Qualitative Theory of Ordinary Differential Equations

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