Mathematics for Dynamic Modeling

Mathematics for Dynamic Modeling

1st Edition - September 10, 1987

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  • Author: Edward Beltrami
  • eBook ISBN: 9781483267869

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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.

Table of Contents

  • 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


Product details

  • No. of pages: 294
  • Language: English
  • Copyright: © Academic Press 1987
  • Published: September 10, 1987
  • Imprint: Academic Press
  • eBook ISBN: 9781483267869

About the Author

Edward Beltrami

Affiliations and Expertise

State University of New York, Stony Brook, U.S.A.

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