Local Semi-Dynamical Systems
- PDF / 6,411,046 Bytes
- 160 Pages / 576 x 785 pts Page_size
- 78 Downloads / 201 Views
90 N. P. Bhatia · O. Hajek Case Western Reserve University, Cleveland, Ohio
Local Semi-Dynamical Systems 1969
'
\...r
'w
Springer-Verlag Berlin· Heidelberg· New York
This work was supported by the National Science Foundation under Grants no. NSF-GP-7447 and NSF-GP-896L The authors are indebted to Mrs. Elizabeth Roach for her meticulous typing, and to Mr. Charles Allen for the preparation of the diagrams.
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer-Verlag Berlin· Heidelberg 1969 Library of Congress Catalog Card Number 70 - 79551 Printed in Germany. Title No. 3696
CONTENTS
.. . .. .... ... ..... ...........
O.
Int roduct ion •......•
1.
Local semi-dynamical systems: and properties ••....
2
basic definitions
. .... .......... . .................... 12
11.
... .. ... .... ........... 27 Invariance ... . . .... .. . . ....... . . .... ..... . .............. 35 Compactness conditions ..... ....... .... ... . .............. 42 Limit sets •• . .. . ............ ............... ........ ..... 49 The positive prolongation .. ..... . ....................... 59 Stability and orbital stability. ........ .. . .............. 74 Attraction . . . .... . . . .. ...... . ..................... 79 Flow near an invariant set . ............................. 94 Liapunov functions. .. .... . ... .. ............ .............100 The start point set .... . . . .... .. ..................... •120
12.
Minimality; characteristic 0 •••••••••••••••••••••••••••• 132
13.
Functional-differential equations
2.
3. 4. 5. 6.
7. 8.
9. 10.
Solutions:
negative continuation
............ ...........141
References
154
Index
156
2
O. 0.1
INTRODUCTION
This paper is devoted to the basic theory of
the so-called local semi-dynamical systems.
These are
oojects related to the classical dynamical systems (see 0.2).
The differences are that development into posi-
tive time only is specified. so that indeterminate behavior into the past is allowed; and it is not assumed that solutions are defined or extendable over all positive times (see 0.6 and 0.7).
Such objects arise from
a "dynamical" interpretation of functional-differential equations with time-lag. and also of evolution-type partial differential equations (see 0.7 to 0.10).
This
is in contrast to dynamical systems which mainly arise from ordinary autonomous differential equations. as described below. 0.2
A dynamical system (sometimes called a global
bilateral dynamical system) on a topological space is a continuous mapping
71":
X
X x R .. X which satisfies
the initial value axiom and the group axiom. X7r0 = x, (X7rt )7I'"S = X7r( t-i-s},
3 for all
x E X and
t,s E R.
line; the value of
at
operator fashion, by
xvt
(R
(x,t)
denotes the real is denoted in
instead of
For
a detailed study of dynamical systems see [15), [6J, [lOJ. 0.3
One of the principal motivations for the
study of dynamical systems is that they describe completely a certain larg
Data Loading...
