We wish to thank the members of our families for their continuing support, suggestions, tolerance, and their humoring of our fluctuat­ing moods as this multi-faceted text, which combines theoretical con­cepts, computer usage, and experimental verification, evolved from an idea to reality.

The Enns Family Karen Russell Jennifer Heather Nicole Robert Justine Gabrielle

The MCGuire Family Lynda Colleen Sheelo Michael Serag Kevin Ruth

Richard H. Enns George C. McGuire

Nonlinear Physics with Mathematica for Scientists and Engineers

EXIRA MATERIALS

extras.springer.com

Springer Science+Business Media, LLC

Richard H. Enns Department of Physics Simon Fraser University Bumaby, BC V5A 1S6 Canada

George C. McGuire Department of Physics University College of the Fraser Valley Abbotsford, BC V2S 7M9 Canada

Library of Congress Cataloging-in-Publication Data

Enns, Richard H. Nonlinear physics with Mathematica for scientists and engineers I Richard H. Enns and

George C. McGuire. p.cm.

Includes bibliographical references and index. ISBN 978-1-4612-6664-8 ISBN 978-1-4612-0211-0 (eBook) DOI 10.1007/978-1-4612-0211-0

1. Nonlinear theories-Data processing. 2. Mathematical physics-Data processing. 3. Mathematica (Computer ftle) 1. McGuire, George, 1940- II. Title.

QC20.7.N6.E57 2001 530.1S-dc21

AMS Subject Classifications: OOA79,70Kxx

Printed on acid-free paper © 2001 Springer Science+Business Media New York Originally published by Birkhäuser Boston in 2001

2001035590 CIP

Ali rights reserved. This work may not be translated or copied in whole or in part without the written pennission ofthe publisher Springer Science+Business Media. LLC. except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information stomge and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe former are not especially identified, is not to be taken as a sign that such names, as understood by the Tmde Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Additional material to this book can be downloaded from http://extras.springer.com.

ISBN 978-1-4612-6664-8 SPIN 10794415

Typeset by the authors Cover design by Jeff Cosloy, Newton, MA

9 876 5 4 3 2 1

Contents

I T~O~ 1

1 Introduction 3 1.1 It's a Nonlinear World . . . . . . . . . . . . . 3 1.2 Symbolic Computation . . . . . . . . . . . . . 6

1.2.1 Examples of Mathematica Operations 7 1.2.2 Getting Mathematica Help ...... 26 1.2.3 Use of Mathematica in Studying Nonlinear Physics. 27

1.3 Nonlinear Experimental Activities 35 1.4 Scope of Part I (Theory) . . . . . . . . . . . . . . . . . . . . 37

2 Nonlinear Systems. Part I 39 2.1 Nonlinear Mechanics . . . 39

2.1.1 The Simple Pendulum 39 2.1.2 The Eardrum. . . . . 46 2.1.3 Nonlinear Damping 48 2.1.4 Nonlinear Lattice Dynamics. 51

2.2 Competition Phenomena. . . . . . . 53 2.2.1 Volterra-Lotka Competition Equations. 54 2.2.2 Population Dynamics of Fox Rabies in Europe 59 2.2.3 Selection and Evolution of Biological Molecules 62 2.2.4 Laser Beam Competition Equations 64 2.2.5 Rapoport's Model for the Arms Race. 66

2.3 Nonlinear Electrical Phenomena ....... 68 2.3.1 Nonlinear Inductance ......... 68 2.3.2 An Electronic Oscillator (the Van der Pol Equation) 69

2.4 Chemical and Other Oscillators . 76 2.4.1 Chemical Oscillators 76 2.4.2 The Beating Heart 80

3 Nonlinear Systems. Part II 81 3.1 Pattern Formation . . . . . . . . . . . . . . . . . 81

3.1.1 Chemical Waves ............. 81 3.1.2 Snowflakes and Other Fractal Structures. 83

vi CONTENTS

3.1.3 Rayleigh-Benard Convection . . . . . . . . . 89 3.1.4 Cellular Automata and the Game of Life. . . 90

3.2 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.2.1 Shallow Water Waves (KdV and Other Equations) 99 3.2.2 Sine-Gordon Equation . . . 103 3.2.3 Self-Induced Transparency. . . . . . 107 3.2.4 Optical Solitons ........... 107 3.2.5 The Jovian Great Red Spot (GRS) . 110 3.2.6 The Davydov Soliton. . . . . . . . . 111

3.3 Chaos and Maps . . . . . . . . . . . 111 3.3.1 Forced Oscillators ....... 112 3.3.2 Lorenz and ROssler Systems. . .. ., . . . 115 3.3.3 Poincare Sections and Maps . . .. .. . . . 117 3.3.4 Examples of One- and Two-Dimensional Maps 119

4 Topological Analysis 125 4.1 Introductory Remarks ........ . 4.2 Types of Simple Singular Points. . . . 4.3 Classifying Simple Singular Points ..

