Patterns of Chaos

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Patterns of Chaos

Overview

For centuries, scientists ignored or avoided chaotic, or nonlinear, data. Real but messy results were often ascribed to experimental error or "noise." In the 1960s, starting with the work of Edward Lorenz (1917- ), new approaches began to reveal the structures, dependencies, and patterns of nonlinear data. These data were found everywhere—in the price of cotton, the rise and fall of animal populations, and the shape of clouds and mountains, for example. Thanks largely to visual approaches that were supported by new, more powerful computers, general principles such as sensitivity to initial conditions and self-similarity were revealed. Chaos theory has given us a deeper understanding of nonlinear systems, both natural and artificial; pointed to solutions in communication, medicine, ecology, and other fields; it has even entered popular culture through images of fractals, the novel and film Jurassic Park, and popular science fiction stories.

Background

Linear equations, such as Newton's Laws, are well behaved. They provide exact answers and can be applied to practical, real-life problems. Much of engineering, for example, is based on linear equations. Nonlinear equations, on the other hand, contain infinite components and cannot be solved exactly; their results are unpredictable. For centuries, most scientists ignored and avoided nonlinear results, which they labeled useless "noise." Ignoring chaos, however, distorts our view of nature and blocks our understanding of many natural phenomena.

In the 1960s, a new way of looking at the world, called chaos theory, began to emerge. Mathematicians, biologists, physicists, and other scientists started to tackle nonlinear equations. The solution of these equations finally revealed order and structure within what looked like noise.

A milestone in chaos theory is the work of meteorologist Edward Lorenz. In 1960, Lorenz was trying to understand, model, and predict weather systems. He found that even when he was doing simple calculations, he could end up with messy and apparently random results—the hallmark of nonlinear equations. Rather than ignore these effects, Lorenz took a closer look and was able to abstract three key ideas: sensitivity to initial conditions, infinite variation, and strange attractors.

The idea of sensitivity to initial conditions came from a rounding error. When Lorenz tried to pick up where he had left off with a calculation, he found that the difference between 0.506 and 0.506127 led to wildly different weather patterns, despite the simplicity of the initial equations. Small effects could have big consequences. Infinite variation came from the discovery that, as results were tracked for certain equations, they stayed within a designated area, but they never repeated themselves. It was like having an infinite line in a box. The third idea, which came to be called strange attractors, indicated that there could be stable points that were unreachable. The typical attractors of a pendulum, for instance, are the regular consistent arc and the straight down, stopped position. Both states are energy minima that a pendulum will return to even if it is bumped and set, temporarily, into another motion. Strange attractors create motion that is just as stable, but totally irregular.

Four cultural conditions helped Lorenz establish chaos theory. First, the 1960s was a decade when authority and exact answers were questioned and challenged. Novelty and nonconformist thinking found expression in science, as well as in politics, music, and culture. Second, large investments were being made in research, much of it resulting from Cold War tensions. Pure research projects, without defined payoffs, were tolerated and, in some cases, even nurtured. For instance, Thomas Watson, CEO of IBM, personally advocated the support of "wild ducks" in his company's research division. These first two conditions provided a level of tolerance, if not acceptance, for thinkers who worked outside the mainstream. Third, as scientists in a variety of disciplines tried to solve particularly difficult problems, they all hit the same limits—a lack of tools to handle nonlinearity. Fourth, just as chaos gained a footing in the scientific world, computers became more powerful and widely available. Since chaos equations require an extreme amount of iterative (repetitive) calculation, computers offered a way through the mind-numbing computations of nonlinear systems. They also produced pictures that made complex patterns visible and accessible to intuition. This helped break down the preference for linear equations and more easily solved problems.

Of course, Lorenz was not the first to observe nonlinear systems. Jules Poincaré (1854-1912) found that the three-body problem (predicting positions of three bodies in space) was chaotic. Even knowing the gravities, velocities, and positions of each body failed to make such predictions practical because unmeasurable differences led to large effects. Experiments done in the 1920s put one of the first cracks in the linear wall. Balthasar van der Pol was looking at frequency changes driven by current changes in a vacuum tube. Because oscilloscopes were not yet available, he listened for a pattern of sounds and he consistently heard irregular noise just before frequencies locked in. Years later, these unusual results were brought to the attention of Stephen Smale (1930- ). Van der Pol's work demonstrated the incorrectness of a conjecture Smale had put forth, that erratic behavior could not be stable. It set his thinking on a track that was, ultimately, to lead him to strange attractors.

