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APPLIED NONLINEAR DYNAMICS Analytical, Computational, and Experimental Methods Ali H. Nayfeh Virginia Polytechnic Institute and State University Balakumar Balachandran University of Maryland WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA

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APPLIED NONLINEAR DYNAMICS

WILEY SERIES IN NONLINEAR SCIENCE Series Editors: Abdullaev Bolotin Jackson Kahn and Zarmi Moon (ed.) Nayfeh Nayfeh Nayfeh and Balachandran Nayfeh and Pai Ott, Sauer, andyorke Pfeiffer and Glocker Qu Vakakis et al. Yamamoto and Ishida ALI H. NAYFEH, Virginia Tech ARUN V, HOLDEN, University of Leeds Theory of Solitons in lnhomogeneous Media Stability Problems in Fracture Mechanics Exploring Nature s Dynamics Nonlinear Dynamics: Exploration through Normal Forms Dynamics and Chaos in Manufacturing Processes Method of Normal Forms Nonlinear Interactions: Analytical, Computational, and Experimental Methods Applied Nonlinear Dynamics Linear and Nonlinear Structural Mechanics Coping with Chaos Multibody Dynamics with Unilateral Contacts Robust Control of Nonlinear Uncertain Systems Normal Modes and Localization in Nonlinear Systems Linear and Nonlinear Rotordynamics: A Modem Treatment with Applications

All books published by Wiley-VCH are carehlly produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: Applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. 0 1995 by John Wiley & Sons, lnc. 0 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - nor transmitted or translated into machine language without written permission from the publishe ;. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of Germany Printed on acid-free paper Printing and Bookbinding buch bucher dd ag, Birkach ISBN-13: 978-0-47 1-59348-5 ISBN-10: 0-47 1-59348-6

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CONTENTS PREFACE xiii 1 INTRODUCTION 1 1.1 DISCRETE-TIME SYSTEMS.... 2 1.2 CONTINUOUS-TIMESYSTEMS... 6 1.2.1 Nonautonomous Systems... 6 1.2.2 Autonomous Systems... 11 1.2.3 Phase Portraits and Flows... 13 1.3 ATTRACTING SETS... 15 1.4 CONCEPTS OF STABILITY... 20 1.4.1 Lyapunov Stability... 20 1.4.2 Asymptotic Stability... 23 1.4.3 Poincarb Stability... 25 1.4.4 Lagrange Stability (Bounded Stability)... 27 1.4.5 Stability Through Lyapunov Function.... 27 1.5 ATTRACTORS... 29 1.6 COMMENTS... 31 1.7 EXERCISES... 31 2 EQUILIBRIUM SOLUTIONS 35 2.1 CONTINUOUS-TIME SYSTEMS... 35 2.1.1 Linearization Near an Equilibrium Solution... 36 2.1.2 Classification and Stability of Equilibrium Sohtions... 39 2.1.3 Eigenspaces and Invariant Manifolds... 47 2.1.4 Analytical Construction of Stable and Unstable Manifolds... 58 vii

