Coaxing Order From Chaos

In 1959, a young meteorologist named Edward Lorenz set out to build himself a world. Naturally, the project entailed a few compromises:

His world was flat, contained only the barest approximation of an atmosphere, and existed only in the memory banks of the Royal-McBee computer inside his office at the Massachusetts Institute of Technology. At the center of this world was a mathematical weather-prediction model, except, instead of the thousands of equations and variables typically used, Lorenz's contained only twelve of each. But Lorenz wasn't interested in improving weather predictions—he wanted to know if prediction was even possible.

Despite the simplicity of his simulation, Lorenz soon succeeded in reproducing some of the unpredictability of daily weather patterns, represented by a series of symbols on his computer printouts. But it took a fortuitous shortcut to reveal the most surprising behavior of his model. One day, wanting to repeat the results of a previous simulation, he input the numbers from an earlier run, started the simulation, and stepped out for coffee. When he returned an hour later, two months had passed in the simulation, but the new numbers coming out of the computer looked nothing like those from the previous run. From the same beginning, his simulation appeared in have produced two completely different patterns of weather.

Tracing pattern: Computer-generated image of the Lorenz attractor reveals the structure in Edward Lorenz's weather model. Computed in Fractint by Wikimol

Lorenz suspected an equipment problem, but eventually realized that the explanation was much simpler. Instead of typing in the entire numbers from the previous run, he had shortened them by three decimal places, substituting, for example, 123.875 for 123.875664. Over time, these tiny initial discrepancies had been magnified, resulting in major changes in the simulated weather patterns. In 1972, Lorenz examined the implications of this result in a now legendary talk entitled "Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?" The phenomenon, which he called "sensitive dependence on initial conditions," has been known as the butterfly effect ever since.

In the years following Lorenz's discovery, the butterfly effect, more than any other property, came to symbolize all that was bizarre and fascinating about what we now call chaotic systems. By demonstrating that some phenomena, no matter how precisely we attempt to measure them, are ultimately beyond predicting, it gave birth to the enduring impression that chaos is something recalcitrant, unwieldy, unpredictable to its core.

And yet, for almost twenty years, researchers like Duke physics professor Dan Gauthier have been engaged in a quest to perfect the use of chaos as a tool, to transform it from an apparent bug in nature's programming into a feature. By turning the logic of the butterfly effect, sensitive dependence, on its head—nudging chaotic systems with a series of tiny feedback adjustments—they have succeeded in coaxing ordered behavior from a staggering range of disparate systems—in Gauthier's case, from the fluctuations of laser beams to the electrical dynamics of the human heart. Chaos, he would tell you, can not only be tamed, it can be harnessed.

In the early 1980s, when Gauthier entered graduate school, chaos was still the province of only a few isolated researchers. His own passion was for lasers. While still in high school, he and a friend had begun experimenting with the devices, donations from a canceled research program at a local ballbearing company. "They're the same lasers that are in supermarkets [as scanners] nowadays," he explains, "but back then, they were very rare." Together, the two of them undertook projects they'd read about in Scientific American, making holograms in Gauthier's parents' garage and repeating experiments that had, 150 years earlier, helped prove that light traveled in waves.

It was an interest that only deepened during Gauthier's undergraduate days at the University of Rochester as he began studying optics, the physics of light, and its interaction with matter. "What I liked about laser physics," he recalls, "was that I could work alone or with one other person and really accomplish something. That's what's a bit different about the optics field—there are almost no theorists. You've got to design the experiment, you've got to build the equipment, you've got to do the experiment, analyze the data, and write the paper. You've got to do a little bit of everything."

Yet, at the same time Gauthier was engrossed in the intricacies of lasers, ideas from chaos theory were rapidly filtering into the general consciousness of physicists, mathematicians, and theoretical biologists. The common element across these disciplines was mathematics, the equations that in abstract form describe phenomena as diverse as the mixing of fluids and the synchronized blinking of fireflies. In many cases, with a simple exchange of variables, two entirely unrelated systems might be described by the same set of equations, a realization that allowed scientists experimenting in all types of systems, to discover general principles underlying chaos.

