and contradictory.

It was easy to see why turbulence resisted analysis. The equations of fluid flow are nonlinear partial differential equations, unsolvable except in special cases. Yet Ruelle worked out an abstract alternative to Landau’s picture, couched in the language of Smale, with images of space as a pliable material to be squeezed, stretched, and folded into shapes like horseshoes.

Instead of a piling up of frequencies, leading to an infinitude of independent overlapping motions, they proposed that just three independent motions would produce the full complexity of turbulence.

Most seductive of all was an image that the authors called a strange attractor.

The strange attractor lives in phase space, one of the most powerful inventions of modern science. Phase space gives a way of turning numbers into pictures, abstracting every bit of essential information from a system of moving parts, mechanical or fluid, and making a flexible road map to all its possibilities.

In phase space the complete state of knowledge about a dynamical system at a single instant in time collapses to a point. That point is the dynamical system—at that instant. At the next instant, though, the system will have changed, ever so slightly, and so the point moves. The history of the system time can be charted by the moving point, tracing its orbit through phase space with the passage of time.

If the system is a swinging, frictionless pendulum, one variable is position and the other velocity, and they change continuously, making a line of points that traces a loop, repeating itself forever, around and around.

Every orbit must eventually end up at the same place, the center: position 0, velocity 0. This central fixed point “attracts” the orbits. Instead of looping around forever, they spiral inward. The friction dissipates the system’s energy, and in phase space the dissipation shows itself as a pull toward the center, from the outer regions of high energy to the inner regions of low energy.

Phase-space portraits of physical systems exposed patterns of motion that were invisible otherwise, as an infrared landscape photograph can reveal patterns and details that exist just beyond the reach of perception.

Even in two dimensions, phase-space portraits had many surprises in store, and even desktop computers could easily demonstrate some of them, turning equations into colorful moving trajectories. Some physicists began making movies and videotapes to show their colleagues, and some mathematicians in California published books with a series of green, blue, and red cartoon-style drawings—“chaos comics,” some of their colleagues said, with just a touch of malice.

Every piece of a dynamical system that can move independently is another variable, another degree of freedom. Every degree of freedom requires another dimension in phase space, to make sure that a single point contains enough information to determine the state of the system uniquely.

Spaces of four, five, or more dimensions tax the visual imagination of even the most agile topologist. But complex systems have many independent variables. Mathematicians had to accept the fact that systems with infinitely many degrees of freedom—untrammeled nature expressing itself in a turbulent waterfall or an unpredictable brain—required a phase space of infinite dimensions.

An attractor can be a single point. For a pendulum steadily losing energy to friction, all trajectories spiral inward toward a point that represents a steady state—in this case, the steady state of no motion at all.

Like so many of those who began studying chaos, David Ruelle suspected that the visible patterns in turbulent flow—self-entangled stream lines, spiral vortices, whorls that rise before the eye and vanish again—must reflect patterns explained by laws not yet discovered.

Certainly the attractor would not be a fixed point, because the flow would never come to rest. Energy was pouring into the system as well as draining out. What other kind of attractor could it be? According to dogma, only one other kind existed, a periodic attractor, or limit cycle—an orbit that attracted all other nearby orbits.

In the short term any point in phase space can stand for a possible behavior of the dynamical system. In the long term the only possible behaviors are the attractors themselves.

By definition, attractors had the important property of stability—in a real system, where moving parts are subject to bumps and jiggles from real-world noise, motion tends to return to the attractor.

Turbulence in a fluid was a behavior of a different order, never producing any single rhythm to the exclusion of others. A well-known characteristic of turbulence was that the whole broad spectrum of possible cycles was present at once.

Ruelle and Takens wondered whether some other kind of attractor could have the right set of properties. Stable—representing the final state of a dynamical system in a noisy world. Low-dimensional—an orbit in a phase space that might be a rectangle or a box, with just a few degrees of freedom. Nonperiodic—never repeating itself, and never falling into a steady grandfather-clock rhythm.

What kind of orbit could be drawn in a limited space so that it would never repeat itself and never cross itself—because once a system returns to a state it has been in before, it thereafter must follow the same path.

Edward Lorenz had attached it to his 1963 paper on deterministic chaos, a picture with just two curves on the right, one inside the other, and five on the left. To plot just these seven loops required 500 successive calculations on the computer. A point moving along this trajectory in phase space, around the loops, illustrated the slow, chaotic rotation of a fluid as modeled by Lorenz’s three equations for convection.

Those loops and spirals were infinitely deep, never quite joining, never intersecting. Yet they stayed inside a finite space, confined by a box. How could that be? How could infinitely many paths lie in a finite space?

Where the spirals appear to join, the surfaces must divide, he realized, forming separate layers in the manner of a flaky mille-feuille.

In Japan the study of electrical circuits that imitated the behavior of mechanical springs—but much faster—led Yoshisuke Ueda to discover an extraordinarily beautiful set of strange attractors.

Indeed, the folding and squeezing of space was a key to constructing strange attractors, and perhaps a key to the dynamics of the real systems that gave rise to them.

To convert these three-dimensional skeins into flat pictures, scientists first used the technique of projection, in which a drawing represented the shadow that an attractor would cast on a surface.

