Ordinary Differential Equations With Applications (2nd Edition) / Edition 2

by Sze-bi Hsu

Overview

During the past three decades, the development of nonlinear analysis, dynamical systems and their applications to science and engineering has stimulated renewed enthusiasm for the theory of Ordinary Differential Equations (ODE).This useful book, which is based on the lecture notes of a well-received graduate course, emphasizes both theory and applications, taking numerous examples from physics and biology to illustrate the application of ODE theory and techniques.Written in a straightforward and easily accessible style, this volume presents dynamical systems in the spirit of nonlinear analysis to readers at a graduate level and serves both as a textbook and as a valuable resource for researchers.This new edition contains corrections and suggestions from the various readers and users. A new chapter on Monotone Dynamical Systems is added to take into account the new developments in ordinary differential equations and dynamical systems.

Product Details

ISBN-13:	9789814452908
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/21/2013
Series:	Series On Applied Mathematics , #21
Edition description:	New Edition
Pages:	312
Product dimensions:	5.80(w) x 9.10(h) x 0.90(d)

Table of Contents

Preface to the First Edition v

Preface to the Second Edition ix

1 Introduction 1

1.1 Where do ODEs arise 1

2 Fundamental Theory 11

2.1 Introduction and Preliminaries 11

2.2 Local Existence and Uniqueness of Solutions of I.V.P. 14

2.3 Continuation of Solutions 23

2.4 Continuous Dependence Properties 26

2.5 Differentiability of I.C. and Parameters 29

2.6 Differential Inequalities 32

2.7 Exercises 37

3 Linear Systems 43

3.1 Introduction 43

3.2 Fundamental Matrices 44

3.3 Linear Systems with Constant Coefficients 49

3.4 Two-Dimensional Linear Autonomous Systems 58

3.5 Linear Systems with Periodic Coefficients 62

3.6 Adjoint Systems 70

3.7 Exercises 74

4 Stability of Nonlinear Systems 81

4.1 Definitions 81

4.2 Linearization 83

4.3 Saddle Point Property 92

4.4 Orbital Stability 102

4.5 Travelling Wave Solutions 110

4.6 Exercises 117

5 Method of Lyapunov Functions 123

5.1 An Introduction to Dynamical Systems 123

5.2 Lyapunov Functions 130

5.3 Simple Oscillatory Phenomena 142

5.4 Gradient Vector Fields 145

5.5 Exercises 147

6 Two-Dimensional Systems 157

6.1 Poincaré-Bendixson Theorem 157

6.2 Levinson-Smith Theorem 168

6.3 Hopf Bifurcation 177

6.4 Exercises 183

7 Second Order Linear Equations 189

7.1 Sturm's Comparison Theorem and Sturm-Liouville BVP 189

7.2 Distributions 197

7.3 Green's Function 199

7.4 Fredholm Alternative 206

7.5 Exercises 209

8 The Index Theory and Brouwer Degree 213

8.1 Index Theory in the Plane 213

8.2 Introduction to the Brouwer Degree in Rⁿ 221

8.3 Exercises 226

9 Perturbation Methods 229

9.1 Regular Perturbation Methods 229

9.2 Singular Perturbation : Boundary Value Problem 235

9.3 Singular Perturbation : Initial Value Problem 241

9.4 Exercises 253

10 Introduction to Monotone Dynamical Systems 255

10.1 Monotone Dynamical System with Applications to Cooperative Systems and Competitive Systems 255

10.2 Uniform Persistence 261

10.3 Application : Competition of Two Species in a Chemostat with Inhibition 270

10.4 Two Species Competition Models 282

10.5 Exercises 286

Bibliography 291

Index 295