## Chaos

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...

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