# Chaos

Last Updated on May 8, 2015, by eNotes Editorial. Word Count: 2313

Underneath the rubric “chaos” lurks a revolution in science. James Gleick, a science writer for *The New York Times,* sets for himself the task of chronicling that revolution in progress. Though there is far from universal agreement on the boundaries of the new science, or even that it should be...

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Underneath the rubric “chaos” lurks a revolution in science. James Gleick, a science writer for *The New York Times,* sets for himself the task of chronicling that revolution in progress. Though there is far from universal agreement on the boundaries of the new science, or even that it should be called chaos, most critics acknowledge fundamental breakthroughs in the mathematical understanding of turbulent systems. Until the advent of chaos research in the 1960’s and 1970’s, and the development of high-speed computers, there were no adequate scientific ways to describe phenomena as common as the turbulence produced by boiling water, mixing cream into coffee, or the collision of air masses. Chaos studied these systems in process. Mathematicians and physicists, ecologists and astronomers, physiologists and those in fluid dynamics—all learned to “see” in a new way, not piecemeal and locally, but systemwide and globally.

The revolutionary character of chaos is not limited to new ways of evaluating systems. Gleick borrows insights from science historian Thomas Kuhn and *The Structure of Scientific Revolutions* (1962, enlarged second edition 1970), picturing chaos as a threat to the old ways of doing science, as well as a risky undertaking for its pioneers: “A few freethinkers working alone, unable to explain where they are heading, afraid even to tell their colleagues what they are doing—that romantic image lies at the heart of Kuhn’s scheme, and it has occurred in real life, time and time again, in the exploration of chaos.” Gleick’s report on chaos is in part the story of such free-thinking scientists and how the interdisciplinary nature of the study of nonlinear systems encroached upon their colleagues’ well-guarded turf. Each of the chapters in *Chaos* follows one or more pioneers of the new science as insight is added to insight. Swirled into the mixture of personalities is Gleick’s explanation of the fundamentals of chaos and its applications. A fitting kind of turbulence prevails throughout the book as each chapter explores new lines of research, which mix with and reinterpret earlier data. Gleick’s account is enlivened by dozens of anecdotes derived from extensive interviews. There is the image of Mitchell Fiegenbaum thinking deep thoughts on long walks at Los Alamos National Laboratory in New Mexico, and the four rebellious researchers who constituted the “dynamical systems collective” at the University of California’s Santa Cruz campus.

The text is supplemented with sections of color and black-and-white illustrations and graphs, though oddly text and illustrations are not always closely coordinated (the detailed index, however, does refer to the appropriate picture under the appropriate concept). Gleick’s prose is simple and basically nonmathematical as he conveys the feeling as well as the content of what it means to “make a new science.” As a result, *Chaos* has been widely praised; it was chosen by *The New York Times Book Review* as one of the best books of 1987.

One of the foundations of chaos research is that nonlinear systems are sensitive to and dependent upon small changes in initial conditions. Linear or straight-line relationships can easily be graphed with a ruler, because the values in a linear equation are proportional to one another. As Gleick notes, if friction did not exist, a simple linear equation could describe how much energy is required to accelerate a hockey puck. Once friction is introduced, however, the straightforwardness in equations disappears. Friction plays a part in how fast the puck is moving, but it is not a constant, linear, factor because friction is more or less significant depending on the present speed of the puck. A nonlinear equation is required. “Nonlinearity means that the act of playing the game has a way of changing the rules.”

This insight was forcefully brought out in the early 1960’s with the work of meteorologist Edward Lorenz. When Lorenz attempted to capture the essence of atmospheric convection using three simple, but nonlinear, equations, he found that tiny changes in the numbers he was using at the beginning produced wild variations as the equations were iterated (that is, as numerical values produced by each equation were plugged back into the equation to simulate constantly changing weather patterns). Perhaps a difference of a degree or two in initial atmospheric temperatures, for example, might mean the difference between a placid day and the advent of a hurricane a month in the future. The difficulty for long-range weather forecasts, however, was that even the most sensitive temperature gauges had only finite accuracy, and only so many could practically be placed in the atmosphere. Thus, small temperature changes overlooked by the instruments, or changes outside the range of the instrument, might be the determinants of the development of a storm weeks ahead. Though it sounds obvious, Lorenz’s discovery violated a basic scientific credo, a kind of modified Newtonianism, that imperfect measurements in the beginning could nevertheless produce close approximations of future events.

