Ode Architect Companion

Cover Page......Page 1
copyright......Page 4
Workbook Preface......Page 5
Acknowledgments......Page 8
Modules/Chapters Overview......Page 9
Contents......Page 14
1. Modeling with the ODE Architect......Page 18
Building a Model of the Pacific Sardine Population......Page 19
The Logistic Equation......Page 27
Introducing Harvesting via Landing Data......Page 29
How to Model in Eight Steps......Page 32
Explorations......Page 34
2. Introduction to ODEs......Page 42
Solutions to Differential Equations......Page 43
Slope Fields......Page 44
Finding a Solution Formula......Page 45
The Juggler......Page 47
The Sky Diver......Page 48
Explorations......Page 52
3. Some Cool ODEs......Page 60
Finding a General Solution......Page 61
Time-Dependent Outside Temperature......Page 63
Air Conditioning a Room......Page 64
The Case of the Melting Snowman......Page 66
Explorations......Page 68
4. Second-Order Linear Equations......Page 74
Undamped Oscillations......Page 75
The Effect of Damping......Page 77
Forced Oscillations......Page 78
Beats......Page 80
Seismographs......Page 81
Explorations......Page 86
5. Models of Motion......Page 94
Vectors......Page 95
Forces and Newton's Laws......Page 96
Dunk Tank......Page 97
Longer to Rise or to Fall?......Page 98
Indiana Newton......Page 99
Ski Jumping......Page 101
Explorations......Page 102
6. First-Order Linear Systems......Page 110
Examples of Systems: Pizza and Video, Coupled Springs......Page 111
Linear Systems with Constant Coefficients......Page 112
Solution Formulas: Eigenvalues and Eigenvectors......Page 114
Calculating Eigenvalues and Eigenvectors......Page 115
Phase Portraits......Page 116
Using ODE Architect to Find Eigenvalues and Eigenvectors......Page 119
Parameter Movies......Page 120
Explorations......Page 122
7. Nonlinear Systems......Page 132
The Geometry of Nonlinear Systems......Page 133
Linearization......Page 134
Separatrices and Saddle Points......Page 137
Behavior of Solutions Away from Equilibrium Points......Page 138
Bifurcation to a Limit Cycle......Page 139
Spinning Bodies: Stability of Steady Rotations......Page 140
The Planar Double Pendulum......Page 143
Explorations......Page 146
8. Compartment Models......Page 152
Lake Polution......Page 153
Allergy Relief......Page 154
Lead in the Body......Page 156
Equilibrium......Page 158
The Autocatalator and a Hopf Bifurcation......Page 159
Explorations......Page 164
9. Population Models......Page 172
The Logistic Model......Page 173
Two-Species Population Models......Page 175
Predator and Prey......Page 176
Species Competition......Page 177
Mathematical Epidemiology: The SIR Model......Page 178
Explorations......Page 180
10. The Pendulum and Its Friends......Page 190
Modeling Pendulum Motion......Page 191
Conservative Systems: Integrals of Motion......Page 193
The Effect of Damping......Page 194
Separatrices......Page 197
Writing the Equations of Motion for Pumping a Swing......Page 199
Geodesics......Page 202
Geodesics on a Surface of Revolution......Page 203
Geodesics on a Torus......Page 205
Explorations......Page 210
11. Applications of Series Solutions......Page 220
Recurrence Formulas......Page 221
Ordinary Points......Page 223
Regular Singular Points......Page 224
Bessel Functions......Page 226
Transforming Bessel's Equation to the Aging Spring Equation......Page 227
Explorations......Page 230
12. Chaos and Control......Page 238
Solutions as Functions of Time......Page 239
Poincare Sections......Page 240
The Unforced Pendulum......Page 241
Tangled Basins, the Wada Property......Page 243
Gaining Control......Page 245
Explorations......Page 248
13. Discrete Dynamical Systems......Page 250
Equilibrium States......Page 252
Linear vs. Nonlinear Dynamics......Page 253
Stability of a Discrete Dynamical System......Page 254
Bifurcations......Page 255
Periodic and Chaotic Dynamics......Page 257
What is Chaos?......Page 258
Complex Numbers and Functions......Page 259
Iterating a Complex Function......Page 260
Julia Sets, the Mandelbrot Set, and Cantor Dust......Page 261
Explorations......Page 266
Glossary......Page 274