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Copyright © 1997 Garnett P. Williams. All rights reserved.
Published by arrangement with Taylor & Francis Ltd.
Reprinted 1999
Printed in Great Britain.
PREFACE
Virtually every branch of the sciences, engineering, economics, and related fields now discusses or refers to
chaos. James Gleick's 1987 book, Chaos: making a new science and a 1988 one-hour television program on
chaos aroused many people's curiosity and interest. There are now quite a few books on the subject. Anyone
writing yet another book, on any topic, inevitably goes through the routine of justifying it. My justification
consists of two reasons:
• Most books on chaos, while praiseworthy in many respects, use a high level of math. Those books have
been written by specialists for other specialists, even though the authors often label them "introductory."
Amato (1992) refers to a "cultural chasm" between "the small group of mathematically inclined initiates
who have been touting" chaos theory, on the one hand, and most scientists (and, I might add, "everybody
else''), on the other. There are relatively few books for those who lack a strong mathematics and physics
background and who might wish to explore chaos in a particular field. (More about this later in the Preface.)
• Most books, in my opinion, don't provide understandable derivations or explanations of many key
concepts, such as Kolmogorov-Sinai entropy, dimensions, Fourier analysis, Lyapunov exponents, and
others. At present, the best way to get such explanations is either to find a personal guru or to put in gobs of
frustrating work studying the brief, condensed, advanced treatments given in technical articles.
Chaos is a mathematical subject and therefore isn't for everybody. However, to understand the fundamental
concepts, you don't need a background of anything more than introductory courses in algebra, trigonometry,
geometry, and statistics. That's as much as you'll need for this book. (More advanced work, on the other hand,
does require integral calculus, partial differential equations, computer programming, and similar topics.)
In this book, I assume no prior knowledge of chaos, on your part. Although chaos covers a broad range of
topics, I try to discuss only the most important ones. I present them hierarchically. Introductory background
perspective takes up the first two chapters. Then come seven chapters consisting of selected important material
(an auxiliary toolkit) from various fields. Those chapters provide what I think is a good and necessary
foundation—one that can be arduous and time consuming to get from other sources. Basic and simple chaosrelated
concepts follow. They, in turn, are prerequisites for the slightly more advanced concepts that make up
the later chapters. (That progression means, in turn, that some chapters are on a very simple level, others on a
more advanced level.) In general, I try to present a plain-vanilla treatment, with emphasis on the idealized case
of low-dimensional, noise-free chaos. That case is indispensable for an introduction. Some real-world data, in
contrast, often require sophisticated and as-yet-developing methods of analysis. I don't discuss those techniques.
The absence of high-level math of course doesn't mean that the reading is light entertainment. Although there's
no way to avoid some specialized terminology, I define such terms in the text as well as in a Glossary. Besides,
learning and using a new vocabulary (a new language) is fun and exciting. It opens up a new world.
My goal, then, is to present a basic, semitechnical introduction to chaos. The intended audience consists of
chaos nonspecialists who want a foothold on the fundamentals of chaos theory, regardless of their academic
level. Such nonspecialists may not be comfortable with the more formal mathematical approaches that some
books follow. Moreover, many readers (myself included) often find a formal writing style more difficult to
understand. With this wider and less mathematically inclined readership in mind, I have deliberately kept the
writing informal—"we'll" instead of "we will," "I'd" instead of ''I would," etc. Traditionalists who are used to a
formal style may be uneasy with this. Nonetheless, I hope it will help reduce the perceived distance between
subject and reader.
I'm a geologist/hydrologist by training. I believe that coming from a peripheral field helps me to see the subject
differently. It also helps me to understand—and I hope answer—the types of questions a nonspecialist has.
Finally, I hope it will help me to avoid using excessive amounts of specialized jargon.
In a nutshell, this is an elementary approach designed to save you time and trouble in acquiring many of the
fundamentals of chaos theory. It's the book that I wish had been available when I started looking into chaos. I
hope it'll be of help to you.
In regard to units of measurement, I have tried to compromise between what I'm used to and what I suspect
most readers are used to. I've used metric units (kilometers, centimeters, etc.) for length because that's what I
have always used in the scientific field. I've used Imperial units (pounds, Fahrenheit, etc.) in most other cases.
I sincerely appreciate the benefit of useful conversations with and/or help from A. V. Vecchia, Brent Troutman,
Andrew Fraser, Ben Mesander, Michael Mundt, Jon Nese, William Schaffer, Randy Parker, Michael Karlinger,
Leonard Smith, Kaj Williams, Surja Sharma, Robert Devaney, Franklin Horowitz, and John Moody. For
critically reading parts of the manuscript, I thank James Doerer, Jon Nese, Ron Charpentier, Brent Troutman, A.
V. Vecchia, Michael Karlinger, Chris Barton, Andrew Fraser, Troy Shinbrot, Daniel Kaplan, Steve Pruess,
David Furbish, Liz Bradley, Bill Briggs, Dean Prichard, Neil Gershenfeld, Bob Devaney, Anastasios Tsonis,
and Mitchell Feigenbaum. Their constructive comments helped reduce errors and bring about a much more
readable and understandable product. I also thank Anthony Sanchez, Kerstin Williams, and Sebastian
Kuzminsky for their invaluable help on the figures.
I especially want to thank Roger Jones for his steadfast support, perseverance, hard work, friendly advice and
invaluable expertise as editor. He has made the whole process a rewarding experience for me. Other authors
should be so fortunate.
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