Miller / Michel Ordinary Differential Equations

1. Auflage 2014
ISBN: 978-1-4832-5910-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 366 Seiten, Web PDF

Ordinary Differential Equations is an outgrowth of courses taught for a number of years at Iowa State University in the mathematics and the electrical engineering departments. It is intended as a text for a first graduate course in differential equations for students in mathematics, engineering, and the sciences. Although differential equations is an old, traditional, and well-established subject, the diverse backgrounds and interests of the students in a typical modern-day course cause problems in the selection and method of presentation of material. In order to compensate for this diversity, prerequisites have been kept to a minimum and the material is covered in such a way as to be appealing to a wide audience. The book contains eight chapters and begins with an introduction the subject and a discussion of some important examples of differential equations that arise in science and engineering. Separate chapters follow on the fundamental theory of linear and nonlinear differential equations; linear boundary value problems; Lyapunov stability theory; and perturbations of linear systems. Subsequent chapters deal with the Poincare-Bendixson theory and with two-dimensional van der Pol type equations; and periodic solutions of general order systems.

Autoren/Hrsg.

Miller, Richard K

Michel, Anthony N.

Inhaltsverzeichnis

1;Front Cover;1
2;Ordinary Differential Equations;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;PREFACE;12
7;ACKNOWLEDGMENTS;14
8;CHAPTER 1. INTRODUCTION ;16
8.1;1.1 Initial Value Problems;16
8.2;1.2 Examples of Initial Value Problems;22
8.3;Problems;50
9;CHAPTER 2. FUNDAMENTAL THEORY;54
9.1;2.1 Preliminaries;55
9.2;2.2 Existence of Solutions;60
9.3;2.3 Continuation of Solutions;64
9.4;2.4 Uniqueness of Solutions;68
9.5;2.5 Continuity of Solutions with Respect to Parameters;73
9.6;2.6 Systems of Equations;78
9.7;2.7 Differentiability with Respect to Parameters;83
9.8;2.8 Comparison Theory;85
9.9;2.9 Complex Valued Systems;89
9.10;Problems;90
10;CHAPTER 3. LINEAR SYSTEMS;95
10.1;3.1 Preliminaries;95
10.2;3.2 Linear Homogeneous and Nonhomogeneous Systems;103
10.3;3.3 Linear Systems with Constant Coefficients;115
10.4;3.4 Linear Systems with Periodic Coefficients;127
10.5;3.5 Linear nth Order Ordinary Differential Equations;132
10.6;3.6 Oscillation Theory;140
10.7;Problems;145
11;CHAPTER 4. BOUNDARY VALUE PROBLEMS;152
11.1;4.1 Introduction;152
11.2;4.2 Separated Boundary Conditions;158
11.3;4.3 Asymptotic Behavior of Eigenvalues;162
11.4;4.4 Inhomogeneous Problems;167
11.5;4.5 General Boundary Value Problems;174
11.6;Problems;179
12;CHAPTER 5. STABILITY;182
12.1;5.1 Notation;183
12.2;5.2 The Concept of an Equilibrium Point;184
12.3;5.3 Definitions of Stability and Boundedness;187
12.4;5.4 Some Basic Properties of Autonomous and Periodic Systems;193
12.5;5.5 Linear Systems;194
12.6;5.6 Second Order Linear Systems;201
12.7;5.7 Lyapunov Functions;209
12.8;5.8 Lyapunov Stability and Instability Results: Motivation;217
12.9;5.9 Principal Lyapunov Stability and Instability Theorems;220
12.10;5.10 Linear Systems Revisited;233
12.11;5.11 Invariance Theory;236
12.12;5.12 Domain of Attraction;245
12.13;5.13 Converse Theorems;249
12.14;5.14 Comparison Theorems;254
12.15;5.15 Applications: Absolute Stability of Regulator Systems;258
12.16;Problems;265
13;CHAPTER 6. PERTURBATIONS OF LINEAR SYSTEMS;273
13.1;6.1 Preliminaries;273
13.2;6.2 Stability of an Equilibrium Point;275
13.3;6.3 The Stable Manifold;280
13.4;6.4 Stability of Periodic Solutions;288
13.5;6.5 Asymptotic Equivalence;295
13.6;Problems;300
14;CHAPTER 7. PERIODIC SOLUTIONS OF TWO-DIMENSIONAL SYSTEMS;305
14.1;7.1 Preliminaries;305
14.2;7.2 Poincaré-Bendixson Theory;307
14.3;7.3 The Levinson–Smith Theorem;313
14.4;Problems;317
15;CHAPTER 8. PERIODIC SOLUTIONS OF SYSTEMS;320
15.1;8.1 Preliminaries;321
15.2;8.2 Nonhomogeneous Linear Systems;321
15.3;8.3 Perturbations of Nonlinear Periodic Systems;327
15.4;8.4 Perturbations of Nonlinear Autonomous Systems;332
15.5;8.5 Perturbations of Critical Linear Systems;334
15.6;8.6 Stability of Systems with Linear Part Critical;339
15.7;8.7 Averaging;345
15.8;8.8 Hopf Bifurcation;348
15.9;8.9 A Nonexistence Result;350
15.10;Problems;353
16;BIBLIOGRAPHY;357
17;INDEX;361