The Qualitative Theory of Ordinary Differential Equations; An Introduction

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  • Edition: Reprint
  • Format: Trade Paper
  • Copyright: 1989-02-01
  • Publisher: Dover Publications

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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

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
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
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
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
Table of Contents provided by Publisher. All Rights Reserved.

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