Bifurcation and Chaos in Engineering

Springer London Berlin Heidelberg New York Barcelona Budapest Hong Kong Milan Paris Santa Clara Singapore Tokyo

Yushu Chen and Andrew Y.T. Leung

Bifurcation and Chaos in Engineering

With 235 Figures

, Springer

Professor Yushu Chen, PhD Department of Mechanics, Tianjin University, Tianjin, China 300072

Professor Andrew Y.T. Leung, DSc, PhD, CEng, FRAeS, MIStructE, MHKIE Manchester School of Engineering, Simon Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK

ISBN -13:978-1-4471-1577 -9 e-ISBN -13:978-1-4471-1575-5 DOl: 10.1007/978-1-4471-1575-5

British Library Cataloguing in Publication Data Chen, Yushu

Bifurcation and chaos in engineering 1. Engineering mathematics 2. Differential dynamical systems 3. Bifurcation theory 4. Chaotic behavior in systems I. Title II. Leung. Andrew 620'.0015'1

ISBN -13 :978-1-4471-1577-9

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of repro graphic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.

© Springer-Verlag London Limited 1998 Softcover reprint of the hardcover 1st edition 1998

The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use.

The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.

Typesetting: Camera-ready by authors

69/3830-543210 Printed on acid-free paper

Preface

For the many different deterministic non-linear dynamic systems (physical, mechanical, technical, chemical, ecological, economic, and civil and structural engineering), the discovery of irregular vibrations in addition to periodic and almost periodic vibrations is one of the most significant achievements of modern science. An in-depth study of the theory and application of non-linear science will certainly change one's perception of numerous non-linear phenomena and laws considerably, together with its great effects on many areas of application. As the important subject matter of non-linear science, bifurcation theory, singularity theory and chaos theory have developed rapidly in the past two or three decades. They are now advancing vigorously in their applications to mathematics, physics, mechanics and many technical areas worldwide, and they will be the main subjects of our concern.

This book is concerned with applications of the methods of dynamic systems and subharmonic bifurcation theory in the study of non-linear dynamics in engineering. It has grown out of the class notes for graduate courses on bifurcation theory, chaos and application theory of non-linear dynamic systems, supplemented with our latest results of scientific research and materials from literature in this field. The bifurcation and chaotic vibration of deterministic non-linear dynamic systems are studied from the viewpoint of non-linear vibration. It is advantageous to do so, because on the one hand both have similar concepts and methods, and on the other hand, the explanation of theoretical methods with a view towards application facilitates not only those engaged in the study the work of vibration but also those devoted to studies in other fields. One can use this book as a reference work on dynamical systems. Although this book is far from an exhaustive monograph on non­linear dynamical and chaos in engineering systems, the authors hope that it will serve as a systematic and practical aid to those who carry out teaching and research work on bifurcation and chaos of non-linear vibration systems in the fields of mathematics, physics, mechanics, ecology, engineering and technology with a stress on civil and structural engineering.

This book can be divided into four main parts. The first part describes the theory of dynamic systems, encompassing Chapter I - dynamical systems, ordinary differential equations and stability of motion, Chapter 2 - calculations of flow, and Chapter 3 - discrete dynamic systems. The second part presents the main methods of the bifurcation theory. The Liapunov-Schmidt method, centre manifold theorem, vector fields, normal form theory, averaging method, singularity theory, Hopf bifurcation theory and the bifurcation theory of 112 subharmonic resonance of non­linear parametrically excited vibration systems, and Euler's buckling problems - are all dealt with in Chapters 4 to 7. The third part consists of Chapter 8, dealing with chaos theory and its application, and Chapter 9, dealing with the construction of chaotic regions. The fourth part consists of Chapter 10, which describes the numerical methods in non-linear dynamics, and Chapter 11, which finally presents some important engineering practice examples of application of bifurcation theory.

We are greatly indebted to Professors William F Langford, Wang Zhaolin and Huang Kelei and Doctors TC Fung, T Ge, SG Mao and SK Chui for their valuable contributions and constructive opinions in the preparation of this book. Thanks are

vi Bifurcation and Chaos in Engineering

also extended to Wang Deshi, Cao Qingjie, Zhang Weiyi, Hu Jindong, Leung Hau Yan and others in the group of the theory and application of non-linear dynamical systems in the Department of Mechanics, Tianjin University and the Department of Civil Engineering, Hong Kong University for their work and comments in the preparation of manuscripts. The secretary support of Mr Li Weidong and Ms B Knight of the University of Manchester in the final production of the camera ready copy of the book is gratefully acknowledged. We are also thankful to the helping hands of the staff of Springer-Verlag to make the publication of the book possible.

Finally, A YTL is grateful to the support of his family, Anna, Colin, Edwin and Johanna throughout the preparation the book.