In another development, Mitchell Feigenbaum (1945- ) determined mathematical formulations of phase boundary conditions. By finding consistency across different scales, he showed that many systems changed phase—went from having one stable point to having two—and did so under consistent, predictable conditions. Feigenbaum's work had neither the rigor of a mathematical proof, nor the demonstrated connection with real-world phenomena that physicists demand. His discovery of the quantitative universality of the behaviors of a variety of different systems might have ended up as an academic exercise but for the work for Albert Libchaber (1934- ). Libchaber did the simplest and most controlled experiment with turbulence ever. In an environment protected from vibration, at a temperature near absolute zero, he measured thermally-induced turbulence in a tiny cell filled with liquid helium. The complexity of the turbulence increased exactly as much and as frequently as Feigenbaum's equations predicted.

Libchaber's work helped to validate the work of others involved in the study of chaos, scientists who were looking at the rise and fall of animal populations, the scaling of earthquakes, and genetic variation. At the same time, the work of Benoit Mandelbrot (1924- ) was helping to elucidate and popularize chaos both inside and outside the scientific community. Mandelbrot had come up with the idea of fractional dimensions—fractals. Fractals provided a way to express the self-similarity of nature across different scales. The quality of the shape of a coastline, for instance, does not vary whether it is seen from space, an airplane, three feet away, or under a microscope. Fractals help explain how you can have Lorenz's infinity in a box. They characterize Feigenbaum's phase changes. They also provide a means to produce appealing, evocative computer visualizations.

Impact

Chaos science has become an accepted discipline with academic departments, journals, degrees, and funding. The study of nonlinear phenomena has established common ground for many diverse scientists, particularly those who share a holistic outlook, to talk across their disciplines.

For all scientists, chaos forces a reanalysis of the meaning of the data when noise, complexity, and anomalous results appear. It demands a reexamination of edge conditions, where approximations may hide the real story. Chaos has revealed that the linear and most easily manipulated aspects of nature are actually the exceptions to a more complicated and interesting reality.

Virtually every area of science, including biology, astrophysics, thermodynamics, ecology, and chemistry has identified and studied chaotic systems. Even medicine and social sciences, such as economics, have used the tools of chaos to provide better understanding, to recognize the limits of linear approaches, and to suggest solutions to complex problems. Heart fibrillations, turbulence, chemical reactions, and the distribution of galaxy clusters are just some of the real-world phenomena that have been shown to be governed by nonlinearity.

Chaos has provided more reliable communications, guidance on vaccination programs, equations for artificial life, and simulations and techniques for encryption. It has led to improved weather predictions and new ways of compressing data.

One frontier of chaos involves public policy issues. Problems like health care, political realignments, and global sustainability have aspects—self-organization, nonlinearity, sensitivity to initial conditions, and decentralization—that seem analogous to natural systems that are known to be chaotic. The hope is that the theories and analyses developed in chaos science (often referred to as complexity in this context) may provide insight and direction in these areas. For instance, there is a growing realization that much of the detail work of economics research is based on an outdated and incorrect linear view of what is essentially a nonlinear phenomenon. Research resources are being reallocated accordingly.

Chaos theory has also changed the way science in general is done. Computers have provided a visual and dynamic means to rapidly explore and understand complex systems and are an important way for a scientist to develop intuition about abstract, complicated phenomena.

Chaos not only benefited from computers, but it became the first discipline to establish computers as valid experimental tools. The acceptance and recognition of computer simulations has since led to an understanding of artificial life, rational drug design, and ecological studies. These are considered to be real experimental contributions to their disciplines, not simply theoretical constructs. New devices, including jet aircraft, have been designed using computers. Simulations have become realistic enough, even at the video game level, to teach skills like flying and urban design. In fact, some popular games, like SimCity, even take advantage of cellular automata, which are chaos-driven artificial life forms.

Perhaps the most dramatic impact on popular culture by chaos has been its inclusion in Michael Crichton's Jurassic Park. This 1990 novel, and the subsequent Hollywood film, both provide popular explanations of chaos concepts and use chaos as a plot device. Fractal art has become a visible instance of chaos and can be found on T-shirts, ties, coffee mugs, and calendars. Fractal equations have become critical to computer animation for commercial movies and games, and have provided a means of data compression. And chaos continues to inspire science fiction writers and to provide rationales for their extrapolations.

PETER J. ANDREWS