The Qualitative Theory of Ordinary Differential Equations: An Introduction

by Fred Brauer, John A. Nohel

Overview

"This is a very good book ... with many well-chosen examples and illustrations." — American Mathematical Monthly
This highly regarded text presents a self-contained introduction to some important aspects of modern qualitative theory for ordinary differential equations. It is accessible to any student of physical sciences, mathematics or engineering who has a good knowledge of calculus and of the elements of linear algebra. In addition, algebraic results are stated as needed; the less familiar ones are proved either in the text or in appendixes.
The topics covered in the first three chapters are the standard theorems concerning linear systems, existence and uniqueness of solutions, and dependence on parameters. The next three chapters, the heart of the book, deal with stability theory and some applications, such as oscillation phenomena, self-excited oscillations and the regulator problem of Lurie.
One of the special features of this work is its abundance of exercises-routine computations, completions of mathematical arguments, extensions of theorems and applications to physical problems. Moreover, they are found in the body of the text where they naturally occur, offering students substantial aid in understanding the ideas and concepts discussed. The level is intended for students ranging from juniors to first-year graduate students in mathematics, physics or engineering; however, the book is also ideal for a one-semester undergraduate course in ordinary differential equations, or for engineers in need of a course in state space methods.

Product Details

ISBN-13:	9780486658469
Publisher:	Courier Corporation
Publication date:	02/01/1989
Series:	Dover Books on Mathematics Series
Edition description:	Dover ed
Pages:	320
Product dimensions:	5.50(w) x 8.50(h) x (d)

Read an Excerpt

PREFACE

In this book we present a self-contained introduction to some important aspects of modern qualitative theory for ordinary differential equations. Since only a minimal background in techniques of solution of differential equations such as is body textly acquired in an elementary undergraduate course is assumed, the book is accessible to any student of physical sciences, mathematics, or engineering who has a good knowledge of calculus and of the elements of linear algebra. In addition, algebraic results are stated as needed; the less familiar ones are proved either in the text or in appendixes.

Our principal objective is the study of stability theory in Chapters 4, 5 and the applications considered in Chapter 6. Here the reader will find a rigorous but elementary treatment of several topics which are of importance in modern engineering and physics. In effect, these methods provide the justification of many approximations ordinarily assumed to be valid. At the same time readers interested in mathematical theory will find here an introduction to several topics of current research interest. Chapters 1, 2, and 3 serve as preparation. Some of this preparatory material has already been presented (but in less detail) in. The reader familiar with Chapters 6 and 7 of can begin the present book with Chapter 4.

There are, of course, several interesting and important topics which we have omitted and for several reasons. For example, the Poincare-Bendixson theory of plane autonomous systems and the theory of the index of a critical point are not included, because we can hardly improve upon the beautiful presentation by W. Hurewicz in. Other topics, such as the use of fixed point theorems and implicit function theorems require a more sophisticated background than we wish to assume. For this reason we have also omitted the important methods of Poincare and of averaging for establishing existence of and approximation to periodic solutions of almost linear perturbed systems. In a different direction, the reader interested in pursuing the study of boundary value problems is referred to Chapter 5 of for an introduction and to Chapters 7 to 12 of or to for a more advanced treatment.

At the University of Wisconsin we have each taught one-semester courses to juniors and seniors (in mathematics, physical sciences, and engineering) based on the first five chapters. There is, however, ample material here for a one-semester course for those students who are prepared to begin with Chapter 4.

Throughout the text and at the end of several chapters, the reader will find numerous exercises, some routine, and some more difficult. These are designed to help follow the argument and to provide a better understanding of the subject and, thus, they form an essential part of the book. The interested reader will find many problems from the first three chapters solved in our book, Problems and Solutions in Ordinary Differential Equations (W. A. Benjamin, Inc. New York, 1968).

It is impossible to acknowledge all the help, direct and indirect, from which we have benefited in the preparation of this book. Certainly, the stimulating influences of Professors Norman Levinson and Earl Coddington, and of our colleagues Charles Conley, Jacob Levin, and Wolfgang Wasow, have been valuable. Professors H. A. Antosiewicz, J. K. Hale, A. Strauss, and A. D. Ziebur read portions of a preliminary version of the manuscript and we acknowledge with pleasure their many useful suggestions. Professors B. Berndt, D. Ferguson, and J. Williamson gallantly served as our assistants in teaching some of this material and we are grateful for their helpful comments. In preparing Chapter 6 we have found valuable ideas in some unpublished lecture notes by Professor Lawrence Markus and we are grateful to him for making them available to us. Finally, it is a pleasure to acknowledge the help of Mrs. Phyllis J. Rickli who has converted almost illegible handwriting into a clean manuscript, and the help of the staff of W. A. Benjamin, Inc., in the construction of this book. Naturally any errors that may remain in spite of all this assistance are our responsibility, and we would appreciate being advised of them.

Fred Brauer John A. Nohel

Madison, Wisconsin October 1968

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Excerpted from "The Qualitative Theory of Ordinary Differential Equations"
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Copyright © 1969 Fred Brauer and John A. Nohel.
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Table of Contents

Preface
Chapter 1. Systems of Differential Equations
1.1 A Simple Mass-Spring System
1.2 Coupled Mass-Spring Systems
1.3 Systems of First-Order Equations
1.4 Vector-Matrix Notation for Systems
1.5 The Need for a Theory
1.6 Existence, Uniqueness, and Continuity
1.7 The Gronwall Inequality
Chapter 2. Linear Systems, with an Introduction to Phase Space Analysis
2.1 Introduction
2.2 Existence and Uniqueness for Linear Systems
2.3 Linear Homogeneous Systems
2.4 Linear Nonhomogeneous Systems
2.5 Linear Systems with Constant Coefficients
2.6 Similarity of Matrices and the Jordan Canonical Form
2.7 Asymptotic Behavior of Solutions of Linear Systems with Constant Coefficients
2.8 Autonomous Systems--Phase Space--Two-Dimensional Systems
2.9 Linear Systems with Periodic Coefficients; Miscellaneous Exercises
Chapter 3. Existence Theory
3.1 Existence in the Scalar Case
3.2 Existence Theory for Systems of First-Order Equations
3.3 Uniqueness of Solutions
3.4 Continuation of Solutions
3.5 Dependence on Initial Conditions and Parameters; Miscellaneous Exercises
Chapter 4. Stability of Linear and Almost Linear Systems
4.1 Introduction
4.2 Definitions of Stability
4.3 Linear Systems
4.4 Almost Linear Systems
4.5 Conditional Stability
4.6 Asymptotic Equivalence
4.7 Stability of Periodic Solutions
Chapter 5. Lyapunov's Second Method
5.1 Introductory Remarks
5.2 Lyapunov's Theorems
5.3 Proofs of Lyapunov's Theorems
5.4 Invariant Sets and Stability
5.5 The Extent of Asymptotic Stability--Global Asymptotic Stability
5.6 Nonautonomous Systems
Chapter 6. Some Applications
6.1 Introduction
6.2 The Undamped Oscillator
6.3 The Pendulum
6.4 Self-Excited Oscillations--Periodic Solutions of the Liénard Equation
6.5 The Regulator Problem
6.6 Absolute Stability of the Regulator System
Appendix 1. Generalized Eigenvectors, Invariant Subspaces, and Canonical Forms of Matrices
Appendix 2. Canonical Forms of 2 x 2 Matrices
Appendix 3. The Logarithm of a Matrix
Appendix 4. Some Results from Matrix Theory
Bibliography; Index