Introduction

Detenninistic motion and the transient process of dynamic systems from one detenninistic motion to another occur by means of vibrations.

A detenninistic motion is one that has repeatability and a certain stability, and the transient process is the whole process that instigates this detenninistic motion. The set of transient processes constitutes the attraction domain of the motion. When a physical parameter in the system under study changes to a certain value, the solution curve branches out a family of curves. This is called bifurcation. If the detenninistic motion changes rapidly enough (i.e. jumps exist), the new state of motion is considered to be caused by hard elements, otherwise by soft elements. Phenomena arising in such a non-linear system are named non-linear vibrations.

Widely used methods of the theory of non-linear vibration, or methods of finding a periodic solution, are the perturbation method, the averaging method and the numerical method. They have developed rapidly since the 1920s. These methods, however, were used in the past only in studying the periodic solution when the parameters in the vibration system are given constants. But in engineering practice some parameters of a system often may be subjected to small variations (also referred to as perturbation). The speed of a vehicle running on a straight rail, the temperature and density in a chemical process, and the damping coefficient and frequency of the excitation in parametrically excited system are only a few examples, and the perturbation of these parameters often tends to cause the appearance of bifurcation of the periodic solutions.

The bifurcation theory penneates different fields of engineering and natural science. Differential equations describing physical systems often contain parameters whose measured values fluctuate within a very small range. If the differential equation simulating a physical system is structurally unstable when the parameters reach some values, then a change will appear in quality in the behaviour of the solution (i.e. bifurcation) when a small variation takes place the right-hand side of the differential equation. As a result, it is necessary to know how the phase portraits respond to the change of parameters.

It is common to describe a mechanical model by means of a huge group of differential equations, for example, a system with numerous variables. In order to proceed or simplify the analysis, the common practice is to regard those variables which change very slightly in the dynamic process as constants, or to omit some of the variables which affect only minor factors. Yet it is often impossible to evaluate the effects of the dropped tenns on the original model according to the simplified equation. In this case, if the omitted tenns are taken as perturbations, they can be dealt with by singularity theory.

The parameter family which varies very slowly with time is tenned a slowly varying family. The theory of relaxation vibration is closely associated with the bifurcation theory in which parameters do not vary with time. There are slowly varying parameters in the "slow-fast" system of the relaxation vibration. When the rate of change of slowly varying parameters is zero, the "slow-fast" system becomes the bifurcation system mentioned previously; when the rate is not zero, a special phenomenon called "dynamic bifurcation" takes place.

Vlli Bifurcation and Chaos in Engineering

Not only does the bifurcation theory interest researchers with its profound and active topics but also its close ties with practical engineering problems reveal its importance. Complicated theoretical bifurcation problems can be found in the snake motion of high-speed vehicles, heat convection in fluid, vibration of pipes and related problems, oscillations in chemical reactions, dynamic buckling of the Euler rod, oscillation of chain bridges, bifurcation of shock waves, bifurcation in the neural network in human bodies, bifurcation in superconductors, bifurcation of dynamic systems in sociology and economics, and so on.

The singularity theory can be used to simplify the form of the bifurcation equations. It can determine the minimum number of unfolding parameters from among numerous control parameters.

Of the many different deterministic non-linear dynamic systems (physical, mechanical, technical, chemical, biological, economic) the discovery of irregular vibrations in addition to periodic and almost-periodic vibrations is one of the important scientific discoveries in recent years. The reason why irregular vibrations (chaotic) are studied from a viewpoint of the vibration theory is mainly that they have similar concepts and methods.

Chaos is a motion occurring over a limited range in the deterministic system. With no regularity, such a motion is similar to a stochastic one and extremely sensitive to the initial value (that is, a small change in the initial value results in a completely different response - it may fall into Liapunov's instability). In other words, chaotic motion has unpredictability over a long time. The deterministic systems are those that can usually be described by differential equations, partial differential equations, difference equations and even simple iterative equations. The coefficients in these equations are deterministic.

In a word, chaotic motion is an unstable bounded stationary motion (i.e. locally unstable and wholly compressible). This definition unfolds the two aspects of chaotic motion: instability (this character can be made exact when the Liapunov's averaging exponent is greater then zero) and finiteness. Or, chaotic motion is a bounded stationary motion without equilibrium, periodicity and almost-periodicity. Here the bounded stationary motion means that the state of motion does not vary with time in a stochastic sense (from the viewpoint of the finite domain as a whole in phase space).

The appearance of bifurcation in non-linear systems and global bifurcation, in particular, will lead to chaotic motion in many cases. The bifurcation theory is one of the theories which explain how the resolvable system transits from a normal orderly state to a chaotic state. As H. Poincare well says: the bifurcation theory is a torch, it lights up the road leading from systems which can be studied to those that cannot be studied. Chaos is different from the stochastic motion in that it has definite modes of motion and fine structures, e.g. self-resemblance structure.