What they found was a world of difference between chaotic systems like Lorenz's weather model and merely random processes like the roll of dice. Lorenz's model only appeared unruly. None of the equations defining it contained any element of chance. Had Lorenz restarted his simulation with numbers accurate to the full six decimal places, he would have observed exactly the same weather as before. Yet, viewed under the microscope of mathematical analysis, data from the Lorenz model disclosed what scientists call a strange attractor, a pattern of organized behavior hidden amid the unpredictability.

In systems ranging from planetary motion to wildlife population data, the discovery of other strange attractors revealed that many types of unpredictable behavior arose from order, not randomness, and had their roots in a common mathematics. Even fractals, geometric patterns like those of fern fronds and coastlines that repeat themselves on both the smallest and largest scales, were found to be produced by simple mathematical rules, a discovery that spurred interest in computer-generated cousins of natural geometries such as the iconic Julia and Mandelbrot sets.

Universal order: Gauthier with circuit board used to generate chaos. Les Todd

These developments did not go unnoticed by the graduate students at Rochester. "There were quite a few of us in graduate school at the time who were very interested in chaos," Gauthier recalls. "A lot of the first papers were starting to come out at a level where a new person could come in and try to understand it. The whole idea of the Mandelbrot set—fractals—was pretty popular around 1984. And so a bunch of us were staying up really late at night, figuring out how to program one of the computers in the building that was hooked up to a color pen plotter. A little machine would come over, pick up the pen, and then draw on the paper, put the felt-tip pen back, and grab another. And so we were making these great color Mandelbrot sets. I was very primed for doing that."

Around this time, Gauthier's adviser, Robert Boyd, began to suggest that he might be able to observe chaos in the particular experimental laser setup he was studying.

Though some of the earliest experiments on chaos had been performed using lasers (like the Lorenz equations, the equations governing the electric field of the beam can exhibit chaos), experiments at the time were most often performed with vats of fluid and swinging pendulums. Yet, as Boyd understood, the same types of equations governing lasers also applied to other chaotic systems, so laser experiments offered the opportunity to test general aspects of chaos at high levels of precision.

Even so, Gauthier did not immediately act on Boyd's suggestion. "Bob would come in and say, 'I think this particular setup might show chaos,' and I would never think much about it," Gauthier says. "Then he would come back a couple of months later—'Have you seen chaos?' He kept coming back and pestering me about it. And then finally, when I did spend some time on it, we did see chaotic behavior, and that forced me to learn a lot about how you take experimental data and verify that it's chaotic."

This challenge, of separating randomness from chaos, became the subject of Gauthier's doctoral thesis. By discovering sensitive dependence and strange attractors in his own system, he opened the door for precision tests of chaos, a development that promised not only to shed light on lasers, but also on the study of planets and populations and pendulums.

By the late 1980s, the revolution that Lorenz had sparked with his weather model was well underway, not only in physics, but across the sciences. Yet it was not until 1990 that Edward Ott, Celso Grebogi, and James Yorke, a trio of re searchers at the University of Maryland, began to argue that the very feature that made prediction impossible for chaotic systems might also make it feasible to control them. Since, they reasoned, very tiny changes in chaotic systems resulted in large-scale alterations in behavior, and since, as mathematicians had shown, every chaotic system always operated within a hair's breadth of order, it stood to reason that only the tiniest push, delivered at precisely the right moment, could be enough to tip chaos back into regularity. Suddenly, it was as if the swinging payload of a tower crane could be stabilized just by blowing on it with a straw.

For Gauthier, newly arrived at Duke, the work suggested radical possibilities for making use of chaos. For example, it offered the hope that some systems like spaceflight trajectories, which engineers had only limited power to alter, might be manipulated through small interventions. In addition, rather than work to eliminate chaos from manmade systems such as sensor networks like burglar alarms or radar, a scientist could make use of sensitive dependence on initial conditions, the butterfly effect, to magnify even the faintest of signals.

