Fred Brauer and John A. Nohel
Klappentext
"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.
Inhalt
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