Taking advantage of this character of the bifurcation theory, Landau and Hopf gave a picture of the transition from laminar to turbulent flow when the Reynolds number increases. In Landau's description, this transition is realised through the tori of the increased number of dimensions. Since the discovery of the strange attractor by Lorenz in 1963, four typical routes leading to chaos have been known. They are periodic doubling bifurcation, intermittent transition, secondary Hopf bifurcation and the breaking of the KAM (Kolmogorov-Amold-Mosor) torus. The mechanisms of

Introduction ix

the first three have all been clear in theory, but that of the fourth remains to be solved. Another important mechanism leading to chaos concern homoclinic and heteroclinic bifurcations. At this point, the stable and unstable manifolds of the saddle points will intersect with each other under disturbances, resulting in a hyperbolic limit set - the Smale horseshoe. This is thought to be one of the basic structures of chaos by mathematicians. In 1963 Milnikov proposed a method of analyzing homo clinic bifurcation. Though important advances have been made in the study on chaos, little has been reported of the chaos in high and infinite dimensional systems. The main difficulty is that it is hard to imagine the visual structure of the stable and unstable manifolds in the high dimensional systems.

The ubiquitous presence of bifurcation and chaos in different fields of engineering and natural sciences gives impetus to the rapid development of the theory of dynamic systems.

At the tum of this century Poincare and others evolved the concept of dynamic systems from the study of classical mechanics and qualitative theory of differential equations. Instead of integrating the differential equations, he took the right-hand side of the equations as a defined vector field and through the form of the known vector field he explored the solution curve family defined by differential equations. (Taking a curve as the path moved through by a particle, we assume that every particle on the curve moves simultaneously, and in this context we call the solution curve an orbit or a trajectory.) Poincare introduced such concepts as equilibrium state, periodic solution, recurrence and studied the behaviour of a solution developing with time as a whole. Poincare's innovation signifies that he regards the state variable of a system not only as a function of the time, but also as a function of the initial conditions, i.e. the continuously varying structure of flow is the total sum of curves passing through all the points in definition domain to value domain.

Modem research work on the differential dynamic system originated from the work of M. M. Peixoto and others in the 1960s. Thanks to the advocacy and promotion of scholars such as V.I. Arnold, S. Smale and many others, research of basic theories of this discipline has made important progress. However, such new tools were only at the disposal of mathematicians before the mid-1970s. In the past decade the study of differential dynamic systems has spread extensively to many different areas of application.

The two main developing directions of dynamic systems are the study of orderly (or regular) motion and chaotic (or irregular) motion. The orderly and disorderly motions of the discrete system are merely behaviours of time, its disorderly behaviour is only time chaos with its geometrical feature being the strange attractor; the orderly and disorderly motions of the continuous system are behaviours of both time and space. Turbulent flow, being a continuous system, is chaos of space as well as chaos of time (its geometrical feature is a saddle invariant set). They are associated with each other, but one cannot be derived from the other.

This book consists of eleven chapters and an introduction.

Chapter 1 introduces the basic concepts of dynamic systems, the important properties of flow, Poincare -Bendixon's theorem, a brief description of the main theories of ordinary differential equations and stability of motion.

x Bifurcation and Chaos in Engineering

Chapter 2 treats the calculation of flow: first the divergence of flow, then the calculation of linear flow, non-linear differential equations and the calculation of its flow, and finally the stable manifold theorem.

Chapter 3 discusses discrete dynamic systems, introducing discrete dynamic systems and the linear map, the non-linear map and the stable manifold theorem of the map, the classification of the generic map, the Poincare map and the structural stability of the vector field.

Chapter 4 deals with the Liapunov-Schmidt reduction, the concepts of bifurcation, the implicit function theorem and singularity theory. Examples include the one-half subharmonic resonance bifurcation of non-linear parametrically excited vibration systems and Ropfbifurcation.

Chapter 5 is focused on the centre manifold theorem and normal form theory of a vector field by the matrix method.

Chapter 6 is devoted to Ropf bifurcation as an example of application of the centre manifold theorem and normal form theory, the complex normal form of Ropf bifurcation, the real normal form, the calculation formulae of the analytical method of the stability of the solution, and lastly, bifurcation problems of systems with double zero eigenvalues.

Chapter 7 introduces the application of the averaging method in bifurcation theory, the averaging method for systems with many degrees of freedom, the geometrical description of the averaging method, averaging method and local bifurcation, and averaging method and global bifurcation.

Chapter 8 presents a brief description of chaos, examples of chaos in non­linear systems, the method of study and the statistical character of chaos.

