Preface
Part 1 First Thoughts on Equilibria and Stability
Chapter One Simple Dynamic Models
1.1 Back and Forth, Up and Down
1.2 The Harmonic Oscillator
1.3 Stable Equilibria, I
1.4 What Comes Out Is What Goes In
1.5 Exercises
Chapter Two Stable and Unstable Motion, I
2.1 The Pendulum
2.2 When Is a Linear System Stable?
2.3 When Is a Nonlinear System Stable?
2.4 The Phase Plane
2.5 Exercises
Chapter Three Stable and Unstable Motion, II
3.1 Liapunov Functions
3.2 Stable Equilibria, II
3.3 Feedback
3.4 Exercises
Chapter Four Growth and Decay
4.1 The Logistic Model
4.2 Discrete Versus Continuous
4.3 The Struggle for Life, I
4.4 Stable Equilibria, III
4.5 Exercises
A Summary of Part 1
Part 2 Further Thoughts and Extensions
Chapter Five Motion in Time and Space
5.1 Conservation of Mass, II
5.2 Algae Blooms
5.3 Pollution in Rivers
5.4 Highway Traffic
5.5 A Digression on Traveling Waves
5.6 Morphogenesis
5.7 Tidal Dynamics
5.8 Exercises
Chapter Six Cycles and Bifurcation
6.1 Self-Sustained Oscillations
6.2 When Do Limit Cycles Exist?
6.3 The Struggle for Life, II
6.4 The Flywheel Governor
6.5 Exercises
Chapter Seven Bifurcation and Catastrophe
7.1 Fast and Slow
7.2 The Pumping Heart
7.3 Insects and Trees
7.4 The Earth's Magnet
7.5 Exercises
Chapter Eight Chaos
8.1 Not All Attractors Are Limit Cycles or Equilibria
8.2 Strange Attractors
8.3 Deterministic or Random?
8.4 Exercises
Chapter Nine There Is a Better Way
9.1 Conditions Necessary for Optimality
9.2 Fish Harvesting
9.3 Bang-Bang Controls
9.4 Exercises
Appendix Ordinary Differential Equations: A Review
First-Order Equations (The Case k = 1)
The Case k = 2
The Case k = 3
References and a Guide to Further Readings
Ordinary Differential Equations
Introductions to Differential Equation Modeling
More Advanced Modeling Books
Hard to Classify
Notes on the Individual Chapters
Index