Source: American Scientist, Vol. 85 #5, September-October 1997, p. 487- 488.
MATHEMATICS AND COMPUTERS
Chaos: An Introduction to Dynamical Systems. Kathleen T. Alligood, Tim D. Sauer and James A. Yorke. 603 pp. Springer-Verlag, 1996. $30.
There is a continuing fascination with chaos theory among members of the general public, and although this is not the book I would recommend to the layperson, it is by far the most complete introduction to chaos theory available in a single text. Alligood, Sauer and especially Yorke are well-known names in the research literature on chaos, fractals and nonlinear dynamics. The book is based on class material taught by the authors at George Mason University and at the University of Maryland. The reader is assuined to be familiar with calculus, differential equations and linear algebra, although some chapters require only a first-year background in calculus.
The first six chapters of the book cover iterated maps in one, two and more dimensions. (Iterated maps describe the time evolution of a system by expressing its state as a function of a previous state. As the mapping is iterated, the system evolves in discrete updates, each corresponding to a step in time.) In a natural progression, chapters seven through nine deal with flows-or differential equations-where updating is continuous and it is the current rate of change of the system that is described as a function of the current state. The last four chapters are extensions of the concepts of stable manifolds, bifurcations and cascades that were introduced in the first two chapters. Two appendices introduce matrix algebra and computer solutions of ordinary differential equations.
Chapter 1 begins easily enough, with one-dimensional maps that develop into the logistic map G(x) = rx(1 - x). This is usual fare, except that the chapter ends with two novel sections, which will reappear at the end of each chapter. The first challenges the reader with current research ideas, references to relevant research literature and exercises with which to practice.
The second, and in my view the most exciting part of the book, is a device called the "Lab visit." Each one centers on a specific topic, such as fractal di- mension in experiments, the leaky faucet or chaos in simple mechanical devices, and includes an account of a visitation, complete with facts, actual data, references, and laboratory experiments and a discussion of their consequences. Taken together they illustrate the broad reach and depth of chaos science.
To elaborate at the end of chapter two on two-dimensional maps, the authors tell the story of the quest for an answer to that old nagging question, "Is the solar system stable?" The extraordinary answer-obtainable only with the use of a specifically designed computer called a Digital Orrery-is that the solar system is unstable, and chaotic too. We must qualify these assertions, however. That the solar system is chaotic does not necessarily mean that it is in the process of disintegrating or that planets are about to start colliding with each other in a fierce allout cataclysm. Mathematical chaos is tame when compared to its mundane biblical sister, and all it says about the solar system is that the predictability of events far in the future is not guaranteed. In other words, we live in a world whose destiny we can only predict for a comparatively short time, after which, whatever happens is beyond our powers to know or even to speculate. This instability occurs because of the fundamental characteristic of a chaotic system: its extreme sensitivity to initial conditions, or the idea that small changes can have a large effects. The French mathematician Henri Poincaré is credited with this discovery, now called the "butterfly effect." In his 1903 essay "Science and Method," Poincaré noted:
If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation in a succeeding moment ... [however even in such hypothetical case] we could still only know the initial situation approximately. If that enables us to predict the succeeding situation with the same approximation, that is all we require.... But is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena.... Prediction becomes impossible.
A disturbing aspect of this result is that an uncertainty of just one kilometer in the measurement of an astronomical position can grow to a full astronomical unit (the sun-to-earth distance) in just 95 million years! Thus, it is impossible for us to predict the future of the solar system beyond a certain-geologically short-period of time, or conversely, it is impossible for us to be sure of the relative positions of planets in the solar system hundreds of millions of years before the present. Because it is believed that earth's climate is strongly influenced by the slow changes exerted by the gravitational pull of the sun, moon and planets on its orbital parameters, it is therefore impossible to calculate with credible precision what these influences were say, 200 million years ago, when large ice sheets covered the continent of Gondwana. If chaos were not intrinsic to the solar system, geologists could attempt to correlate the record of those ancient ice ages with the calculated changes in the earth's orbital motions to search for cause-effect relationships. If the system is chaotic, however, there is little hope.
I found little difficulty becoming totally absorbed by this book. My favorite sections deal with flows, or differential equations. The topics touched here are of general interest, such as the Lotka-Volterra models of population dynamics, the Lorenz and Rossler attractors, the synchronization circuit of Chua and forced oscillations. The final chapters on bifurcations, cascades and time-series analyses contain some of the most difficult material in the book, but the authors use easy and well-illustrated examples to motivate and guide the reader through the material. In general the prose is clear, and the profuse illustration helps enormously, especially when the learning gets tough, as it often does.
-J. A. Rial, Geology, University of North Carolina at Chapel Hill