viii CONTENTS 2.2 FIXEDPOINTSOFMAPS... 61 2.3 BIFURCATIONS OF CONTINUOUS SYSTEMS... 68 2.3.1 Local Bifurcations of Fixed Points.... 70 2.3.2 Normal Forms for Bifurcations... 81 2.3.3 Bifurcation Diagrams and Sets.... 83 2.3.4 Center Manifold Reduction... 96 2.3.5 The Lyapunov-Schmidt Method.... 108 2.3.6 The Method of Multiple Scales... 108 2.3.7 Structural Stability... 115 2.3.8 Stability of Bifurcations to Perturbations... 116 2.3.9 Codimension of a Bifurcation... 119 2.3.10 Global Bifurcations... 121 121 2.4 BIFURCATIONS OF MAPS... 2.5 EXERCISES... 128 3 PERIODIC SOLUTIONS 147 3.1 PERIODIC SOLUTIONS... 147 3.1.1 Autonomous Systems... 148 3.1.2 Nonautonomous Systems... 156 3.1.3 Comments... 158 3.2 FLOQUET THEORY... 158 3.2.1 Autonomous Systems... 159 3.2.2 Nonautonomous Systems... 169 3.2.3 Comments on the Monodromy Matrix... 171 3.2.4 Manifolds of a Periodic Solution... 172 3.3 POINCAREMAPS... 172 3.3.1 Nonautonomous Systems... 176 3.3.2 Autonomous Systems... 181 3.4 BIFURCATIONS... 187 3.4.1 Symmetry-Breaking Bifurcation... 189 3.4.2 Cyclic-Fold Bifurcation... 195 3.4.3 Period-Doubling or Flip Bifurcation... 200 3.4.4 Transcritical Bifurcation... 204 3.4.5 Secondary Hopf or Neimark Bifurcation... 205 3.5 ANALYTICAL CONSTRUCTIONS... 208 3.5.1 Method of Multiple Scales... 209 3.5.2 Center Manifold Reduction... 212

CONTENTS ix 3.5.3 General Case... 217 3.6 EXERCISES... 219 4 QUASIPERIODIC SOLUTIONS 231 4.1 POINCARE MAPS... 233 4.1.1 Winding Time and Rotation Number... 238 4.1.2 Second-Order Poincarb Map... 240 4.1.3 Comments... 241 4.2 CIRCLE MAP... 242 4.3 CONSTRUCTIONS... 248 4.3.1 Method of Multiple Scales... 249 4.3.2 Spectral Balance Method... 251 4.3.3 PoincarC Map Method... 253 4.4 STABILITY... 254 4.5 SYNCHRONIZATION... 255 4.6 EXERCISES... 269 5 CHAOS 277 5.1 MAPS... 278 5.2 CONTINUOUS-TIME SYSTEMS... 288 5.3 PERIOD-DOUBLING SCENARIO... 295 5.4 INTERMITTENCY MECHANISMS... 296 5.4.1 Type I Intermittency... 300 5.4.2 Type I11 Intermittency... 305 5.4.3 Type I1 Intermittency... 311 5.5 QUASIPERIODIC ROUTES... 314 5.5.1 Ruelle-Takens Scenario... 315 5.5.2 Torus Breakdown... 317 5.5.3 Torus Doubling... 331 5.6 CRISES... 334 5.7 MELNIKOV THEORY... 356 5.7.1 Homoclinic Tangles... 356 5.7.2 Heteroclinic Tangles... 5.7.3 Numerical Prediction of Manifold Intersections.. 363 5.7.4 Analytical Prediction of Manifold Intersections.. 366 5.7.5 Application of Melnikov s Method... 374 5.7.6 Comments... 390 359

X CONTENTS 5.8 BIFURCATIONS OF HOMOCLINIC ORBITS... 390 5.8.1 Planar Systems... 391 397 5.8.2 Orbits Homoclinic to a Saddle... 5.8.3 Orbits Homoclinic to a Saddle Focus... 402 5.8.4 Comments... 407 5.9 EXERCISES... 410 6 NUMERICAL METHODS 423 6.1 CONTINUATION OF FIXED POINTS... 423 6.1.1 Sequential Continuation... 425 6.1.2 Davidenko-Newton-Raphson Continuation... 428 6.1.3 Arclength Continuation... 428 6.1.4 Pseudo-Arclength Continuation... 432 6.1.5 Comments... 435 6.2 SIMPLE TURNING AND BRANCH POINTS... 436 6.3 HOPF BIFURCATION POINTS... 438 6.4 HOMOTOPY ALGORITHMS... 441 6.5 CONSTRUCTION OF PERIODIC SOLUTIONS... 445 6.5.1 Finite-Difference Method... 446 6.5.2 Shooting Method... 449 6.5.3 Poincard Map Method... 455 6.6 CONTINUATION OF PERIODIC SOLUTIONS... 455 6.6.1 Sequential Continuation... 456 6.6.2 Arclength Continuation... 456 6.6.3 Pseudo-Arclength Continuation... 458 6.6.4 Comments... 460 7 TOOLS TO ANALYZE MOTIONS 401 7.1 INTRODUCTION... 462 7.2 TIME HISTORIES... 465 7.3 STATE SPACE... 472 7.4 PSEUDO-STATE SPACE... 478 7.4.1 Choosing the Embedding Dimension... 483 7.4.2 Choosing the Time Delay... 495 7.4.3 Two or More Measured Signals... 500 7.5 FOURIER SPECTRA... 502 7.6 POINCARE SECTIONS AND MAPS... 514

