Chaotic Modelling and Simulation Analysis of Chaotic Models, Attractors and Forms Christos H. Skiadas Charilaos Skiadas Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-7900-5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher can- not assume responsibility for the validity of all materials or the consequences of their use. 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CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Skiadas, Christos H. Chaotic modelling and simulation: analysis of chaotic models, attractors and forms / Christos H. Skiadas, Charilaos Skiadas. p. cm. Includes bibliographical references and index. ISBN 978-1-4200-7900-5 (hardcover : alk. paper) 1. Chaotic behavior in systems. 2. Mathematical models. I. Skiadas, Charilaos. II. Title. Q172.5.C45S53 2009 003’.857--dc22 2008038183 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Preface “Is there a chaotic world? To what extent? Mathematics, geometry or simply chaos?” The above questions arise time and time again when dealing with chaotic phenom- ena and the related attractors and chaotic forms. During the lengthy preparation of this manuscript, computers were used in the same way that geometers of old used geometric tools. Simple or complicated ideas are designed, estimated and tested us- ing this new tool. The result is more than 500 graphs and illustrations that fill the pages in front of you. Verbal explanations are kept to a minimum, and a lot of effort is devoted to ensur- ing that the presentation of ideas progresses from the most elementary to the most advanced, in a clear and comprehensible way, while requiring little previous knowl- edge of mathematics. In this way, we hope that a more general audience will benefit from the material of this book, especially from the large variety of chaotic attractors included. A lot of new material is presented alongside the classical forms and attractors appearing on the literature on chaos. However, most of the illustrations are based on new simulation methods and techniques. A rather lengthy introduction provides the reader with an overview of the subsequent chapters, and the interconnection between them. Our inspirations came from various disciplines, including geometry from the an- cient Greek and Alexandrian period (a great deal of which is taught in the Greek school system), mathematics from the developments of the last centuries, astronomy and astrophysics from recent developments (the Conference on Chaos in Astronomy organized by George Contopoulos in 2002 was very stimulating) and the amazing chaotic illustrations of Hubble and Chandra and the vortex movements from fluid flow theory. Chaotic theory is developing in a new way that influences the world around us, and consequently also influences our ways of approaching, analyzing and solving problems. It is not surprising that one of the central models in the chaos literature, the Henon-Heiles´ model, is presented in a paper with the title “The applicability of the third integral of motion: Some numerical experiments.” Numerical experiments in 1964 were the basis for many significant changes in astronomy in the decades that followed. In 1963 Edwin Lorenz, in his pioneering work on “Deterministic Nonpe- riodic Flow,” proposed a more prominent title for chaotic modelling, by including the term “deterministic.” His work spearheaded numerous studies on chaotic phe- nomena. Fifteen years later, Feigenbaum explored the rich chaotic properties of the simple discrete Logistic model, known since 1845 following the work of Verhulst on the continuous version of this model, and introduced scientists to the chaotic field. Our intention in this work was to present the main models developed by the pi- oneers of chaos theory, along with new extensions and variations to these models. The plethora of new models that can arise from the existing ones through analysis and simulation was surprising. This book is suitable for a wide range of readers, including systems analysts, math- ematicians, astronomers, engineers and people in any field of science and technology whose work involves modelling of systems. It is our hope that this book will prompt more people to become involved in the rapidly advancing field of chaotic models and will inspire new developments in the field. Christos H. Skiadas Charilaos Skiadas List of Figures 1.1 The (x, y) diagram . 3 1.2 Continuous and discrete Rossler¨ systems . 4 1.3 Bifurcations of the logistic and the Gaussian . 5 1.4 Delay models . 6 1.5 The Henon´ and Holmes attractors . 7 1.6 The Lorenz and autocatalytic attractors . 8 1.7 A chemical reaction attractor . 9 1.8 Non-chaotic portraits . 9 1.9 A rotation chaotic map . 10 1.10 The Ikeda attractor . 11 1.11 Ikeda attractors . 12 1.12 Translation-reflection attractors . 13 1.13 Symmetric forms . 14 1.14 Separation of chaotic forms . 15 1.15 Flowers . 16 1.16 Orbits in two-body motion . 17 1.17 Motion in the plane . 17 1.18 Motion in space . 18 1.19 Ring systems . 19 1.20 Spiral galaxies . 20 1.21 Spiral galaxies . 21 1.22 Reflection forms . 22 1.23 Spiral Patterns . 23 1.24 Henon-Heiles´ systems . 24 1.25 A two-armed spiral galaxy . 25 1.26 Chaotic paths in the Henon-Heiles´ system (E = 1/6) . 26 1.27 The Contopoulos system: orbits and rotation forms at resonance ratio 4/1 (left) and 2/3 (right) . 27 2.1 The solution to the logistic equation . 35 2.2 The two Gompertz models . 37 2.3 Chaotic forms . 41 2.4 A snail’s pattern . 42 2.5 Chaos as randomness . 43 3.1 Tracing iterations . 48 3.2 Second order stationary points . 49 3.3 The logistic and the inverse logistic map for b = 3.5 . 50 3.4 The third order cycle of the logistic and the inverse logistic map for b = 1 + √8.............................. 50 3.5 Chaotic region for the logistic and the inverse logistic map at b = 3.9.................................. 51 3.6 The logistic map . 53 3.7 The logistic map, b 3 ....................... 54 3.8 The logistic map, b >≤ 3 ....................... 54 3.9 Order-4 bifurcation of the logistic map; real time and phase space diagrams . 55 3.10 Order-8 bifurcation of the logistic map; real time and phase space diagrams . 56 3.11 Order-16 bifurcation and real time and phase space diagrams of the logistic . 56 3.12 Order-32 bifurcation and real time and phase space diagrams of the logistic . 56 3.13 The order-5 bifurcation and real time and phase space diagrams . 57 3.14 The order-6 bifurcation and real time and phase space diagrams . 57 3.15 The bifurcation diagram of the logistic . 59 3.16 Order-3 bifurcations . 60 3.17 The order-3 bifurcation . 61 3.18 The May model: an order-4 bifurcation (left) and the bifurcation diagram (right) . 62 3.19 Bifurcation diagrams of the Gaussian model . 63 3.20 Bifurcation diagram of the Gaussian model (a = 17) . 63 3.21 Bifurcation diagram of the G-L model . 63 3.22 Cycles in the G-L model . 64 3.23 Patent applications in the United States . 65 3.24 The GRM1 model . 67 3.25 Spain, oxygen steel process (1968–1980) . 69 3.26 Italy oxygen steel process (1970–1980) . 70 3.27 Luxemburg oxygen steel process (1962–1980) . 71 3.28 Bulgaria oxygen steel process (1968–1978) . 72 3.29 The modified logistic model (ML) . 73 3.30 The modified logistic model (ML) for b = 1.3 and e = 0.076 . 73 3.31 A chaotic stage of the modified logistic model (ML) for b = 1.3 and e = 0.073 . 73 3.32 A third order cycle of the GRM1 model for b = 1.3 and σ = 0.025 74 3.33 A third order cycle of the GRM1 model for b = 1.2 and σ = 0.01175 . 75 3.34 A third order cycle of the modified logistic model for b = 1.3 and e = 0.0719 .