But many of these applications, as Gauthier understood, required the elimination of a crucial bottleneck: "All the previous experiments that had been done before were on rather slow systems," he recalls. In other words, if experimenters wanted to control the experimental equivalent of the swinging crane payload, they had plenty of time to measure its speed and direction, feed the data into a computer, calculate how hard and where to blow, and carry out the plan, all before the payload had completed a swing.

However, electronic circuits fluctuate much faster, on the order of a billionth of a second. To stabilize these systems, the correction would need to come even faster, and the comparatively sluggish computer eliminated entirely. Together with another recent hire, Josh Socolar, now an associate professor of physics, Gauthier began to brainstorm ideas for a control system that would operate continuously, not just at the precise moment Ott, Grebogi, and Yorke had prescribed.

He and Socolar eventually settled on using time-delayed feedback from the circuits themselves to perform the correction, a sort of echo effect that obliterated the chaos. Their scheme opened up the possibility of controlling electronics, lasers, and other fast devices without the aid of a computer, and suggested the possibility of self-monitoring, self-stabilizing chaotic systems.

Indeed, the implications of Gauthier and Socolar's work extended far beyond the realm of electronic circuits. In 1992, a team of researchers from several different universities published a paper in Science reporting promising results from a study of intervention to correct arrhythmia in heart tissue. Because the processes that govern the natural flow of electrical ions through the heart allow for the possibility of chaotic behavior, small abnormalities in the system can lead to spasms, destroying the synchronous contraction necessary for proper blood flow. Pacemakers work by applying a hard shock to this system, jolting it back into regularity. What the study's authors offered was evidence that techniques of chaotic control might allow for smaller, less drastic corrections, saving power and dramatically prolonging a pacemaker's life.

Like many other scientists, Gauthier read the cardiac chaos paper with interest, seeing a potential application for his and Socolar's feedback control scheme. Then, a chance encounter with biomedical engineering professor Wanda Krassowska Neu Ph.D. '87 provided the opportunity to work on the problem directly.

The week the cardiac chaos paper was published, Neu participated in a site review for Duke's Engineering Research Center for Emerging Cardiovascular Technologies. One of the members of the site-review team had taken a copy of with him on the plane and, by the time he arrived, was showing it around, asking everyone, "Why isn't someone here at Duke working on this?" Not long afterward, Neu approached Gauthier and asked him what he knew about controlling chaos. Soon the two of them, along with Socolar, were hard at work on the chaotic dynamics of the heart.

They found that regulating the behavior of an electrical signal that varied not just in time, but also over the entire surface of the heart, required a whole new order of sophistication in chaotic control. Throughout the 1990s, experimenters across the country tried to translate the Science findings from the laboratory to the emergency room, with limited success. The Duke researchers, who began with experiments that attempted to subdue chaotic spasms in the hearts of sheep, have, for the last few years, shifted to smaller-scale trials in millimeter-size chunks of muscle.

"We've just been focused on small pieces of tissue," Gauthier says. "This is much harder than any of the other controlling-chaos problems that people have been doing."

Just as it would be nearly impossible to try calming a pool full of turbulent water with a single oar, so the Duke group has found that applying shocks at only a handful of points does little to stabilize a runaway electrical signal in the heart. While small shocks might have transient effects locally, they do little to settle down the system as a whole. So far, Gauthier and his collaborators have had success in correcting a few pathological patterns, like the aberrant weak beats of pulsus alternans, but have not developed a chaotic pacemaker. After spending more than a decade on the problem, Gauthier still sees potential in the work, but, lately, has come to appreciate the commitment necessary to make it a reality. "In order to do that, I'd really have to drop everything else," he says. Right now, there are other ideas calling.