Yet while small changes at the beginning produced vast differences later on in a system of nonlinear equations, the variations were not random. Hidden within Lorenz’s three nonlinear equations was an elegant “butterfly” pattern: As lines showing the progress of the system were graphed in three dimensions, two “wings” emerged with the pattern traced over and over again, yet never exactly the same as any other pattern. Lorenz found that a small change in the initial values used in his equations meant a complete reshuffling of the wing pattern as the equation was iterated. Thus, though different initial values of nonlinear equations could produce chaotic differences later, at a deeper level the system was patterned, even though the pattern might be reshuffled. In the midst of chaos locally, as others later put it, the system was stable globally. The Great Red Spot in the Jovian atmosphere is a case in point. In the light of chaos, older theories that tried to account for the Red Spot’s stability by tying it to a surface feature can be discarded. “The spot is a self-organizing system, created and regulated by the same nonlinear twists that create the unpredictable turmoil around it. It is stable chaos.”

Robert May began his career as a theoretical physicist in Australia, but it is as a biologist that he made his mark on the emerging science of chaos. In a mathematical exploration of changes in the boom-and-bust cycles of wildlife populations, May discovered that when the boom-and-bust parameter is low, the population becomes extinct. There may not be sufficient animal population to overcome predators or establish its own niche, and so the population is doomed. As the parameter increased, however (as May expected), the population increases at a steady rate. If the system is driven harder, (that is, if the parameter is increased), the boom-and-bust rate breaks in two, indicating an alternating higher and lower population in alternating years. The system is still regular, though more complex, when at a higher parameter there appear four alternating population levels. As the parameter is increased, the number of population levels doubles, then doubles again. At a certain parameter value, chaos intervenes. That is, in nonlinear equations governing the rate of animal population, changes from year to year appear to be totally random. Yet as the parameter was further increased, out of “randomness” emerged new areas of periodicity. The population would have three stable levels, then six, then twelve, with an eventual return to chaos—only to be followed by new areas of stability or patterned behavior. Another old scientific intuition—that order gave rise to order, randomness to randomness—had been dashed. In May’s first crude work with a hand calculator, it seemed that even simple nonlinear equations gave rise to chaos, which appeared random but which was infinitely finely structured.

May’s findings were not mere theoretical curiosities. Gleick points out that records of measles epidemics in New York display a similar kind of “deterministic chaos.” When inoculations were first introduced, the number of measles cases, contrary to intuition, did not slowly subside but rather took on a pattern of seeming random oscillation. An inoculation program could actually produce a short-term increase in the disease, even while the disease was being eliminated over the long run.

The mathematical investigation of the patterns within chaos drew on what became known as fractal geometry. First developed by Benoit Mandelbrot at IBM, fractal (for fractional dimension) geometry became a tool of nonlinear dynamics, used alike by seismologists and Hollywood’s special effects artists. Robert May had discovered a patterned chaos in population behavior equations; Mandelbrot was working on the problem of noise during computer interfaces. Whenever one computer talked with another computer over telephone lines, it seemed that little random bursts of noise would produce transmission errors. Mandelbrot, working on the problem in the 1960’s, discerned that no matter how short a time span, the noise was never continuous. Between one burst of noise and another there were always times of clean transmission. It was that way on the largest and the smallest scales. Mathematically speaking, the “dust” of noise was infinitely small but ever present. (Mandelbrot’s advice to IBM was to increase the redundancy in a transmission and not to try to overpower the noise which seeped in everywhere.)

