Methods of A. M. Lyapunov and their Application

FOREWORD*......Page 7
FOREWORD......Page 8
FROM THE AUTHOR......Page 11
TABLE OF CONTENTS......Page 17
INTRODlJCTION......Page 19
l. Metric space......Page 27
2. Operators and functionals in R......Page 28
3. Neighborhood of a set......Page 29
5. Normed linear spaces......Page 30
6. Dynamical systems inn-dimensional Euclidean space En......Page 32
7. Examples of dynamical systems in functional spaces......Page 35
8. Dynamical systems in metric space......Page 38
9. Formulation of the problem of stability of invariant setsPrincipal definitions......Page 40
10. Uniformly asymptotically stable and uniformly attracting invariantsets of a dynamical system f(p, t)......Page 45
II. Qualitative description, from the point of view of the Lyapunovstability, of a neighborhood of an invariant set......Page 49
12. Necessary and sufficient conditions for stability......Page 59
13. Necessary and sulficient conditions for instability......Page 66
14. Necessary and sufficient conditions for the existence of uniformlyasymptotically stable and uniformly attracting invariant sets......Page 70
IS. Method of estimating the distance from the motion f(p, t) to acertain invariant set M with the aid of functionals......Page 79
I. Stationary systems of differential equations......Page 85
2. Case of analytic right members of the system (2.1)......Page 105
3. Systems of differential equations with homogeneous right members......Page 122
4. Case of k zero roots......Page 148
5. Case of several pairs of pure imaginary roots......Page 162
6. System of non-stationary differential equations......Page 170
1. Auxiliary theorems from the theory of partial differential equations......Page 184
2. Representation of solutions of systems of ordinary dilferentialequatiort,s in a neighborhood of a singular point......Page 191
3. Analytic representation of the 0-curves of a special system ofdilferential equations......Page 197
I. General systems. Basic definitions......Page 214
2. Conditions for stability and instability of an invariant set M of ageneral system in metric space......Page 220
3. Certain applications to non-stationary systems of ordinarydifferential equations......Page 242
I. Some general propositions......Page 253
2. Stability of solutions of quasi-linear systems of special type......Page 258
3. Stability of the trivial solution of a system of partial ditferentialequations......Page 266
BIBLIOGRAPHY......Page 277
INDEX......Page 280