4.3.1 Poincare's Theorem for the Vortex (Center) .. 4.4 Examples of Phase Plane Analysis . . . .

4.4.1 The Simple Pendulum .......... . 4.4.2 The Laser Competition Equations . . . . 4.4.3 Example of a Higher Order Singularity. .

4.5 Bifurcations............. 4.6 Isoclines............... 4.7 3-Dimensional Nonlinear Systems .

5 Analytic Methods 5.1 Introductory Remarks ....... . 5.2 Some Exact Methods ..... .

5.2.1 Separation of Variables ... 5.2.2 The Bernoulli Equation . . . . .. ... . . . 5.2.3 The Riccati Equation ............ . 5.2.4 Equations of the Structure d2y/dx2 = f(y) ..

5.3 Some Approximate Methods .............. . 5.3.1 Mathematica Generated Taylor Series Solution 5.3.2 The Perturbation Approach: Poisson's Method 5.3.3 Lindstedt's Method ..... .

5.4 The Krylov-Bogoliubov (KB) Method 5.5 Ritz and Galerkin Methods . . . . . .

6 The Numerical Approach 6.1 Finite-Difference Approximations. 6.2 Euler and Modified Euler Methods

6.2.1 Euler Method ....... . 6.2.2 The Modified Euler Method . . .

6.3 Runge-Kutta (RK) Methods ..... .

125 129 133 137 138 138 141 147 154 157 159

167 167

..... 168 171 175 177 179 193 193 196 203 210 216

223 224 227 228 232 238

CONTENTS vii

6.3.1 The Basic Approach . . . . . . . . . . 238 6.3.2 Examples of Common RK Algorithms 241

6.4 Adaptive Step Size . . . . . . . . . . 247 6.4.1 A Simple Example . . . . . . 247 6.4.2 The Step Doubling Approach 249 6.4.3 The RKF 45 Algorithm . . . 250

6.5 Stiff Equations . . . . . . . . . . . . 253 6.6 Implicit and Semi-Implicit Schemes. 258 6.7 Some Remarks on NDSolve . . . . . 263

7 Limit Cycles 265 7.1 Stability Aspects . . . . . . 265 7.2 Relaxation Oscillations. . . 273 7.3 Bendixson's First Theorem 278

7.3.1 Bendixson's Negative Criterion 278 7.3.2 Proof of Theorem. . . . . . 278 7.3.3 Applications .......... 280

7.4 The Poincare-Bendixson Theorem . . 281 7.4.1 Poincare-Bendixson Theorem. 282 7.4.2 Application of the Theorem . . 282

7.5 The Brusselator Model. . . . . . . . . 285 7.5.1 Prigogine-Lefever (Brusselator) Model. 285 7.5.2 Application of the Poincare-Bendixson Theorem 286

7.6 3-Dimensional Limit Cycles . . . . . . . . . . . . . . . . 291

8 Forced Oscillators 293 8.1 Duffing's Equation . . . . . . . . . . . . 293

8.1.1 The Harmonic Solution ..... 296 8.1.2 The Nonlinear Response Curves 298

8.2 The Jump Phenomenon and Hysteresis. 304 8.3 Subharmonic & Other Periodic Oscillations 308 8.4 Power Spectrum .......... 316 8.5 Chaotic Oscillations . . . . . . . . 324 8.6 Entrainment and Quasiperiodicity 335

8.6.1 Entrainment ....... 335 8.6.2 Quasiperiodicity . . . . . 337

8.7 The Rossler and Lorenz Systems 339 8.7.1 The Rossler Attractor 339 8.7.2 The Lorenz Attractor .. 340

8.8 Hamiltonian Chaos . . . . . . . . 342 8.8.1 Hamiltonian Formulation of Classical Mechanics 342 8.8.2 The Henon-Heiles Hamiltonian. . . . . . . . . . 343

9 Nonlinear Maps 355 9.1 Introductory Remarks 355 9.2 The Logistic Map. . . 356

9.2.1 Introduction . 356 9.2.2 Geometrical Representation. 358

viii

9.3 Fixed Points and Stability . . . . . . . . 9.4 The Period-Doubling Cascade to Chaos 9.5 Period Doubling in the Real World 9.6 The Lyapunov Exponent. 9.7 Stretching and Folding . 9.8 The Circle Map .. . 9.9 Chaos versus Noise .. . 9.10 2-Dimensional Maps ..

9.10.1 Introductory Remarks 9.10.2 Classification of Fixed Points 9.10.3 Delayed Logistic Map 9.10.4 Mandelbrot Map ...... .

