## Table of contents for Computer algebra recipes : an advanced guide to scientific modeling / Richard H. Enns, George C. McGuire.

Bibliographic record and links to related information available from the Library of Congress catalog
Note: Electronic data is machine generated. May be incomplete or contain other coding.

```PREFACE                                                           v
INTRODUCTION                                                      1
A. Computer Algebra Systems .......................             1
B. Computer Algebra Recipes  .......................            2
C. Introductory Recipe: Boys Will Be Boys . . . . . . . . . . . . . . .  3
D. Maple Help  ..................        .. ...........     .   8
E. How to Use This Text .........................               9
I THE APPETIZERS                                                 11
1 Phase-Plane Portraits                                          13
1.1  Phase-Plane Portraits  ...................        .....   13
1.1.1  Romeo and Juliet . . . . . . . . . . . . . . ..... . . .  .  18
1.1.2  There's No Damping Vectoria's Romantic Heart . . . . . 23
1.1.3  Van der Pol's Limit Cycle . . . . . . . . . . . . . . ....  28
1.2 Three-Dimensional Autonomous Systems . . . . . . . . .  .... 32
1.2.1  The Period-Doubling Route to Chaos . . . . . . . .. . .  33
1.2.2  The Oregonator  .......................            40
1.2.3  R6ssler's Strange Attractor . . . . . . . . . . . . . .  .  44
2 Phase-Plane Analysis                                           47
2.1 Phase-Plane Analysis ..... . ........ .     . ..... .   . 47
2.1.1  Foxes Munch Rabbits .................... 51
2.1.2  The Mona Lisa of Nonlinear Science . . . . . . . . .  .  58
2.1.3  Mike Creates a Higher-Order Fixed Point  .... . . . . 67
2.1.4  The Gnus and Sung of Erehwon  . . . . . . . . . .....  73
2.1.5  A Plethora of Points . . . . . . . . . . . . . . ..... ..  78
2.2 Three-Dimensional Autonomous Systems . . . . . . . . . ..... 82
2.2.1  Lorenz's Butterfly ...................        ...  82
2.3 Numerical Solution of ODEs ..................... 88
2.3.1  Finite Difference Approximations . . . . . . . . . . ....  89
2.3.2  Rabbits and Foxes: The Sequel . . . . . . . . . . . ..  91
2.3.3  Glycolytic Oscillator .................. ..       96
2.3.4  Fox Rabies Epidemic ................      ... ..101
II THE ENTREES                                                 107
3 Linear ODE Models                                            109
3.1 First-Order Models .......................... 110
3.1.1  How's Your Blood Pressure? . . . . . . . . . . . . .... 110
3.1.2  Greg Arious Nerd's Problem  . . . . . . . . . . . . .... 115
3.2 Second-Order Models . . . . . . . . . . . . . . ..... ......  118
3.2.1  Daniel Encounters Resistance .. . . . . . . . . . . . . ... 118
3.2.2  Meet Mr. Laplace ...................... 121
3.2.3  Jennifer's Formidable Series . . . . . . . . . . . . . ....   126
3.3 Special Function Models ....................... 130
3.3.1  Jennifer Introduces a Special Family . . . . . . . . .... 131
3.3.2  The Vibrating Bungee Cord . . . . . . . . . . . . . .... 137
3.3.3  Mathieu's Spring ....... :................. 142
3.3.4  Quantum-Mechanical Tunneling .. . . . . . . . . .  . 144
4 Nonlinear ODE Models                                         149
4.1 First-Order Models .......................... 150
4.1.1  An  Irreversible Reaction  . . . . . . . . . . . . . . . ...  150
4.1.2  The Struggle for Existence. . . . . . . . . . . . . ..  152
4.1.3  The Bad Bird Equation ..........        ..... ...161
4.2  Second-Order Models ......................... 164
4.2.1  Patches Gives Chase  ....................        164
4.2.2  Oh What Sounds We Hear! . . . . . . . . . . . . . ... 168
4.2.3  Vectoria Feels the Force and Hits the Bottle . . . . . ... 175
4.2.4  Golf Is Such an "Uplifting" Experience . . . . . . . ... 179
4.3  Variational Calculus Models . . . . . . . . . . . . . . ..  ... .  185
4.3.1  Dress Design, the Erehwonese Way . . . . . . . . . .... 185
4.3.2  Queen Dido Wasn't a Dodo . . . . . . . . . . . . . ...  191
