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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.
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
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
Chapter Three Stable and Unstable Motion, II
3.1 Liapunov Functions
3.2 Stable Equilibria, II
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
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
5.7 Tidal Dynamics
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
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
Chapter Eight Chaos
8.1 Not All Attractors Are Limit Cycles or Equilibria
8.2 Strange Attractors
8.3 Deterministic or Random?
Chapter Nine There Is a Better Way
9.1 Conditions Necessary for Optimality
9.2 Fish Harvesting
9.3 Bang-Bang Controls
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
- No. of pages:
- © Academic Press 1987
- 10th September 1987
- Academic Press
- eBook ISBN:
State University of New York, Stony Brook, U.S.A.