9.11 Mandelbrot and Julia Sets ..... . 9.12 Nonconservative versus Conservative Maps 9.13 Controlling Chaos ......... . 9.14 3-Dimensional Maps: Saturn's Rings

10 Nonlinear PDE Phenomena 10.1 Introductory Remarks .. . 10.2 Burgers' Equation .... . 10.3 Backlund Transformations .

10.3.1 The Basic Idea ... 10.3.2 Examples . . . . . . 10.3.3 Nonlinear Superposition

10.4 Solitary Waves . . . . . . . . 10.4.1 The Basic Approach. 10.4.2 Phase Plane Analysis 10.4.3 KdV Equation .... 10.4.4 Sine-Gordon Equation 10.4.5 The Three-Wave Problem

11 Numerical Simulation 11.1 Finite Difference Approximations 11.2 Explicit Methods ........ .

11.2.1 Diffusion Equation ... . 11.2.2 Fisher's Nonlinear Diffusion Equation 11.2.3 Klein-Gordon Equation .... 11.2.4 KdV Solitary Wave Collisions .

11.3 Von Neumann Stability Analysis 11.3.1 Linear Diffusion Equation 11.3.2 Burgers' Equation

11.4 Implicit Methods . . . . . . . . 11.5 Method of Characteristics . . .

11.5.1 Colliding Laser Beams . 11.5.2 General Equation ... 11.5.3 Sine-Gordon Equation .

11.6 Higher Dimensions ...... . 11.6.1 2-Dimensional Reaction-Diffusion Equations

CONTENTS

363 366 369 372 375 378 383 388 388 390 391 392 394 396 398 404

413 413 414 425 425 425 429 432 432 433 437 442 445

451 451 457 457 466 467 470 473 473 474 476 479 479 482 484 486 487

CONTENTS

11.6.2 3-Dimensional Light Bullet Collisions

12 Inverse Scattering Method 12.1 Lax's Formulation .... 12.2 Application to KdV Equation

12.2.1 Direct Problem ....

II

12.2.2 Time Evolution of the Scattering Data . 12.2.3 The Inverse Problem .

12.3 Multi-Soliton Solutions. . . . . . . . . . 12.4 General Input Shapes ......... . 12.5 The Zakharov-Shabatj AKNS Approach

EXPERIMENTAL ACTIVITIES

Introduction to Nonlinear Experiments

1 Magnetic Force

2 Magnetic Tower

3 Spin Toy Pendulum

4 Driven Eardrum

5 Nonlinear Damping

6 Anharmonic Potential

7 Iron Core Inductor

8 Nonlinear LRC Circuit

9 Tunnel Diode Negative Resistance Curve

10 Tunnel Diode Self-Excited Oscillator

11 Forced Duffing Equation

12 Focal Point Instability

13 Compound Pendulum

14 Damped Simple Pendulum

15 Stable Limit Cycle

16 Van der Pol Limit Cycle

17 Relaxation Oscillations: Neon Bulb

ix

488

491 492 495 495 497 500 501 503 505

511

513

517

521

525

529

533

537

543

547

553

559

563

569

575

577

579

587

591

x

18 Relaxation Oscillations: Drinking Bird

19 Relaxation Oscillations: Tunnel Diode

20 Hard Spring

21 Nonlinear Resonance Curve: Mechanical

22 Nonlinear Resonance Curve: Electrical

23 Nonlinear Resonance Curve: Magnetic

24 Subharmonic Response: Period Doubling

25 Diode: Period Doubling

26 Five-Well Magnetic Potential

27 Power Spectrum

28 Entrainment and Quasiperiodicity

29 Quasiperiodicity

30 Chua's Butterfly

31 Route to Chaos

32 Driven Spin Toy

33 Mapping

Bibliography

Index

CONTENTS

597

601

605

609

613

617

621

623

627

633

637

639

641

645

649

651

655

669

Preface

Philosophy of the Text This text presents an introductory survey of the basic concepts and applied mathematical methods of nonlinear science as well as an introduction to some simple related nonlinear experimental activities. Students in engineering, phys­ics, chemistry, mathematics, computing science, and biology should be able to successfully use this book. In an effort to provide the reader with a cutting edge approach to one of the most dynamic, often subtle, complex, and still rapidly evolving, areas of modern research-nonlinear physics-we have made extensive use of the symbolic, numeric, and plotting capabilities of a powerful computer algebra software system applied to examples from these disciplines.

Currently, the two dominant computer algebra or symbolic computation soft­ware systems are Mathematica and Maple. In an effort to introduce nonlinear physics to as wide an audience as possible, we have created two different ver­sions of this text, an earlier edition making use of Maple having already been published l . This edition is based on Mathematica 4.1, the current Mathemat­ica release at the time of writing. Since the two software systems have different strengths and the subject of nonlinear physics and the interests of the authors continues to evolve, this Mathematica version is not simply a verbatim transla­tion of the earlier Maple text into Mathematica. For example, in this text we have introduced new nonlinear computer algebra files which make use of Math­ematica's extensive programming, graphics, and sound production capabilities and have included three additional experimental activities.