At the center of each of the pair of rooms that make up Dan Gauthier's lab space in the physics department sits a half-ton, bump-proof optical table, its honeycombed surface sprouting long-stemmed lenses and silvered mirrors like the quills of a steel porcupine. From an overhead shelf, strips of plastic hang veil-like, filtering out stray dust from the crisscrossing glow of laser beams. Their target is a tiny chamber filled with rubidium gas cooled to less than a degree above absolute zero. Cabling hangs everywhere, over the equipment and above the table—wires, coaxial connectors, fiber optics. Everything has the look of being rearrangeable at a moment's notice.

To the side of all of this, in a pile of nondescript hardware, sits Gauthier's latest experiment, a palm-sized circuit board ringed by a trio of gold connectors. Hooked into the lab's high-speed oscilloscope, it produces a thin, wiggling line of a signal, its points alternating between high and low voltages like Ping Pong balls between a pair of invisible paddles. Fluctuating almost too fast to follow, the readout appears jittery, patternless, a surprising product for a device of pure logic. Gauthier and Socolar are calling the phenomenon "Boolean chaos," from the binary logic that gives rise to its dynamics. The purpose of this circuit, however, is not to generate computation, but chaos.

The idea behind it is rooted in the fundamental electronics of computing: In theory, each of a computer's logic gates is binary—entirely "off" or "on"—flipping instantly from one to the other in response to its inputs. Wired together and left to run freely, the activity of these idealized units is expected to settle down into fixed, repeating patterns.

In reality, however, the gates require a short time to make the transition between the two states, during which their voltages lie somewhere between the two extremes. For this reason, computers must "clock" their hardware, synchronizing the flips so that all components have a chance to complete their transitions before the next command is executed. But remove that restriction, and the gates begin to flip unpredictably, giving rise to the apparently random readout on the oscilloscope screen. Yet, as postdocs Rui Zhang and Hugo Cavalcante have recently shown, the activity of their hardware is not at all random. Rather, it possesses all the mathematical hallmarks of chaos.

The phenomenon is simple enough to be realized with a single logic gate (the working model uses three) and can be built for less than $50, most of which goes toward the high-speed connectors necessary for data collection. And while Gauthier is interested in electronics applications of the circuits (like improved intruder detection sensors), the mathematics that describes them is also, intriguingly, the mathematics of genes and gene networks, with individual logic gates standing in for bits of DNA.

Like logic gates, most genes are well described as either "on" (actively being read and producing protein) or "off" (suppressed and producing no protein). However, unlike logic gates, genes interact with each other chemically, a slower process that results in more gradual flips and numerous units functioning at intermediate levels. Here, the question is how the body conspires to eliminate chaos (suffice it to say, you want stability, not sensitive dependence, at work in your genes), and whether it might be possible to correct genetic dysfunction through the methods of chaotic control. And once again, because mathematics is the key ingredient for understanding the system's dynamics, whatever lessons Gauthier and company learn in circuit boards are likely to have at least some carryover to cells.

For the moment, though, it is still too early to tell whether Boolean chaos will ultimately give rise to gene therapies, just as chaotic control has yet to deliver the perfect pacemaker. At present, Gauthier concedes, there is no "killer app" for chaos, no indispensable application of the techniques he and so many others have been working to develop. Fifty years after Lorenz's discovery, the butterfly effect is still more marvel than technology. Even so, Gauthier has recently begun toying with the idea of an intruder system built from arrays of chaotic devices, a network so sensitive that living things would register amid manmade structures like flares in a desert night.

It could happen. At the very least it won't be boring. And for a scientist like Dan Gauthier, why shouldn't that be enough? After all, chaos is the puzzle of a million guises, each its own strange attractor. From lasers to logic gates to living, beating tissue, a curiosity cabinet of patterns in the noise.                                     

Pearson is a postdoctoral researcher in the department of neurobiology. His story on the neuroscience of risk appeared in the November-December 2008 issue of Duke Magazine.