Mathematics for Dynamic Modeling (eBook)

Edward Beltrami (Autor)

eBook Download: PDF

2014 | 1. Auflage
294 Seiten
Elsevier Science (Verlag)
978-1-4832-6786-9 (ISBN)

Infos

Mathematics for Dynamic Modeling provides an introduction to the mathematics of dynamical systems. This book presents the mathematical formulations in terms of linear and nonlinear differential equations. Organized into two parts encompassing nine chapters, this book begins with an overview of the notions of equilibrium and stability in differential equation modeling that occur in the guise of simple models in the plane. This text then focuses on nonlinear models in which the limiting behavior of orbits can be more complicated. Other chapters consider the problems that illustrate the concepts of equilibrium and stability, limit cycles, chaos, and bifurcation. This book discusses as well a variety of topics, including cusp catastrophes, strange attractors, and reaction-diffusion and shock phenomena. The final chapter deals with models that are based on the notion of optimization. This book is intended to be suitable for students in upper undergraduate and first-year graduate course in mathematical modeling.

Inhalt

Front Cover 1
Mathematics for Dynamic Modeling 4
Copyright Page 5
Table of Contents 8
Dedication 6
Preface 14
Part 1: First Thoughts on Equilibria and Stability 18
Chapter One. Simple Dynamic Models 20
1.1 Back and Forth, Up and Down 20
1.2 The Harmonic Oscillator 23
1.3 Stable Equilibria, I 25
1.4 What Comes Out Is What Goes In 29
1.5 Exercises 30
Chapter Two. Stable and Unstable Motion, I 34
2.1 The Pendulum 34
2.2 When Is a Linear System Stable? 36
2.3 When Is a Nonlinear System Stable? 39
2.4 The Phase Plane 43
2.5 Exercises 53
Chapter Three. Stable and Unstable Motion, II 56
3.1 Liapunov Functions 56
3.2 Stable Equilibria, II 65
3.3 Feedback 69
3.4 Exercises 75
Chapter Four. Growth and Decay 78
4.1 The Logistic Model 78
4.2 Discrete Versus Continuous 83
4.3 The Struggle for Life, I 85
4.4 Stable Equilibria, ill 91
4.5 Exercises 95
A Summary of Part 1 98
Part 2: Further Thoughts and Extensions 100
Chapter Five. Motion in Time and Space 102
5.1 Conservation of Mass, II 102
5.2 Algae Blooms 106
5.3 Pollution in Rivers 112
5.4 Highway Traffic 118
5.5 A Digression on Traveling Waves 128
5.6 Morphogenesis 132
5.7 Tidal Dynamics 142
5.8 Exercises 148
Chapter Six. Cycles and Bifurcation 154
6.1 Self-Sustained Oscillations 154
6.2 When Do Limit Cycles Exist? 160
6.3 The Struggle for Life, II 172
6.4 The Flywheel Governor 179
6.5 Exercises 184
Chapter Seven. Bifurcation and Catastrophe 188
7.1 Fast and Slow 188
7.2 The Pumping Heart 199
7.3 Insects and Trees 206
7.4 The Earth's Magnet 213
7.5 Exercises 219
Chapter Eight. Chaos 224
8.1 Not All Attractors Are Limit Cycles or Equilibria 224
8.2 Strange Attractors 231
8.3 Deterministic or Random? 235
8.4 Exercises 244
Chapter Nine. There Is a Better Way 246
9.1 Conditions Necessary for Optimality 246
9.2 Fish Harvesting 253
9.3 Bang-Bang Controls 260
9.4 Exercises 268
Appendix: Ordinary Differential Equations: A Review 272
First-Order Equations (The Case k = 1) 273
The Case k = 2 274
The Case k = 3 278
References and a Guide to Further Readings 280
Ordinary Differential Equations 281
Introductions to Differential Equation Modeling 281
More Advanced Modeling Books 282
Hard to Classify 283
Notes on the Individual Chapters 284
Index 292

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik ► Maschinenbau
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Planung / Organisation
• ISBN-10: 1-4832-6786-5 / 1483267865
• ISBN-13: 978-1-4832-6786-9 / 9781483267869