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DIFFERENTIAL EQUATIONS LABORATORY WORKBOOK A Collection of Experiments, Explorations and Modeling Projects for the Computer

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DIFFERENTIAL EQUATIONS LABORATORY WORKBOOK A Collection of Experiments, Explorations and Modeling Projects for the Computer

Robert L. Borrelli Harvey Mudd College

Courtney S. Coleman Harvey Mudd College

William E. Boyce Rensselaer Polytechnic Institute

John Wiley & Sons, Inc. New York. Chichester. Brisbane. Toronto. Singapore

Copyright 1992 by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of thi s work beyond that permitted by Sections 107 and 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons.

ISBN 0-471-55142-2

Printed in the United States of America

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Contents

Preface IX

Notes for the Instructor xiii

CHAPTERS

1 Introduction: Learning About Your Hardware/Software

2 Solution Curves and Numerical Methods 39

3 First Order Equations 73

4 Second Order Equations 113

5 Planar Systems 149

6 Higher Dimensional Systems 209

APPENDICES

A Team Laboratory Reports 257 Some hints on writing team reports

B Mathematical Modeling 259 Derivation of the differential equations that appear in the modeling experiments throughout the workbook

B.l Population and Rate Models 260 B.2 Mechanics 264 B.3 Chemical Reactions 275 B.4 Circuits 282 B.S Scaling and Dimensionless Variables 286

C The Atlas 291

INDEX

A collection of graphs of solutions of differential equations

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List of Experiments

Experiments listed in boldface are team experiments.

Chapter 1 Plotting Orbits and Solution Curves 5

1.1 Direction Fields and Solution Curves 9 1.2 ODEs in Non Normal Form 10 1.3 Direction Fields and Orbits 11 1.4 Solvers and IVPs 12

Generating Atlas Plots 13 1.5 Generating Atlas Plots 14

First Order Rate Laws 15 1.6 Population Growth and Decay 19 1.7 Radioactive Decay: Carbon-14 Dating 21 1.8 Potassium-Argon Dating 22

Falling Bodies 23 1.9 Falling Bodies Near the Earth's Surface 27 1.10 Escape Velocities 28

Aliasing and Other Phenomena 29 1.11 Aliasing and Other Phenomena 36

Chapter 2 39 Properties of Orbits and Solution Curves 41

2.1 Fundamentals 45 2.2 Sign Analysis 46

Equilibrium Solutions and Sensitivity 47 2.3 Pitchfork Bifurcation 48

Solutions That Escape to Infinity 49 2.4 Solutions That Escape to Infinity 51

Picard Process for Solving IVPs 53 2.5 Picard Process for Solving IVPs 56

Euler's Method for Solving IVPs 57 2.6 Euler's Method and Explicit Solutions 61 2.7 Limitations of Euler 's Method 62

Euler Solutions to the Logistic Equation 63 2.8 Convergent Euler Sequences 66 2.9 Period Doubling and Chaos: Graphical Evidence 67 2.10 Period Doubling and Chaos: Theory 68

Chapter 3 73 Linear First Order ODEs: Properties of Solutions 75

3.1 Superposition 77

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3.2 Singularities 78 Linear First Order ODEs: Data 79

3.3 Dependence on Data 81 3.4 Bounded Input/Bounded Output 82

Separable ODEs: Implicit Solutions 83 3.5 Separable ODEs: Implicit Solutions 86

Nonlinear ODEs: Homogeneous Functions 87

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3.6 Nonlinear ODEs: Homogeneous Functions 88 The ODE: M(x, y)dx + N(x, y)dy = 0 89

3.7 Planar Systems and M dx + N dy = 0 91 3.8 Construction of Integral Curves: The Cat 92

ODEs in Polar Coordinates 93 3.9 ODEs in Polar Coordinates 95

Comparison of Solutions of Two ODEs 97 3.10 Comparison of Solutions of Two ODEs 98

Harvesting a Species 9.9 3.11 Constant Rate Harvesting 101 3.12 Variable Rate Harvesting 103 3.13 Intermittent Harvesting 104

Salt Levels in a Brine 105 3.14 Linear Brine Models 107 3.15 Nonlinear Brine Models 108

Bimolecular Chemical Reactions 109 3.16 Quadratic Rates as Bimolecular Models 111 3.17 Modeling a Bimolecular Reaction 112

Chapter 4 113 Properties of Solutions 115

4.1 Properties of Solutions 117 Constant Coefficient Linear ODEs: Undriven 119

4.2 Constant Coefficient Linear ODEs: Undriven 121 Constant Coefficient Linear ODEs: Driven 123

4.3 Beats and Resonance 125 4.4 General Driving Forces 126

Frequency Response Modeling 127 4.5 Parameter Identification 130 4.6 Gain and Phase Shift 131

