Chaos In Discrete Dynamical Systems. A Visual Introduction In 2 Dimensions

Chaos in Discrete Dynamical Systems A VISUAL INTRODUCTION IN 2 DIMENSIONS RALPH H. ABRAHAM University of California S.;mta Cruz LAURA GARDII NI University of Urbino CHRISTIAN MIIRA Institut National des Sciences Appliquees de Toulouse with 153 illustrations made with the assistance of Scott Hotton Companion CD-ROM by Ronald J. Record and Ralph H. Abraham EXIRA MATERIALS extras.springer.com Ralph H. Abraham University of California Santa Cruz P. O. Box 1378 Santa Cruz, CA 95064 USA Laura Gardini Institute di Scienze Econom iche Universita di Urbino Urbino 61029 ltaly Christian Mira Institut National des Sciences Appliquees de Toulouse Dept. of Control Engineering Toulouse 31077 France Library of Congress Cataloging-in-Publication Data Abraham, Ralph. Chaos in discrete dynamical systems :a visual introduction in 2 dimensions 1 Ralph H.Abraham, Laura Gardini, Christian Mira. p. cm. lncludes bibliographical references and index. Additional material to this book can be downloaded from http://extras.springer.com ISBN 978-1-4612-7347-9 ISBN 978-1-4612-1936-1 (eBook) DOI 10.1007/978-1-4612-1936-1 1. Differentiable dynamical systems. 2. Chaotic behavior in systems. 1. Gardini, L (Laura) 11. Mira, C. III. Title. QA614.8.A268 1997 003'.857-dc21 96-37581 © 1997 Springer Science+Business Media New York Originally published by Springer-Verlag New York, lnc. in 1997 Softcover reprint of the hardcover 1st edition 1997 FOREWORDTOTHE PROJECT You are looking at the outcome of a three-year project, a unique experiment in electronic publishing. For lack of a better word, we call this a package. It has three intertwined components: a book, a CDROM, and a website. It is perhaps the first such multimedia package devoted to an advanced branch of mathematics. The book is the primary component, and it is extensively illustrated with monochrome computer graphics. The CD-ROM is devoted mainly to twelve computer graphic animations in color, which animate and expand the graphics in the book. The user interface to the CD-ROM is made in the style, and with the technology, of the World Wide Web. Therefore, it integrates seamlessly with the website devoted to the book and CD-ROM, which is maintained at the Visual Math Institute. This website also connects outward with the resources of the World Wide Web. The motivation for this package is the conviction that this style of electronic publication is the ideal medium for mathematical communication, and especially for the branch of mathematics known as dynamical systems theory, including our subject: noninvertible discrete chaos theory in two dimensions. The essence of this communicattve style is the dynapic technique, in which a drawing is developed stroke-by-stroke, along with a carefully coordinated spoken commentary. This is the traditional method used by most mathematicians, when speaking among themselves: Visual Math! We will now introduce the three components separately. Ralph H. Abraham Santa Cruz, California October 1996 ix PREFACETOTHE BOOK This book is a visual introduction to chaos and bifurcations in noninvertible discrete dynamical systems in two dimensions, using the method of critical curves. Historical Background Dynamical systems theory is a classical branch of mathematics which began with Newton circa 1665. It provides mathematical models for systems which evolve in time according to a rule, originally expressed in analytical form as a system of ordinary differential equations. These models are called continuous dynamical systems. They are also called flows, as the points of the system evolve by flowing along continuous curves. In the 1880s, Poincare studied continuous dynamical systems in connection with a prize competition on the stability of th