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THE ESSENCE OF CHAOS Edward N.Lorenz Copyright © 1993 by the University of Washington Press First published in 1993. This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Paperback edition first published in the United Kingdom in 1995. Second impression 1995. Published in the United States by University of Washington Press as a volume in the Jessie and John Danz Lecture Series. Published in the UK by: UCL Press Limited 1 Gunpowder Square London EC4A 3DE The name of University College London (UCL) is a registered trade mark used by UCL Press with the consent of the owner. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. ISBN 0-203-21458-7 Master e-book ISBN ISBN 0-203-27116-5 (Adobe eReader Format) ISBN: 1-85728-454-2 PB Contents Preface vii CHAPTER 1 Glimpses of Chaos 3 It Only Looks Random Pinballs and Butterflies It Ain’t Got Rhythm Zeroing In on Chaos CHAPTER 2 A Journey into Chaos 25 Chaos in Action The Heart of Chaos Broken Hearts Chaos of Another Species In and Out of Chaos CHAPTER 3 Our Chaotic Weather 77 Prediction: A Tale of Two Fluids Meteorology: Two Tales of One Fluid The Unperformable Experiment Voices from Dishpans The Five-Million-Variable Dynamical System The Consequences CHAPTER 4 Encounters with Chaos 111 Prologue Recognition In Limbo Searching The Strange Attractor The Ubiquity of Chaos Make Your Own Chaos Is Randomness Chaos? CHAPTER 5 What Else Is Chaos? 161 Nonlinearity Complexity Fractality APPENDIXES 1. The Butterfly Effect 179 2. Mathematical Excursions 183 3. A Brief Dynamical-Systems Glossary 203 Bibliography 211 Index 217 Preface IN THE SPRING OF 1990 I received an invitation from the University of Washington to deliver a set of lectures, as part of the series that had been inaugurated a generation earlier through the benevolence and farsightedness of Jessie and John Danz. The lectures were to be delivered before a general audience, and I was free to choose a subject. Some thirty years previously, while conducting an extensive experiment in the theory of weather forecasting, I had come across a phenomenon that later came to be called “chaos”—seemingly random and unpredictable behavior that nevertheless proceeds according to precise and often easily expressed rules. Earlier investigators had occasionally encountered behavior of this sort, but usually under rather different circumstances. Often they failed to recognize what they had seen, and simply became aware that something was blocking them from solving their equations or otherwise completing their studies. My situation was unique in that, as I eventually came to realize, my experiment was doomed to failure unless I could construct a system of equations whose solutions behaved chaotically. Chaos suddenly became something to be welcomed, at least under some conditions, and in the ensuing years I found myself turning more and more toward chaos as a phenomenon worthy of study for its own sake. It was easy to decide what topic the lectures should cover. I accepted the invitation, and chose as a title “The Essence of Chaos.” Eventually a set of three lectures took shape. The first one defined chaos and illustrated its basic properties with some simple examples, and ended by describing some related phenomena— nonlinearity, complexity, and fractality—that had also come to be called “chaos.” The second lecture dealt with the global weather as a complicated example of a chaotic system. The final one presented an account of our growing awareness of chaos, offered a prescription via which one could design one’s own chaotic systems, and ended with some philosophical speculations. In keeping with the anticipated make-up of the audience, I displayed no mathematical formulas, and avoided technical terms except for some that I defined as I went along. The present volume, with the same title, is written in the spirit of the Danz Lectures. It contains the same material, together with additions written to fill in the many gaps that were inevitably present in a limited oral presentation. The leading lecture has been expanded to become Chapters 1, 2, and 5, while the second one has been made into Chapter 3. The final lecture, with its historical account that begins with the discovery of Neptune, proceeds through the work of Henri Poincaré and his successors, and pauses to tell of my own involvement with chaos, has become Chapter 4. My decision to convert the lectures into a book has been influenced by my conviction that chaos, along with its many associated concepts—strange attractors, basin boundaries, period- doubling bifurcations, and the like—can readily be understood and relished by readers who have no special mathematical or other scientific background, despite the occasionally encountered references to chaos as a branch