Linear Systems And Exponential Dichotomy And Structure Of Sets Of Hyperbolic Points

By: Zhensheng Lin, Yan-Xia Lin

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Hardcover | 3 May 2000

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220 Pages

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Historically, the theory of stability is based on linear differential systems, which are simple and important systems in ordinary differential equations. The research on differential equations and on the theory of stability will, to a certain extent, be influenced by the research on linear differential systems. For differential linear equation systems, there are still many historical open questions attracting mathematicians. This text deals with the theory of linear differential systems developed around the notion of exponential dichotomies. The first author advanced the theory of stability through his research in this field. Several important results on linear differential systems are presented. They concern exponential dichotomy and the structure of the sets of hyperbolic points. The book contains five chapters. Chapter One introduces some necessary classical results on the linear differential systems, and the following chapters discuss exponential dichotomy, spectra of almost periodic linear systems, the Floquet theory for quasi periodic linear systems and the structure of sets of hyperbolic points. The book should be useful as a reference in the area of the stability theory of ordinary differential equations and the theory of dynamic systems.

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Foreword
Preface
Notation
Linear Differential Systems	p. 1
General Properties of Linear Differential Systems	p. 1
Linear Differential System with Constant Coefficients	p. 5
Linear Differential System with Periodic Coefficients	p. 9
Exponential Dichotomy	p. 17
Exponential Dichotomy Theory	p. 18
First Approximate Theory	p. 43
Local Linearization	p. 59
The Structural Stability of Linear Systems	p. 63
Spectra of Almost Periodic Linear Systems	p. 77
Definitions and Spectra	p. 77
A Simple Disproof of Millionscikov's Example	p. 78
The Reducibility of Quasi Periodic Linear Systems	p. 85
Spectra of Almost Periodic Linear Systems	p. 92
Spectra of Almost Periodic Linear Systems (continued)	p. 105
Floquet Theory for Quasi Periodic Linear Systems	p. 125
Almost Periodic Linear Systems and Floquet Theory	p. 125
Quasi Periodic Systems and Floquet Theory	p. 127
The Characteristics Exponential Roots of Quasi Periodic Linear Systems	p. 133
Floquet Theory for Quasi Periodic Linear Systems	p. 138
Structure of Sets of Hyperbolic Points	p. 145
A New Definition of the Set of Hyperbolic Points	p. 146
C[superscript r] Closing Lemma	p. 155
Structural Stability of Vector Fields	p. 175
A Simple Proof of Kupka-Smale Generic Theorem	p. 186
Bibliography	p. 199
Index	p. 203
Table of Contents provided by Blackwell. All Rights Reserved.

Linear Systems And Exponential Dichotomy And Structure Of Sets Of Hyperbolic Points

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Hardcover

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