4.3.3  The Human Fly Plans His Escape Route . . . . . . . ... 195
4.3.4  This Would Be a Great Amusement Park Ride .  . . . . . 201
5 Linear PDE Models. Part 1                                    207
5.1  Checking  Solutions  .......................... 207
5.1.1  The Palace of the Governors . . . . . . . . . . . ....  207
5.1.2  Play  It, Sam  .........................         211
5.1.3  Three  Easy  Pieces  . . . . . .  . .  . .  . .  .  . ..  .  .  . . .. 215
5.1.4  Complex, Yet Simple . . . . . . . . . . . . . . ... . .   . .  220
5.2 Diffusion and Laplace's Equation Models . . . . . . . . . . .... 223
5.2.1  Freeing  Excalibur .................     ..  .. .  ... 223
5.2.2  Aussie Barbecue ...................         .. ..227
5.2.3  Benny's Solution ...................        .. ..231
5.2.4  Hugo and the Atomic Bomb . . . . . . . . . . . . . .  . 236
5.2.5  Hugo Prepares for His Job Interview . . . . . . . . . . . . 241
6 Linear PDE Models. Part 2                                     247
6.1 Wave Equation Models ...................          ..... 247
6.1.1  Vectoria Encounters Simon Legree ...... . . . . . . . 247
6.1.2  Homer's Jiggle Test  . . . . . . . . . . . . . . . . . . . . 251
6.1.3  Vectoria's Second Problem . . . .. . . . . . . . . .  254
6.1.4  Sound of Music? .................. .       ...... 257
6.2 Semi-infinite and Infinite Domains  ...... . . . . . . . . . . 261
6.2.1  Vectoria's Fourth Problem .. . . . . . . . . . . . ......  261
6.2.2  Assignment Complete! .. . . . . . . . . . . . . . .....  263
6.2.3  Radioactive Contamination . . . . . . . . . . . . . . . . . 266
6.2.4  "Play It, Sam" Revisited  . . ....... . . . . . ......  270
6.3 Numerical Simulation of PDEs . . . . . . . . . . . . . . . . . . . 274
6.3.1  Freeing Excalibur the Numerical Way . . . . . . . . . . . 275
6.3.2  Enjoy the Klein-Gordon Vibes ..  . . . . . . . . . . . . 278
6.3.3  Vectoria's Secret  . . . . . . . . . . . . . .  .. . . . .  .  281
III  THE DESSERTS                                               285
7 The Hunt for Solitons                                         287
7.1 The Graphical Hunt for Solitons . . . . . . . . . . . . . . . . . . 290
7.1.1  Of Kinks and Antikinks ..............     ..... 290
7.1.2  In  Search  of Bright Solitons  . . . . . . . . . . . . . . . . . 293
7.1.3  Can Three Solitons Live Together? . . . . . . . . . .  . 296
7.2  Analytic Soliton  Solutions  ............ ..........299
7.2.1  Follow  That Wave! ......................300
7.2.2  Looking for a Kinky Solution . . . . . . . . . . . ....  304
7.2.3  We Have Solitons! ......... .............306
7.3  Simulating Soliton Collisions  ..... . . . . . . . . ...... .  308
7.3.1  To Be or Not to Be a Soliton ................308
7.3.2  Are Diamonds a Kink's Best Friend? ..... . . . . . . . 312
8 Nonlinear Diagnostic Tools                                    319
8.1 The Poincare Section ......... .   ............... 319
8.1.1  A  Rattler Signals Chaos  ...................320
8.1.2  Hamiltonian  Chaos  . . . . ... . . . . . . . . . . . .  . . .  323
8.2  The Power Spectrum  .................. ........          329
8.2.1  Frank N. Stein's Heartbeat . . . . . . . . . . . . .....  332
8.2.2  The Rattler Returns . . . . . . . . . . . . . . ...... ..  334
8.3  The Bifurcation Diagram  . . . . . . . . . . . . . . .  . .337
8.3.1  Pitchforks and Other Bifurcations . . . . . . . . .....  338
8.4 The Lyapunov Exponent ......   ...... ...........342
8.4.1  Mr. Lyapunov  Agrees  ................... . 343
8.5  Reconstructing an Attractor . . . . . . . .... .   . . . . . . . . .  345
8.5.1  Putting Humpty Dumpty Together Again . . . . . . . . . 346
8.5.2  Random Is Random   .................. ..349
8.5.3  Butterfly Reconstruction. . . . . . . . . . . . . . ......  351
Epilogue  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ..353
Bibliography                                                   355

```

Library of Congress subject headings for this publication: Differential equations Numerical solutions Computer programs, Maple (Computer file)Science Mathematical models