CHAOS Bifurcations ond Fractals flround Us A brief introduction WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A. MONOGRAPHS AND TREATISES Volume 28: Applied Nonlinear Dynamics & Chaos of Mechanical Systems with Discontinuities Edited by M. Wiercigroch & B. de Kraker Volume 29: Nonlinear & Parametric Phenomena* V. Damgov Volume 30: Quasi-Conservative Systems: Cycles, Resonances and Chaos A. D. Morozov Volume 31: CNN: A Paradigm for Complexity L O. Chua Volume 32: From Order to Chaos II L P. Kadanoff Volume 33: Lectures in Synergetics V. I. Sugakov Volume 34: Introduction to Nonlinear Dynamics* L Kocarev & M. P. Kennedy Volume 35: Introduction to Control of Oscillations and Chaos A. L Fradkov & A. Yu. Pogromsky Volume 36: Chaotic Mechanics in Systems with Impacts & Friction B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak & J. 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Lamarque *Forthcoming * I WORLD SCIENTIFIC SERIES ON p" . . . . . NONLINEAR SCIENCE series A vol. 47 Series Editor: Leon 0. Chua CHAOS Bifurcations and Fractals firound Us A brief introduction Wanda szemplinska-stupnicka Institute of Fundamental Technological Research, Polish Academy of sciences World Scientific NEWJERSEY • LONDON • SINGAPORE • SHANGHAI • HONGKONG • TAIPEI • BANGALORE Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. CHAOS, BIFURCATIONS AND FRACTALS AROUND US Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic ormechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-238-689-0 This book is printed on acid-free paper. Printed in Singapore by Mainland Press Contents 1. Introduction 1 2. Ueda's "Strange Attractors" 5 3. Pendulum 11 3.1. Equation of motion, linear and weakly nonlinear oscillations 11 3.2. Method of Poincare map 18 3.3. Stable and unstable periodic solutions 20 3.4. Bifurcation diagrams 24 3.5. Basins of attraction of coexisting attractors 28 3.6. Global homoclinic bifurcation 33 3.7. Persistent chaotic motion — chaotic attractor 39 3.8. Cantor set — an example of a fractal geometric object 46 4. Vibrating System with Two Minima of Potential Energy 49 4.1. Physical and mathematical model of the system 50 4.2. The single potential well motion 53 4.3. Melnikov criterion 57 4.4. Fractal boundaries of basins of attraction and transient chaos in the region of principal resonance 62 4.5. Oscillating chaos and unpredictability of the final state after destruction of the resonant attractor 71 4.6. Boundary crisis of the oscillating chaotic attractor 75 4.7. Persistent cross-well chaos 79 4.8. Lyapunov exponents 82 4.9. Intermittent transition to chaos 84 4.10. Large Orbit and the boundary crisis of the cross-well chaotic attractor 87 4.11. Various types of attractors of the two-well potential system 94 5. Closing Remarks 98 Bibliography 101 Index 105 v Chapter 1 Introduction When we observe evolution in time of various phenomena in the macroscopic world that surrounds us, we often use the terms "chaos", or "chaotic", meaning that the changes in time are without pattern and out of control, and hence are unpredictable. The most frustrating phenomena are those, which concern long-term weather forecasting. We can never be sure about the change of weather patterns. The temperature, barometric pressure, wind direction, amount of precipitation and other important factors come as a surprise contradicting predictions made a few days ago. Sometimes we are caught in a storm, sometimes in a heat-wave. The world stock market prices are also an example of a system that fluctuates in time in a random-like, irregular way, and the long-term prognosis does not often come true. The two examples mentioned belong to the category of huge and complicated dynamical systems, with a huge number of variables. The unpredictability of the evolution in time of these interesting events is intuitively natural. Simultaneously, it also seems natural that evolution of physical processes in simple systems, the systems governed by simple mathematical rules, should be predictable far into the future. Suppose we consider a small heavy ball, which can move along a definite track, so that the position of the ball is determined by a single coordinate. Due to Newton's Second Law, the motion of the ball is governed by the second order differential equation. The well known physical system the pendulum belongs to this class of oscillators. We were told that if the forces acting on the ball, as well as its initial position and velocity are given, one could predict the motion, i.e. the 1 2 Chaos, Bifurcations and Fractals Around Us history of the system forever into the future, at least if the powers of our computers are big enough. The scientific researchers were taken by surprise, some of them were unable to agree with the idea that even this type of system may exhibit an irregular motion, sensitive to initial conditions and though unpredictable in time, the motion is labeled as chaotic. This book is aimed at presenting and exploring the chaotic phenomena in the single-degree-of-freedom, nonlinear driven oscillators. The oscillators considered belong to the class of dissipative deterministic dynamical systems. The term "dissipative" means that drag forces act on the ball during motion (aerodynamic forces, friction forces and others), so that the free oscillations always decay in time, and the undriven system tends to its equilibrium position. The other essential feature is that all the forces acting on the ball are determined in time. Such systems are labeled deterministic. For a long time, researchers were deeply convinced that deterministic systems always give a deterministic output. Early discovery of chaotic output in deterministic systems came into view in the field of mathematical iteration equations of the type x \=f(x ), n = 0,1,2, n+ n The formula states that the quantity x at the "instant of time" denoted n+\ can be calculated, if the previous quantity x is known. Interpretation n of the parameter n as "instant of time" is useful in applications. One of the fundamental models of this type has its roots in ecology. Ecologists wanted to know the population growth of a given species in a controlled environment, and to predict the long-term behavior of the population. One of the simple rules used by ecologists is the logistic equation x i = kx(l -x ), n = 0,1,2, n+ n n Here, the "instants" n = 0, 1, 2,.... correspond to the end of each generation. Using this formula one can deduce the population in the succeeding generation x +i from the knowledge of only the population in n the preceding generation x and the constant k. The results obtained for a n wide range of values of the constant k were surprising. As long as k did not exceed the value of about 3.5, the behavior of the population changed in a regular way. But at higher ^-values, in particular within the interval ~3.6 < k < 4.0, strange results were obtained. Namely, the consecutive Introduction 3 values xo, xj, x , , x i looked like an irregular, random-like process, 2 n+ whose essential property was that the fluctuations were sensitive to the initial value x . 0 Dynamical systems generated by the iterative formulae belong to the category of dynamical systems with discrete time. In contrast, the physical systems governed by differential equations are labeled as dynamical systems with continuous time. In the latter case, the sought changes in time of the values of position and velocity can be found by numerical integration of the equation of motion. Indeed, we can apply a numerical procedure that enables us to obtain discrete values of position and velocity. For instance, we may record the sought values in selected instants of time, say, at intervals equal to the period of excitation T. Thus, a series of sought quantities in the discrete time 0, T, 2T, 3T, , nT, .... would be obtained. Yet, we are not able to find an analytical iterative formula for the relation between the position and velocity values at the instant n+l as function of the previous values. That is why the analytical results obtained by mathematicians for dynamical systems with discrete time are not always applicable in the continuous time systems. Yet, the fundamental new concepts of nonlinear dynamics are common for both types of systems. The book is addressed to general Readers, also to those who, although are interested in the fascinating chaotic phenomena encountered in our every day life, do not have a solid mathematical background. To make the book easily accessible, we try to reduce the mathematical approach to minimum, and to apply a simplified version of presentation of the very complex chaotic phenomena. The Reader may even skip the portions of material where equations of motion are derived, and confine his/her attention to the presented physical model. Instead of a mathematical approach, the book is based on geometric interpretation of numerical results. The effort is focused on an explanation of both the theoretical concepts and the physical phenomena, with the aid of carefully selected examples of computer graphics.