No prior knowledge of Mathematica or programming is assumed, the reader being gently introduced to Mathematica as an auxiliary tool as the concepts of nonlinear science are developed. Just as the Maple version was not intended to teach you everything you would like to know about programming in Maple, this text will not begin to cover the vast number of commands and options that are available in Mathematica. The CD-ROM provided with this book gives a wide variety of illustrative nonlinear examples solved with Mathematica, the command structures being introduced on a need to know basis. In addition, nu­merous annotated examples are sprinkled throughout the text and also placed on the CD. An accompanying set of experimental activities keyed to the theory developed in Part I of the book is given in Part II. These activities allow the student the option of "hands on" experience in exploring nonlinear phenomena in the REAL world. Although the experiments are easy to perform, they give

1 Nonlinear Physics with Maple for Scientists and Engineers, Birkhiiuser, Boston

XlI Preface

rise to experimental and theoretical complexities which are not to be underes­timated.

The Level of the Text The essential prerequisites for the first nine chapters of the theory portion of this text would normally be one semester of ordinary differential equations and an intermediate course in classical mechanics. The last three chapters of Part I are mathematically more sophisticated and assume that the student has some familiarity with partial derivatives, has encountered the wave, diffusion, and Schrodinger equations, and knows something about their solutions.

Most of the experimental activities in Part II may be approached on three levels:

• simplest-for non-physicists-investigate the features of the nonlinear phe­nomena with the minimum of data gathering and analysis,

• moderate-for physics majors and engineers-more emphasis on data gath­ering and analysis,

• complex-for experimentalists-<ieeper and more profound analysis re-quired with modifications suggested to stimulate ideas for research projects.

The material in this text has been successfully used to introduce nonlinear physics to students ranging from the junior year to first year graduate level. The book is designed to permit the instructor to pick and choose topics accord­ing to the level and background of the students as well as to their inclination towards theory or experiment.

Suggestions to the Student We suggest that you do not just passively read the material. The book is dy­namic in that it asks you to actively participate. This is obvious if you choose to do some of the Experimental Activities in Part II, but it is also true in progressing through Part I. In the theory part, this means carefully studying the worked-out Mathematica examples appearing in the text, running the addi­tional Mathematica files that appear on the CD-ROM, and doing the associated problems as they are encountered. If this technique is followed, it will provide you with a more profound and broader understanding of the material. The Mathematica code in the files and in the text can be used to produce all the text's plots. The code can provide you with help when you do the problems, and more importantly it allows you to explore and investigate the frontiers of nonlinear science. Since we do not presuppose any knowledge of Mathematica or computer programming skills, it is essential that the text examples be studied and the Mathematica files used. In this way you will acquire the Mathematica programming skills and the confidence to use Mathematica as an auxiliary tool to help you understand the concepts and to do the problems.

Suggestions to the Instructor This book, with its three-pronged approach of developing the necessary nonlin­ear theory, introducing the reader to the Mathematica computer algebra system,

Preface xiii

and providing detailed writeups of associated experimental activities, can be ef­fectively used in a variety of ways. Here are several possibilities:

1. A one-or two-semester course where the instructor explains the underly­ing concepts in short "talk and chalk" sessions that are interspersed with longer interludes during which each student at his or her own computer terminal runs the given Mathematica files, creates new files, solves prob­lems, and explores nonlinear systems. During this time the instructor is free to visit each station and provide individual help and/or answer questions. The sage has abandoned the stage! In this approach, if de­sired, many of the Experimental Activities of Part II are short and simple enough to be done as demonstrations or assigned as take-home projects.

2. A mainly experimental approach that requires the students to concentrate on doing the Experimental Activities referenced in the theory portion of the text and covered in detail in Part II. In this approach, Part I is used to deepen the understanding of the theoretical concepts underlying the experiments. Engineers might find this method of learning more satisfying and appealing. It is essential that even in this approach the students have easy access to computer terminals containing Mathematica so that they can check their experimental results and run their files.

3. A combination of the two approaches listed above for universities that require that all physics courses have a lab component.

4. Finally, the text could be effectively used in a "distance learning" or in a self-taught mode where financial and personal constraints make one or more of these modes mandatory.

Whatever approach to covering the material in this text is used, we hope that you will enjoy this introduction to one of the most exciting topics of contempo­rary science. Best wishes from the authors on your excursion into the wonderful world of nonlinear physics.

Richard H. Enns and George C. McGuire

Nonlinear Physics with Mathematica

for Scientists and Engineers