Dichotomies and Stability in Nonautonomous Linear Systems

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ISBN: 9780415272216 | 0415272211
Cover: Hardcover
Copyright: 10/10/2002

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Linear nonautonomous equations arise as mathematical models in mechanics, chemistry, and biology. The investigation of bounded solutions to systems of differential equations involves some important and challenging problems of perturbation theory for invariant toroidal manifolds. This monograph is a detailed study of the application of Lyapunov functions with variable sign, expressed in quadratic forms, to the solution of this problem. The authors explore the preservation of invariant tori of dynamic systems under perturbation. This volume is a classic contribution to the literature on stability theory and provides a useful source of reference for postgraduates and researchers.

Introduction to the Series

Preface

An Overview of the Book

xiii

Exponentially Dichotomous Linear Systems of Differential Equations and Lyapunov Functions of Variable Sign

(92)

Introduction

(1)

Exponential Dichotomy Criterion for Linear Systems in Terms of Quadratic Forms

(15)

Decomposition over the Whole R Axis of Linear Systems of Differential Equations Exponentially Dichotomous on Semiaxes R+ and R-

(13)

Degeneracy of the Quadratic Form Possessing a Definite-Sign Derivative along the Solutions of the System (1.1.1)

(11)

Integral Representation of Weakly Regular Systems Bounded on the Whole R Axis

(9)

Complement to the Exponentially Dichotomous of Weakly Regular on R Linear Systems

(17)

Regularity of Linear Systems of the Block-Triangular Form

(9)

Perturbation of the Block-Triangular Form Linear Systems which are Regular and Weakly Regular on the Whole R Axis

(10)

Exponentially Dichotomous Linear Systems with Parameters

(6)

Comments and References

(1)

Linear Extension of Dynamical Systems on a Torus

(160)

Introduction

(1)

Necessary Existence Conditions for Invariant Tori

(13)

The Green Function. Sufficient Existence Conditions for an Invariant Torus

107

(4)

Existence Conditions for an Exponentially Stable Invariant Torus

111

(5)

Uniqueness Conditions for the Green Function and its Properties

116

(9)

Sufficient Conditions for Exponential Dichotomy of the Invariant Torus

125

(7)

Necessary Conditions for Exponential Dichotomy of the Invariant Torus

132

(6)

Existence Criterion for the Green Function

138

(15)

The Non-unique Green Function and the Properties of the System Implied by its Existence

153

(16)

Invariant Tori of Linear Extensions with Slowly Changing Phase

169

(8)

Preserving the Green Function under Small Perturbations of Linear Expansions on a Torus

177

(14)

On the Smoothness of an Exponentially Stable Invariant Torus

191

(12)

On the Dependence of Green Functions on Parameters

203

(9)

Continuity and Differentiability of the Green Function

212

(10)

Invariant Tori of Linear Extensions with a Degenerate Matrix at the Derivatives

222

(9)

Bounded Invariant Manifolds of Dynamical Systems and their Smoothness

231

(19)

Comments and References

250

(3)

Splittability of Linear Extensions of Dynamical Systems on a Torus

253

(54)

Introduction

253

(1)

Sufficient Conditions for Splittability of Linear Extensions of Dynamical Systems on a Torus

254

(9)

Reversibility of the Theorem on Splittability

263

(8)

On Triangulation and the Relationship of C'-block Splittability of a Linear System with the Problem on r-frame Complementability up to the Periodic Basis in Rn

271

(7)

Reducing of Linearized Systems to a Diagonal Form

278

(9)

On the Relationship of Exponentially Dichotomous Linear Expansions with the Algebraic System Solvability

287

(6)

Three Block Divisibility of Linear Extensions and Lyapunov Functions of Variable Sign

293

(8)

Algebraic Problems of k-blocked Divisibility of Linear Extensions on a Torus

301

(4)

Comments and References

305

(2)

Problems of Perturbation Theory of Smooth Invariant Tori of Dynamical Systems

307

(48)

Introduction

307

(1)

Solution Variations on the Manifold M

308

(8)

Exponential Stability and Dichotomy Conditions for Linear Extensions of Dynamical Systems on a Torus

316

(4)

Roughness Conditions for the Green Function of the Linear Extension of a Dynamical System on a Torus with the Index of Smoothness l

320

(9)

A Theorem of Perturbation Theory of an Invariant Torus of a Dynamical System

329

(1)

Green Function for a Linear Matrix Equation

330

(5)

On the Problem of Structure of Some Regular Linear Extensions of Dynamical Systems on a Torus

335

(8)

Invariant Manifolds of Autonomous Differential Equations and Lyapunov Functions with Alternating Signs

343

(10)

Comments and References

353

(2)

References

355

(12)

Index

367

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Dichotomies and Stability in Nonautonomous Linear Systems

Summary

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Dichotomies and Stability in Nonautonomous Linear Systems

Summary

Table of Contents

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