of mathematics or a new science. As in the lectures, I have presented the chaos story in nontechnical language, except where, to avoid excessive repetition of lengthy phrases, I have introduced and defined a number of standard terms. I have placed the relevant mathematical equations and their derivations in an appendix, which need not be read for an understanding of the main text, but which may increase the volume’s appeal to the mathematically minded reader. Of course one cannot maintain that there is no mathematics at all in the main text, except by adopting a rather narrow view of what constitutes mathematics. For example, merely noting that one illustration shows two boards sliding thirty meters down a slope, starting ten centimeters apart and ending up ten meters apart, can be looked upon as a mathematical observation; a verbal description of what the illustration depicts is then a mathematical statement. In any event, a good deal of less simple mathematics has gone into the production of the illustrations; most of them are end products of mathematical developments, subsequently converted into computer programs. The reader nevertheless need not confront the formulas, nor the programs, to be able to absorb the messages that the illustrations contain. For their aid during the preparation of this work I am indebted to many persons. First of all I must thank the Danz Foundation, without whose sponsorship of my lectures I would never have taken the first step. I must likewise thank the University of Washington for choosing me as a lecturer. I am deeply indebted to the Climate Dynamics Program of the Atmospheric Sciences Section of the National Science Foundation, and to the program’s current director, Jay Fein, for supporting my research in chaos and its applications to the atmosphere over many years, and, most immediately, for making it possible for me to write the numerous computer programs and to perform the subsequent computations that have resulted in the illustrations in this volume. I wish to thank Joel Sloman for typing and otherwise assisting with not only the final manuscript but also the innumerable intermediate versions, Diana Spiegel for her ever-present aid in dealing with the vagaries of our computer system, and Jane McNabb for bearing the bulk of the administrative burden that otherwise would have fallen on me. Thanks go to Dave Fultz of the University of Chicago for supplying the photographs of his dishpan experiments, and to him and the American Meteorological Society for permission to reproduce them. Thanks also go to Robert Dattore and Wilbur Spangler of the National Center for Atmospheric Research for preparing and making available the lengthy tape containing the many years of recorded upper-level weather data at Singapore. I must give special recognition to Merry Caston, who has gone over the manuscript page by page, and whose pertinent comments have led me to incorporate a good many clarifying additions and other amendments. There are many other persons with whom I have had brief or in some cases extensive conversations, which have exerted their influence on the words that I have written or the ideas that I have expressed. In this connection I must particularly mention Robert Cornett, James Curry, Robert Devaney, Alan Faller, Robert Hilborn, Philip Merilees, Tim Palmer, Bruce Street, Yoshisuke Ueda, J.Michael Wallace, and James Yorke. To still others who may have similarly influenced me without my being aware of it, and also to some anonymous reviewers, I can only say that their names ought to have been included. Finally, I am most grateful to my wife, Jane, who has supplied moral support throughout the preparation of this volume and has accompanied me on numerous travels in search of chaotic material, and to my children Nancy, Edward, and Cheryl—lawyer, economist, and psychologist—who have perfectly filled the role of the intelligent layperson and have subjected the manuscript in various stages of completion to their closest scrutiny. CHAPTER 1 Glimpses of Chaos It Only Looks Random WORDS are not living creatures; they cannot breathe, nor walk, nor become fond of one another. Yet, like the human beings whom they are destined to serve, they can lead unique lives. A word may be born into a language with just one meaning, but, as it grows up, it may acquire new meanings that are related but nevertheless distinct. Often these meanings are rather natural extensions of older ones. Early in our own lives we learn what “hot” and “cold” mean, but as we mature we discover that hot pursuit and cold comfort, or hot denials and cold receptions, are not substances or objects whose temperatures can be measured or estimated. In other instances the more recent meanings are specializations.
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