The Qualitative Theory of Ordinary Differential Equations An Introduction

by Brauer, Fred; Nohel, John A.

• ISBN13: 9780486658469
• ISBN10: 0486658465
• Edition: Reprint
• Format: Paperback
• Copyright: 1989-02-01
• Publisher: Dover Publications

Summary

Superb, self-contained graduate-level text covers standard theorems concerning linear systems, existence and uniqueness of solutions, and dependence on parameters. Major focus on stability theory and its applications to oscillation phenomena, self-excited oscillations and regulator problem of Lurie. Bibliography. Exercises.

Table of Contents

Preface

Systems of Differential Equations

A Simple Mass-Spring System

Coupled Mass-Spring Systems

Systems of First-Order Equations

Vector-Matrix Notation for Systems

The Need for a Theory

Existence, Uniqueness, and Continuity

The Gronwall Inequality

Linear Systems, with an Introduction to Phase Space Analysis

Introduction

Existence and Uniqueness for Linear Systems

Linear Homogeneous Systems

Linear Nonhomogeneous Systems

Linear Systems with Constant Coefficients

Similarity of Matrices and the Jordan Canonical Form

Asymptotic Behavior of Solutions of Linear Systems with Constant Coefficients

Autonomous Systems--Phase Space--Two-Dimensional Systems

Linear Systems with Periodic Coefficients

Miscellaneous Exercises

Existence Theory

Existence in the Scalar Case

Existence Theory for Systems of First-Order Equations

Uniqueness of Solutions

Continuation of Solutions

Dependence on Initial Conditions and Parameters

Miscellaneous Exercises

Stability of Linear and Almost Linear Systems

Introduction

Definitions of Stability

Linear Systems

Almost Linear Systems

Conditional Stability

Asymptotic Equivalence

Stability of Periodic Solutions

Lyapunov's Second Method

Introductory Remarks

Lyapunov's Theorems

Proofs of Lyapunov's Theorems

Invariant Sets and Stability

The Extent of Asymptotic Stability--Global Asymptotic Stability

Nonautonomous Systems

Some Applications

Introduction

The Undamped Oscillator

The Pendulum

Self-Excited Oscillations--Periodic Solutions of the Lienard Equation

The Regulator Problem

Absolute Stability of the Regulator System

Generalized Eigenvectors, Invariant Subspaces, and Canonical Forms of Matrices

Canonical Forms of 2 x 2 Matrices

The Logarithm of a Matrix

Some Results from Matrix Theory

Bibliography

Index

Table of Contents provided by Publisher. All Rights Reserved.