Chapter 9 deals with the construction of chaotic regions. A new method of numerical simulation is given to describe the characteristics of the solutions of dynamic systems. The characteristics of the solutions, such as the number of solutions, the type and periodicity of the solutions, and, more importantly, the existence of chaotic solutions in a physical parametric space are of interest.

Chapter 10 introduces some numerical methods in common use, such as the construction of normal form, symplectic numerical integration and Toeplitz matrices.

Chapter 11 presents the application of bifurcation theory to non-linear structural dynamics, in which are included the bifurcation analysis of oscillations with piecewise-linear characteristics and other applications of bifurcation theory.

Contents

Chapter 1 Dynamical Systems, Ordinary Differential Equations and Stability of Motion 1

1.1 Concepts of Dynamical Systems 1 1.2 Ordinary Differential Equations 5 1.3 Properties of Flow 14 1.4 Limit Point Sets 17 1.5 Liapunov Stability of Motion 23 1.6 Poincare-Bendixson Theorem and its Applications 29

Chapter 2 Calculation of Flows 35 2.1 Divergence of Flows 35 2.2 Linear Autonomous Systems and Linear Flows

and the Calculation of Flows about the IVP 38 2.3 Hyperbolic Operator (or Generality) 47 2.4 Non-linear Differential Equations and the Calculation of their Flows 55 2.5 Stable Manifold Theorem 60

Chapter 3 Discrete Dynamical Systems 66 3.1 Discrete Dynamical Systems and Linear Maps 66 3.2 Non-linear Maps and the Stable Manifold Theorem 68 3.3 Classification of Generic Systems 71 3.4 Stability of Maps and Poincare Mapping 73 3.5 Structural Stability Theorem 76

Chapter 4 Liapunov-Schmidt Reduction 84 4.1 Basic Concepts of Bifurcation 84 4.2 Classification of Bifurcations of Planar Vector Fields 88 4.3 The Implicit Function Theorem 91 4.4 Liapunov-Schmidt Reduction 93 4.5 Methods of Singularity 102 4.6 Simple Bifurcations 119 4.7 Bifurcation Solution of the 112 Subharmonic Resonance Case of

Non-linear Parametrically Excited Vibration Systems 127 4.8 HopfBifurcation Analyzed by Liapunov-Schmidt Reduction 143

Chapter 5 Centre Manifold Theorem and Normal Form of Vector Fields 154 5.1 Centre Manifold Theorem 154 5.2 Saddle-Node Bifurcation 166 5.3 Normal Form of Vector Fields 169

Chapter 6 Hopf Bifurcation 6.1 Hopf Bifurcation Theorem 6.2 Complex Normal Form of the HopfBifurcation 6.3 Normal Form of the HopfBifurcation in Real Numbers 6.4 Hopf Bifurcation with Parameters 6.5 Calculating Formula for the HopfBifurcation Solution

176 176 179 182 185 192

xii Bifurcation and Chaos in Engineering

6.6 Stability of the HopfBifurcation Solution 194 6.7 Effective Method for Computing the HopfBifurcation

Solution Coefficients 198 6.8 Bifurcation Problem Involving Double Zero Eigenvalues 203

Chapter 7 Application of the Averaging Method in Bifurcation Theory 230 7.1 Standard Equation 230 7.2 Averaging Method and Poincare Maps 237 7.3 The Geometric Description of the Averaging Method 241 7.4 An Example of the Averaging Method-the Duffing Equation 248 7.5 The Averaging Method and Local Bifurcation 255 7.6 The Averaging Method, Hamiltonian Systems

and Global Behaviour 261

Chapter 8 Brief Introduction to Chaos 265 8.1 What is Chaos? 265 8.2 Some Examples of Chaos 268 8.3 A Brieflntroduction to the Analytical Method of Chaotic Study 273 8.4 The Hamiltonian System 289 8.5 Some Statistical Characteristics 303 8.6 Conclusions 305

Chapter 9 Construction of Chaotic Regions 311 9.1 Incremental Harmonic Balance Method (IHB Method) 312 9.2 The Newtonian Algorithm 317 9.3 Number of Harmonic Terms 318 9.4 Stability Characteristics 318 9.5 Transition Sets in Physical Parametric Space 319 9.6 Example of the Duffing Equation with Multi-Harmonic Excitation 320

Chapter 10 Computational Methods 341 10.1 Normal Form Theory 341 10.2 Symplectic Integration and Geometric Non-Linear Finite

Element Method 359 10.3 Construction of the Invariant Torus 375

Chapter 11 Non-linear Structural Dynamics 399 11.1 Bifurcations in Solid Mechanics 399 11.2 Non-Linear Dynamics of an Unbalanced Rotating Shaft 406 11.3 Galloping Vibration Analysis for an Elastic Structure 421 11.4 Other Applications of Bifurcation Theory 431

References 436 fud6 ~1