A more revelatory technique was to make a return map, or a Poincaré map, in effect, taking a slice from the tangled heart of the attractor, removing a two-dimensional section just as a pathologist prepares a section of tissue for a microscope slide.

When to sample—where to take the slice from a strange attractor—is a question that gives an investigator some flexibility. The most informative interval might correspond to some physical feature of the dynamical system: for example, a Poincaré map could sample the velocity of a pendulum bob each time it passed through its lowest point.

The strange attractor above—first one orbit, then ten, then one hundred—depicts the chaotic behavior of a rotor, a pendulum swinging through a full circle, driven by an energetic kick at regular intervals. By the time 1,000 orbits have been drawn (below), the attractor has become an impenetrably tangled skein.

THE MOST ILLUMINATING STRANGE ATTRACTOR, because it was the simplest, came from a man far removed from the mysteries of turbulence and fluid dynamics. He was an astronomer, Michel Hénon of the Nice Observatory on the southern coast of France.

Systems that lose energy to friction are dissipative. Astronomical systems are not: they are conservative, or Hamiltonian.

Many astronomers have long and happy careers without giving dynamical systems a thought, but Hénon was different. He was born in Paris in 1931, a few years younger than Lorenz but, like him, a scientist with a certain unfulfilled attraction to mathematics.

Each body—the earth and the moon, for example—travels in a perfect ellipse around the system’s joint center of gravity. Add just one more gravitational object, however, and everything changes. The three-body problem is hard, and worse than hard. As Poincaré discovered, it is most often impossible.

But the equations cannot be solved analytically, which means that long-term questions about a three-body system cannot be answered. Is the solar system stable? It certainly appears to be, in the short term, but even today no one knows for sure that some planetary orbits could not become more and more eccentric until the planets fly off from the system forever.

And astronomers realized that globular clusters generally must not be stable. Binary star systems tend to form inside them, stars pairing off in tight little orbits, and when a third star encounters a binary, one of the three tends to get a sharp kick. Every so often, a star will gain enough energy from such an interaction to reach escape velocity and depart the cluster forever; the rest of the cluster will then contract slightly.

The core of a cluster would collapse, gaining kinetic energy and seeking a state of infinite density. This was hard to imagine, and furthermore it was not supported by the evidence of clusters so far observed. But slowly Hénon’s theory—later given the name “gravothermal collapse”—took hold.

By abstracting only the essence of his system, he made discoveries that applied to other systems as well, and more important systems. Years later, galactic orbits were still a theoretical game, but the dynamics of such systems were under intense, expensive investigation by those interested in the orbits of particles in high-energy accelerators and those interested in the confinement of magnetic plasmas for the creation of nuclear fusion.

Three-dimensional orbits are as hard to visualize when the orbits are real as when they are imaginary constructions in phase space. So Hénon used a technique comparable to the making of Poincaré maps. He imagined a flat sheet placed upright on one side of the galaxy so that every orbit would sweep through it, as horses on a race track sweep across the finish line. Then he would mark the point where the orbit crossed this plane and trace the movement of the point from orbit to orbit.

Such orbits are not completely regular, since they never exactly repeat themselves, but they are certainly predictable, and they are far from chaotic. Points never arrive inside the curve or outside it. Translated back to the full three-dimensional picture, the orbits were outlining a torus, or doughnut shape, and Hénon’s mapping was a cross-section of the torus.

First the egg-shaped curve twisted into something more complicated, crossing itself in figure eights and splitting apart into separate loops. Still, every orbit fell on some loop. Then, at even higher levels, another change occurred, quite abruptly. “Here comes the surprise,” Hénon and Heiles wrote. Some orbits became so unstable that the points would scatter randomly across the paper.

With greater magnification, they suggested, more islands would appear on smaller and smaller scales, perhaps all the way to infinity. Mathematical proof was needed—“but the mathematical approach to the problem does not seem too easy.”

Once again, he decided to throw out all reference to the physical origins of the system and concentrate only on the geometrical essence he wanted to explore.

The key, he believed, was the repeated stretching and folding of phase space in the manner of a pastry chef who rolls the dough, folds it, rolls it out again, folds it, creating a structure that will eventually be a sheaf of thin layers. Hénon drew a flat oval on a piece of paper. To stretch it, he picked a short numerical function that would move any point in the oval to a new point in a shape that was stretched upward in the center, an arch.

At first the points appear to jump randomly around the screen. The effect is that of a Poincaré section of a three-dimensional attractor, weaving erratically back and forth across the display. But quickly a shape begins to emerge, an outline curved like a banana. The longer the program runs, the more detail appears. Parts of the outline seem to have some thickness, but then the thickness resolves itself into two distinct lines, then the two into four, one pair close together and one pair farther apart. On greater magnification, each of the four lines turns out to be composed of two more lines—and so on, ad infinitum.

But the eerie effect of the strange attractor can be appreciated another way when the shape emerges in time, point by point. It appears like a ghost out of the mist. New points scatter so randomly across the screen that it seems incredible that any structure is there, let alone a structure so intricate and fine.

The points wander so randomly, the pattern appears so ethereally, that it is hard to remember that the shape is an attractor. It is not just any trajectory of a dynamical system. It is the trajectory toward which all other trajectories converge.