CONTENTS Xi 7.6.1 Systems of Equations... 514 7.6.2 Experiments... 516 7.6.3 Higher-Order Poincark Sections... 519 7.6.4 Comments... 519 7.7 AUTOCORRELATION FUNCTIONS... 520 7.8 LYAPUNOV EXPONENTS... 525 7.8.1 Concept of Lyapunov Exponents... 525 7.8.2 Autonomous Systems... 529 7.8.3 Maps... 531 7.8.4 Reconstructed Space... 534 7.8.5 Comments.... 537 7.9 DIMENSION CALCULATIONS... 538 7.9.1 Capacity Dimension... 538 7.9.2 Pointwise Dimension... 541 7.9.3 Information Dimension... 545 7.9.4 Correlation Dimension... 547 7.9.5 Generalized Correlation Dimension... 548 7.9.6 Lyapunov Dimension... 549 7.9.7 Comments... 549 550 7.10 HIGHER-ORDER SPECTRA... 7.11 EXERCISES... 557 8 CONTROL 563 8.1 CONTROL OF BIFURCATIONS... 563 8.1.1 Static Feedback Control... 564 568 8.1.2 Dynamic Feedback Control... 8.1.3 Comments... 571 8.2 CHAOS CONTROL... 571 8.2.1 The OGY Scheme... 572 8.2.2 Implementation of the OGY Scheme... 577 8.2.3 Pole Placement Technique... 580 8.2.4 Traditional Control Methods... 582 8.3 SYNCHRONIZATION... 584 BIBLIOGRAPHY 589 SUBJECT INDEX 663

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PREFACE Systems that can be modeled by nonlinear algebraic and/or nonlinear differential equations are called nonlinear systems. Examples of such systems occur in many disciplines of engineering and science. In this book, we deal with the dynamics of nonlinear systems. PoincarC (1899) studied nonlinear dynamics in the context of the n-body problem in celestial mechanics. Besides developing and illustrating the use of perturbation methods, PoincarC presented a geometrically inspired qualitative point of view. In the nineteenth and twentieth centuries, many pioneering contributions were made to nonlinear dynamics. A partial list includes those due to Rayleigh, Duffing, van der Pol, Lyapunov, Birkhoff, Krylov, Bogoliubov, Mitropolski, Levinson, Kolomogorov, Andronov, Arnold, Pontryagin, Cartwright, Littlewood, Smale, Bowen, Piexoto, Ruelle, Takens, Hale, Moser, and Lorenz. While studying forced oscillations of the van der Pol oscillator, Cartwright and Littlewood (1945) observed a constrained random-like behavior, which is now called chaos. Subsequently, Lorenz (1963) studied a deterministic, third-order system in the context of weather dynamics and showed through numerical simulations that this deterministic system displayed random-like behavior too. Unaware of Lorenz s work, Smale (19G7) introduced the horseshoe map as an abstract prototype to explain chaos-like behavior. No doubt PoincarC knew about chaos too, but it is only through numerical simulations on modern computers and experiments with physical system that the presence of chaos has been discovered to be pervasive in many dynamical systems of physical interest. The observation of Poincar6 that small differences in the initial conditions may produce great changes in the final phenomena is now known to be a characteristic of systems that xiii