Dynamical Systems: Stability Theory and Applications
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35 N. P. Bhatia
Western Reserve University· Cleveland, Ohio, USA
G. P. Szego Universita degli Studi di Milano· Milano, Italy
1967
Dynamical Systems: Stability Theory and Appl ications
Springer-Verlag· Berlin· Heidelberg· New York
All rights. especially that of translation into foreign languages. reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard) or by other procedure without written permission from Springer Verlag. @) by Springer-Verlag Berlin' Heidelberg 1967. Library of Congress Catalog Card Number 67 - 25757 Title No.7355.
PREFACE
This book began as a series of lecture notes of the course given by N. P. Bhatia at the Western Reserve University during the Spring of 1965 and the lecture notes of the courses given by G. P. Szeg8 at the University of Milan during the year 1964 - 65 and at Case Institute of Technology during the summer of 1965. These courses were meant for different audiences, on one side graduate students in mathematics, and on the other graduate students in systems theory and physics. }-Iowever in the process of developing these notes we have found a number of other results of interest which we decided to include ( See 1.9,
2.7,2.8,2.11,2.14,3.3,3.4,3.5,3.7,3.8,3.9). Therefore, this monograph is of a dual nature involving both a systematic compilation of known results in dynamical systems and differential equations and a presentation of new Theorems and points of view. As a result, a certain lack of organizational unity and overlapping are evident. The reader should consider this monograph not as a polished, finished product, but rather as a complete survey of the present state of the art including many new open areas and new problems. Thus, we feel that these notes fit the special aims of this Springer-Verlag series. We do hope that this monograph will be appropriate for a one year graduate course in Dynamical Systems. This monograph is still devoted to a mixed audience so we have tried to make the presentation of Chapter I (Dynamical Systems in Euclidean Space) as simple as poss ible, using the most simple mathematical techniques and proving in detail all statements, even those which may be obvious to more mature readers. Chapter 2 (Dynamical Systems in Metric Spaces) is more advanced. Chapter 3 has a mixed composition: Sections 3.1, 3.2, 3.6,
3.7 aJ;ld 3.8 are quite elementary, while the remaining part of the chapter
is advanced. In this latter part we mention many problems which are still in an early developmental stage. A sizeable number of the results contained in this monograph have never been published in book form before. We would like to thak Prof. Walter Leighton of Western Reserve University, Prof. Mi hailo Mesarovic of Case Institute of Technology, and Prof. Monroe Martin, Director of Institute for Fluid Dynamics and Applied Mathematics of the University of Maryland, under whose sponsorship the authors had the chance of writing this monograph. We wish to thank several students at
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