Oscillations in Nonlinear Systems

by Hale, Jack K.

• ISBN13: 9780486673622
• ISBN10: 0486673626
• Edition: Reprint
• Format: Paperback
• Copyright: 2015-05-20
• Publisher: Dover Publications

Summary

By focusing on ordinary differential equations that contain a small parameter, this concise graduate-level introduction to the theory of nonlinear oscillations provides a unified approach to obtaining periodic solutions to nonautonomous and autonomous differential equations. It also indicates key relationships with other related procedures and probes the consequences of the methods of averaging and integral manifolds.
Part I of the text features introductory material, including discussions of matrices, linear systems of differential equations, and stability of solutions of nonlinear systems. Part II offers extensive treatment of periodic solutions, including the general theory for periodic solutions based on the work of Cesari-Halel-Gambill, with specific examples and applications of the theory. Part III covers various aspects of almost periodic solutions, including methods of averaging and the existence of integral manifolds. An indispensable resource for engineers and mathematicians with knowledge of elementary differential equations and matrices, this text is illuminated by numerous clear examples.

Author Biography

A longtime professor at Brown University, Jack K. Hale is Regents Professor Emeritus at the Georgia Institute of Technology and the author of Dover's Ordinary Differential Equations.

Table of Contents

Preface
Introduction and Background Material
Introduction	p. 3
Matrices	p. 11
Linear Systems of Differential Equations	p. 16
Stability of Solutions of Nonlinear Systems	p. 22
Periodic Solutions
Noncritical Cases	p. 27
Periodic Solutions of Equations in Standard Form - Critical Cases	p. 34
Practical Methods of Computing a Periodic Solution and Examples	p. 46
Characteristic Exponents of Linear Periodic Systems	p. 68
Periodic Solutions of Nonautonomous Systems	p. 82
Periodic Solutions of Autonomous Systems	p. 89
Generalizations	p. 95
Almost Periodic Solutions and Integral Manifolds
Almost Periodic Functions and Multiply Periodic Functions	p. 113
Almost Periodic Solutions - Noncritical Case	p. 121
Periodic Solutions Revisited	p. 130
Integral Manifolds - Averaging	p. 134
Integral Manifolds - Noncritical Case	p. 141
Most Periodic Solutions - Critical Case	p. 154
Integral Manifolds - Critical Case	p. 160
Appendix: Principle of Contraction Mappings	p. 171
Bibliography	p. 173
Index	p. 179
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