Indiana University  Research & Creative Activity Spring 2002 • Volume XXIV Number 3



by Hal Kibbey

Fractal images by Eric Bedford

The batter swings, and the crowd roars as the ball heads for deep center field. As soon as the ball leaves the bat, the center fielder turns and starts running at full speed. Player and ball converge on the same spot. At the last moment, the player reaches over his head, and the ball lands in his glove for a long out.

In a room where the baseball game is on television, a child picks up a new balloon not yet inflated. She blows it up, holds it out, releases it, then laughs delightedly as the balloon sputters and darts around the room. Again and again she fetches the balloon, blows it up, releases it. Each time, the balloon races around the room in a different crazy path.

The baseball and balloon are following Newton’s laws of motion, yet the baseball’s flight is predictable, while the balloon’s flight is erratic and unpredictable. How can the same laws produce such different results?

photo of Eric Bedford
Eric Bedford is professor of mathematics at Indiana University Bloomington. photo © 2001 Tyagan Miller

Both ball and balloon represent a dynamical system, a system that changes in time. The baseball arcing through the air is a stable system, while the balloon buzzing wildly around the room is an unstable one. At Indiana University Bloomington, mathematics professors Eric Bedford and Kevin Pilgrim study the behavior of dynamical systems over the long term.

Chaos rules

In popular parlance, Bedford and Pilgrim study chaos. “The proper name for chaos theory is the study of dynamical systems, a subject which has been around for well over 100 years,” says Pilgrim.

Bedford and Pilgrim both specialize in unstable dynamical systems, the ones that seem chaotic. Chaos is visible all around us, from the way water swirls down a drain to the way a plume of smoke rises in still air. But in technical terms, chaos means something quite the opposite of our everyday definition of randomness, confusion, or disorder. In the world of mathematics, a chaotic system is a well-defined system that can be represented by equations. A mathematically chaotic system is totally deterministic—meaning, in the short term, that its behavior is predictable.

When it is followed for any length of time, however, such a system’s behavior gets wildly more complicated. In a predictable system, things that start close together will stay close together as time passes, but in a chaotic system, two points that start close together eventually become far apart as time passes.

“In a chaotic system, as time goes forward, everything is moving away from everything else,” Bedford explains. “You can start as close as you want to something, and after a certain amount of time, you’re no longer close to it. Everything is independent of everything else.”

This ever-widening divide means that if you are trying to predict the future behavior of a chaotic system, errors in initial measurements become overwhelming as time progresses. In a chaotic system, if there is any error at all in your initial measurements of the system, your long-term predictions will be absurdly wrong.

Consider the weather. Although it may be hard to believe, weather is deterministic, not random. But because the atmosphere has so many variables, and because it is so difficult to measure them all simultaneously and accurately, reliable predictions cannot be made for more than a few days into the future. Small errors at the beginning become larger and larger as time goes on. We can talk only about probabilities, such as a 30 percent chance of rain on a particular day.

Simple beginnings

photo of Kevin Pilgrim
Kevin Pilgrim isassistant professor of mathematics at IU Bloomington. photo © 2001 Tyagan Miller

Bedford and Pilgrim concentrate on how to understand systems in which predictability and unpredictability coexist. Our solar system is a familiar example. Kepler’s laws of planetary motion give precise predictions about the motion of a single planet about the sun, if no other planets are considered. But when the solar system is examined with all of its planets, the motion becomes very complicated as the planets interact over long periods of time.

To tackle the general problem of understanding predictable and chaotic behavior combined, Bedford and Pilgrim rely on equations that contain both “real” and “imaginary” terms. Real numbers can be thought of as points on a line. Complex numbers, on the other hand, correspond to points somewhere in a two-dimensional plane. A complex number has two parts, which for historical reasons are called real and imaginary.

“You’d think it would be much simpler just to look at the real line,” Bedford says. “But in fact, when you’re looking at the dynamics of a system, the dynamics on the real line are not complete, and you could be missing a lot. You don’t know for sure what’s missing, because what’s missing is off in the complex plane. When you allow numbers to become complex, you get a more complete picture of what is happening.”

In the field of dynamical systems, the Mandelbrot set refers to a collection of points in the complex plane. With its disks, bulges, spirals, and filaments, the Mandelbrot set is immensely complicated, but the information required to produce the set can be written down in a short computer program. The Mandelbrot set’s combination of complexity and simplicity is symbolic of chaos theory, in which dynamical systems of great complexity can evolve from simple equations.

Phenomenal fractals

One of the remarkable discoveries of the study of dynamical systems is that the long-term behavior of a chaotic system traces fantastically interesting and beautiful shapes as the system moves and evolves in time. The geometric complexity of fractal images is, Pilgrim says, “a visual manifestation of how complicated a problem is.”


The study of chaotic systems and fractals has a long history, but the introduction of computer graphics has made it possible to see fractals and gain new understanding of them. By calculating each value of each point in a chaotic system as time progresses and then displaying the results on a screen, a computer quickly shows what happens as a fractal is generated. A computer can function as a kind of microscope for a mathematician, showing the intricate structure of sections of curves and their edges at whatever scale is desired. When one part of the fractal is magnified, the same complex shapes will reappear. No matter how much magnifying is done, the resulting image is never simpler because fractals have no “smallest parts.” Nested within one another, each smaller structure of a fractal is a miniature version of the larger form.

“Fractals are shapes in which every piece looks like every other piece,” Bedford says. “Put it under a microscope and blow it up as much as you want, and it looks exactly the same.”

Just as chaotic systems can be seen all around us, so fractals appear commonly in nature. Clouds are one of the most conspicuous examples. Ferns, seashells, mountain surfaces, coastlines, the distribution of craters on the moon—all have fractal properties. We even have fractals inside ourselves, in the dendrites at the ends of our nerve cells, the branching of our blood vessels, and the inner surfaces of our lungs. Fractals are found at all scales in nature, because they have nothing to do with scale. The only information needed to produce them is a repeating process of bifurcation and development.

The irony of chaos

Using mathematical models, Bedford and Pilgrim are peering deeply into chaos to gain some understanding of what seems random. Their strategy is to try to find very simplified models, then understand them in great detail.

“The simple form of dynamical systems allows us to obtain a close understanding of chaotic behavior,” Bedford says. “The irony of chaotic systems is that when you can reduce a deterministic system to something probabilistic (characterized by uncertainty), then you’ve learned something. You understand it.”

Studying simple dynamical systems gives researchers an idea of what kinds of questions to ask and what kinds of phenomena to expect when more complicated systems present themselves.

“Developing connections between the simplest dynamical systems, other areas of math, and other areas of science is useful,” Pilgrim says, “because the experience gained by attacking a problem in one realm sheds light on the others.”

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