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E-raamat: Dichotomies and Stability in Nonautonomous Linear Systems

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Dichotomies and Stability in Nonautonomous Linear SystemsSuurem pilt
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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.

Arvustused

"This volume will be of great interest to researchers and students dealing with nonautonomous systems." - Zentralblatt fur Mathematik, Vol. 1026

Introduction to the Series ix
Preface xi
An Overview of the Book xiii
Exponentially Dichotomous Linear Systems of Differential Equations and Lyapunov Functions of Variable Sign
1(92)
Introduction
1(1)
Exponential Dichotomy Criterion for Linear Systems in Terms of Quadratic Forms
2(15)
Decomposition over the Whole R Axis of Linear Systems of Differential Equations Exponentially Dichotomous on Semiaxes R+ and R-
17(13)
Degeneracy of the Quadratic Form Possessing a Definite-Sign Derivative along the Solutions of the System (1.1.1)
30(11)
Integral Representation of Weakly Regular Systems Bounded on the Whole R Axis
41(9)
Complement to the Exponentially Dichotomous of Weakly Regular on R Linear Systems
50(17)
Regularity of Linear Systems of the Block-Triangular Form
67(9)
Perturbation of the Block-Triangular Form Linear Systems which are Regular and Weakly Regular on the Whole R Axis
76(10)
Exponentially Dichotomous Linear Systems with Parameters
86(6)
Comments and References
92(1)
Linear Extension of Dynamical Systems on a Torus
93(160)
Introduction
93(1)
Necessary Existence Conditions for Invariant Tori
94(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
Yu. A. Mitropolsky, A.M. Samoilenko, V.L. Kulik
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