Springs 133 4.7 Linear Springs 137 4.8 Hard and Soft Springs 139 4.9 Aging Springs 140

Circuits 141 4.10 Simple RLC Circuit 147 4.11 Tuning a Circuit 148

Chapter 5 149 Portraits of Planar Systems 151

5.1 Portraits of Planar Systems 152 Autonomous Linear Systems 153

5.2 Gallery of Pictures 157 5.3 Stability 158

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Driven Linear Systems 159 5.4 Driven Linear Systems 161

Interacting Species 163

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5.5 Predator-Prey Models: Harvesting, Overcrowding 165 5.6 Competing Species 166

The Pendulum 167 5.7 The Undriven Pendulum: Linear Model 171 5.8 The Undriven Nonlinear Pendulum 173 5.9 The Driven Upended Pendulum 174

Duffing's Equation 175 5.10 Duffing's Equation 178

Planetary Motion 179 5.11 Elements of Orbital Mechanics 184

Stability and Lyapunov Functions 185 5.12 The Effect of a Perturbation 191 5.13 Stability and Lyapunov Functions 192

Cycles and Limit Cycles 193 5.14 The van der Pol System 196 5.15 Systems with Cycles and Limit Cycles 197

The Poincare-Bendixson Alternatives 199 5.16 The Poincare-Bendixson Alternatives 200

The Hopf Bifurcation 201 5.17 The Hopf Bifurcation 205 5.18 Satiable Predation: Bifurcation to a Cycle 206

Chapter 6 209 Undriven Linear Systems 211

6.1 Portraits of Undriven Linear Systems 213 6.2 Asymptotic Behavior and Eigenelements 215

Driven Linear Systems 217 6.3 Transients, Steady States, Resonance 219 6.4 Coupled Oscillators 221

A Compartment Model: Lead in the Body 223 6.5 A Compartment Model: Lead in the Body 225

Lorenz System: Sensitivity 227 6.6 Inducing Chaos 230 6.7 Search for Cycles 231

Rossler System: Period-doubling 233 6.8 Rossler System: Sensitivity 234

The Rotational Stability of a Tennis Racket 235 ' 6.9 The Rotational Stability of a Tennis Racket 240

Nonlinear Systems and Chemical Reactions 241 6.10 Approach to Equilibrium: Five Species 243 6.11 Approach to Equilibrium: Four Species 244

Oscillating Chemical Reactions 245 6.12 On/Off Oscillations: Autocatalator 247 6.13 Persistent Oscillations: The Oregonator 248

Bifurcations and Chaos in a Nonlinear Circuit 249 6.14 Bifurcations and Chaos in a Nonlinear Circuit 251

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Preface

Differential equations arise in connection with mathematical models in diverse settings in mathematics, engineering, the sciences (physics, ,chemistry, biology, geology, astronomy, etc.)-and even in the social sciences. Since solutions of ordinary differential equations (ODEs) define curves, much can be learned about the behavior of solutions by graphing them. Such graphs, generated with easy-to-use interactive numerical solvers, provide com-pelling visual evidence of theoretical deductions and an understanding of the qualitative properties of solutions. The main focus of this workbook is on computer experiments that support and amplify the topics usually found in introductory ODE texts. The workbook is intended as a supplement, not as a textbook.

Prerequisites The material in the workbook presumes a knowledge of single and multivariable calculus

and some linear algebra. It is expected that the student is concurrently enrolled in a course involving ODEs, and that the student has available a software package that can numerically solve systems of differential equations and present the solutions in graphical form on suitable hardware platforms. Many excellent ODE solvers do not require knowledge of computer programming, and little or none is presumed. The experiments do not usually require special features of any particular text or platform.

Mathematical Models and Computers The essential ingredient in the application of mathematical techniques to the real world is

the construction of a mathematical model for the system of interest. Modeling practitioners believe that a "good" model actually "describes" the system they wish to study, and since mathematical models are amenable to mathematical analysis, great progress can be made in understanding the system. Before the computer age, modelers were forced to keep their models simple enough to allow solution by the analytical techniques of the day (many quite ingenious and sophisticated). Modern computers and software have liberated modelers from this constraint. Models need no longer be analytically tractable, hence no longer require unrealistic or artificial assumptions. Computers can numerically solve (with remarkable accuracy and speed) complicated systems of ODEs and, together with striking graphical displays, allow modelers an insightful look at complex systems.

Overview The approach in the workbook closely parallels what goes on in science and engineer-

ing laboratories. Each computer problem set is a combination of pencil-and-paper and computer work selected from the tasks given in an experiment. The work may be straight-forward and explicit, but the approach is often open-ended and exploratory. The computer experiments provide students with "hands-on" exper