Mandelbrot found that the noise in the circuits, while seemingly random, nevertheless formed an elegant mathematical pattern. The idea that there was a scattering of noise, no matter how finely sliced the time period, had a spatial analogy. One might imagine observing an entire coastline, and then examining a tiny portion of the coastline with a magnifying glass. The same kind of protrusions and inlets are present at both scales. The roughness of the coastline extends at least down to the atomic level, and perhaps beyond. A given stretch of coastline is thus, paradoxically, infinitely long. Mandelbrot was able to calculate the degree of roughness of objects and found that, whatever the scale, the degree of roughness remained constant. Degrees of roughness were measured in what Mandelbrot called fractional dimensions. The curve of a snowflake, or a cloud, or a coastline, “implies an organizing structure that lies hidden among the hideous complication of such shapes.” A simple triangle, and a formula that adds a triangle half its size to each side of the original triangle, and so on, when iterated enough times produces the kind of roughness that mimics the real coastline, or the real snowflake. Simplicity yields complexity.

In the late 1970’s, Mandelbrot plotted a strange fractal shape that now bears his name. The Mandelbrot set is infinitely complex but based on the iteration of a simple nonlinear equation. Plotted on the complex plane (with real numbers on one axis and imaginary numbers on the other), each dot represented the behavior of each number, regardless of whether in the iterated equation the values tended toward infinity. Those numbers that did not were in the set; those that did were outside. High-speed computers were needed for the plotting, for only after, say, ten thousand iterations might it be apparent the equation tended toward infinity. It was like determining which volume levels produced the squeal of microphone-loudspeaker feedback and which did not. When plotted as groups of colored points by a computer, the Mandelbrot set resembles a buglike object with what seem to be wisps of fine hair emanating from its body. Yet the computer can perform ever more calculations, thus “magnifying” any portion of the set, revealing a strange world of infinite complexity and pattern. Here are bizarre seahorses, diamond-studded starfish, fiery plumed squid. And yet, as “magnification” between any two points increases, “one picture seemed more and more random, until suddenly, unexpectedly, deep in the heart of a bewildering region, appeared a familiar oblate form, studded with buds: the Mandelbrot set, every tendril and every atom in place.” The lesson in these abstract pictures was twofold: First, engineers could no longer simply assume that some approximation of the ideal in a nonlinear equation was sufficiently “safe”—the Mandelbrot set demonstrated that in some equations, chaos may be only an infinitely small point away.

The other lesson had to do with “universality.” This discovery is credited to Mitchell Feigenbaum, who in the mid-1970’s found that, as one researcher put it, “there were structures in nonlinear systems that are always the same if you looked at them the right way.” The reappearance of the Mandelbrot set, deep within the Mandelbrot set itself, was a good example. Some of the structures in nonlinear equations were independent of the particular equations. Again, this was more than theory. In the late 1970’s, when Albert Libchaber experimented with the turbulent behavior of liquid helium, he observed “the universal Feigenbaum constants turning in that instant from a mathematical ideal to a physical reality.”

The message for researchers was that while Nature may be chaotic, she is chaotic only in certain ways, only in or through certain patterns. Leaves, clouds, coastlines, all look as they do through the outworking of relatively simple patterning. Evolution itself might be explained as “chaos with feedback.” As chaos has shown, incredible beauty and complex orders can come out of “randomness.” The new scientific credo is this: “Simple systems give rise to complex behavior. Complex systems give rise to simple behavior. And most important, the laws of complexity hold universally, caring not at all for the details of a system’s constituent atoms.”

As a report on science-in-the-making, Gleick’s account necessarily lacks historical perspective. He seems to take most of his sources at face value and succeeds in involving the reader in the social and political struggles that produced chaos. Whether chaos will become merely another discipline, with its own well-guarded turf, or the face of science will truly be changed and barriers shattered must be left to future historians. For his part, Gleick is a compassionate and reliable guide through the turbulent beginnings of a new science.

# Bibliography

Last Updated on May 8, 2015, by eNotes Editorial. Word Count: 20

*Chicago Tribune Books.* November 22, 1987, p. 10.

*The New York Times.* October 15, 1987, p. 11.

*The New York Times Book Review.* XCII, October 25, 1987, p. 11.