Preface Dynamics is a field emerging somewhere between mathematics and the sciences. In our view, it is the most exciting event on the concept horizon for many years. The new concepts appearing in dynamics extend the conceptual power of our civilization and provide new understanding in many fields. We discovered, while working together on the illustrations for a book in 1978,1 * that we could explain mathematical ideas visually, within an easy and pleasant working partnership. In 1980, we wrote an expository article on dynamics and bifurcations,2 using hand-animation to emulate the dynamicpicture technique universally used by mathematicians in talking among themselves: a picture is drawn slowly, line by line, along with a spoken narrative - the dynamic picture and the narrative tightly coordinated. Our efforts inevitably exploded into four volumes, now combined into this book. The dynamic picture technique, evolved through our work together, and in five years of computer graphic experience with the Visual Math Project at the University of California at Santa Cruz, is the basis of this work. The majority of the book is devoted to visual representations, in which four colors are used according to a strict code. Math symbols have been kept to a minimum. In fact, they are almost completely suppressed. Our purpose is to make the book work for readers who are not practiced in symbolic representations. We rely exclusively on visual representations, with brief verbal explanations. Some formulas are shown with the applications, as part of the graphics, but are not essential. However, this strategy is exclusively pedagogic. We do not want anyone to think that we consider symbolic representations unimportant in mathematics. On the contrary, this field evolved primarily in the symbolic realm throughout the classical period. Even now, a full understanding of our subject demands a full measure of formulas, logical expressions, and technical intricacies from all branches of mathematics. A brief introduction to these is included in the Appendix. We have created this book as a short-cut to the research frontier of dynamical systems: theory, experiments, and applications. It is our goal - we know we may fail to reach it - to provide any interested person with an acquaintance with the basic concepts: *Footnotes refer to the Notes, which follow the Appendix. x Preface xi . state spaces: manifolds - geometric models for the virtual states of a system . attractors: static, periodic, and chaotic - geometric models for its local asymptotic behavior . separatrices: repellors, saddles, insets, tangles - defining the boundaries of regions (basins) dominated by different behaviors (attractors), and characterizing the global behavior of a system . bifurcations: subtle and catastrophic - geometric models for the controlled change of one system into another. The ideas included are selected from the literature of dynamics: Part One, "Periodic Behavior," covers the classical period from 1600 to 1950. Part Two, "Chaotic Behavior," is devoted to recent developments, 1950 to the present, on the chaotic behavior observed in experiments. Part Three, "Global Behavior," describes the concept of structural stability, discovered in 1937, and the important generic properties discovered since 1959, relating to the tangled insets and outsets of a dynamical system. These are fundamental to Part Four, "Bifurcation Behavior." In fact, the presentation in Part Four of an atlas of bifurcations in dynamical schemes with one control parameter was the original and primary goal of this whole book, and all of the topics in the first three parts have been selected for their importance to the understanding of these bifurcations. For we regard the response diagram, a molecular arrangement of the atomic bifurcation events described here, as the most useful dynamical model available to a scientist. We assume nothing in the way of prior mathematical training, beyond vectors in three dimensions, and complex numbers. Nevertheless, it will be tough going without a basic understanding of the simplest concepts of calculus. Our first attempt at the pictorial style used here evolved in the first draft of Dynamics: A Visual Introduction, during the summer of 1980. Our next effort, the preliminary draft of Part Two of this book, was circulated among friends in the summer of 1981. Extensive feedback from them has been very influential in the evolution of this volume, and we are grateful to them: Fred Abraham George Francis Jerry Marsden Rob Shaw Ethan Akin Alan Garfinkel Nelson Max Mike Shub Michael Arbib John Guckenheimer Jim McGill Steve Smale Jim Crutchfield Moe Hirsch Kent Morrison Joel Smoller Larry Cuba Phil Holmes Charles Musès Jim Swift Richard Cushman Dan Joseph Norman Packard Bob Williams Larry Domash Jean-Michel Kantor Tim Poston Art Winfree Jean-Pierre Eckman Bob Lansdon Otto Rossler Marianne Wolpert Len Fellman Arnold Mandell Lee Rudolph Gene Yates Katie Scott Chris Zeeman We are especially grateful to Tim Poston and Fred Abraham for their careful reading of the manuscript; to the Dynamics Guild (J. Crutchfield, D. Farmer, N. Packard, and R. Shaw) for their computer plots used in many places in this book; to Richard Cushman for history lessons; to Phyllis Wright and Claire Moore of TypaGraphix for their care in typesetting and production; to Laura Logan of Aerial Press for her expert assistance in the production process; to Diane Rigoli for her splendid final drawings based on our rough sketches for Part Four; and to Rob Shaw for providing photos for xii Preface Section 16.3 and computer plots for Section 17.3. The generosity and goodwill of many dynamicists has been crucial in the preparation of this book; we thank them all. We are grateful to Tom Jones, Andre Leroi-Gourhan, Preston James, Goeffrey Martin, and their publishers for permitting the reproduction of their illustrations. Finally, it is a pleasure to thank the National Science Foundation for financial support. Ralph H. Abraham Christopher D. Shaw Santa Cruz, California October, 1991