Table Of ContentLectu re Notes in Mathematics
A collection of informal reports and seminars
Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
35
N. P. Bhatia
Western Reserve University· Cleveland, Ohio, USA
G. P. Szego
Universita degli Studi di Milano· Milano, Italy
Dynamical Systems:
Stability Theory
and Appl ications
1967
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 our
universities, in particular, A. Cellina, P. Fallone, C. Sutti and G. Kramerich
for checking parts of the manuscript. We are also indebted to Prof.A.Strauss
and Prof. O. Hajek for many helpful suggestions and inspiring discussions
and to Prof. J. Yorke for alloWing to present his new results in Sec. 3.4.
We wish also to express our appreciation to Mrs. Carol Smith of TECH - TYPE
Corp., who typed most of the manuscript.
The work of the first author has been supported by the National Science
Foundation under Grants NSF-GP-4921 and NSF - GP-7057,while the work
of the second author has been sponsored by the CNR, Comitato per la
Matematica, Gruppo N° 11, and by the National Foundation under Grant NSF-GP-
6114.
The authors
March 1967
TABLE OF CONTENTS
o Notation , TerminoLogy and PreLiminary Lemmas Page
0.1 Notation 1
0.2 Terminology 4
0.3 Preliminary Lemmas 6
1 DynamicaL Systems in a EucUdean Space
1.1 Definition of a continuous dynamical system 9
1.2 Elementary concepts 13
1.3 Limit sets of trajectories 28
1.4 Prolongations. 41
1.5 Lagrange and Liapunov stability for compact sets 46
1.6 Liapunov stability for sets 65
1.7 Stability and Liapunov functions 85
1.8 Topological methods 96
1.9 Topological properties of attractors 99
1.10 From periodic motions to Poisson stability 105
1.11 Stability of motions 108
2 Dynamiaal Systems in Metria Spaces
2.1 Definition of a dynamical system and related notation 114
2.2 Elementary Concepts: trajectories and their limit sets 116
2.3 The (first) (positive) prolongation and the prolongational
limit set 121
2.4 Self-intersecting trajectories 127
2.5 Lagrange and Poisson stability 129
2.6 Attraction, stability, and asymptotic stability of compact
sets 134
2.7 Liapunov functions ,and asymptotic stability of compact
sets 143
2.8 Topological properties of Aw(M) , A(M) and Pa(M) 157
2.9 Minimal sets and recurrent motions 163
2.10 Stability of a motion and almost periodic motions 171
2.11 Paralle1izable dynamical systems 182
2.12 Stability and asymptotic stability of closed sets 201
2.13 Higher prolongations and stability 220
2.14 Higher prolongationa1 limit sets and generalized recurrence 235
2.15 Relative stability and relative prolongations 242
3 The second method of Liapu:nov for ordinary differential equations
3.1 Dynamical systems defined by ordinary differential equations 246
3.2 Further properties of the solutions of ordinary differential
equations without uniqueness 282
3.3 Continuous flows without uniqueness 298
3.4 Further results on nonuniqueness by James A.Yorke 307
3.5 Dynamical systems and nonautonomous differential equations 324
3.6 Classical results on the investigation of the stability properties
of flows defined by the solutions of ordinary differential
equations via the second method of Liapunov 330
3.7 New results with relaxed conditions 342
3.8 The extension theorem 351
3.9 The use of higher derivatives of a Liapunov function 365
References 368
Index 407
1
CHAPTER 0
Notation ~ TenminoLogy and P~Liminary Lemmas
0.1 Notation
T: topological space
X: metric space with metric p
E: real euclidean space of n-dimensions
E2: the real euclidean plane
G: group
R: set of real nunbers.
R+ : non-negative real numbers
R-: non-positive real numbers
I: set of integers
r+: set of non-negative integers
1-: set of non-positive integers
In the sequel, when not otherwise stated, capital letters will denote
matrices and sets, small latin letters vector (notable exceptions t,s,k,v and w
which have been used to denote real nunbers), small greek letters real numbers
(notable exception 1T, which denotes a mapping) •
x (xl' ... ,x Ilxll
If = ) £ E, will denote the euclidean norm of x i.e.,
n
n
O.l.l Ilxll - 0: x:)1/2
'1 i 1.
while
0.1.2 Ixl = max (lXii, i = l, ... ,n)
Given two points x,y ~ E p(x,y) will denote the euclidean distance
between x and y, i.e.,
2
0.1.3
If M is a non-empty subset of X, x EX, and a > 0, then we
write
0.1.4 p(x,M) inf{p (x,y):y E. M},
0.1.5 S(M,a) ... {x€X: p(x,M) < a},
0.1.6 S[M,al {x€X: p(x,M) ~ a},
0.1. 7 H(M,a) ... {x Ex: p (x,M) al.
S(M,a) , S[M,al, and H(M,a) will sometimes be referred to as the
open sphere, the closed sphere, and the spherical hyper surface (of radius a about
M) •
c:
The closure, boundary, complement, and interior of any set M X is
denoted respectively by H, aM, C (M), and 1 (M) •
If {x } is any sequence such that lim x ... x, then this fact is simply
n n
n-+ oo
denoted by x -+ x.
n
We shall frequently be concerned with transformations Q from. X to aX
x
(the set of all subsets of X). Given Q:X -+ 2, and M c: X, we write
0.1.8 Q(M) ... U{Q(;x.):x E M}.
where
0.1.9 U{Q(x):x~M} ... x'l.M {Q(x)}
is a family of transformations from X to aX with I
as an index set, then
3
0.1.10
X
denotes the transfoDDation fran X to 2 defined by
0.1.11 Q(x) .. U{Qi (x) :i (-I}.
Given two sets M, NeX, their difference is denoted by M'N. Given two map~
lTt, and lT with IT'1 Tr2 we will denote the composition map_
t 0
Sometimes we will use the logic symbols 3 ,E,), V and =1:1- meaning "there exists",
"belonging to", "such that", "for all" and "implies".
Sometimes the following simplified symbols will be used
t'~L
and It;, .::. I
U
U(t(x) : x~M) , tCx) •
x'tM
4
0·.2 TernrinoZogy
0.2.1 DEFINITION
Given a oompaot set M c:. E" a oontinuous soalar' funotion v = ip(x)"
defined in an open neighborhood N(M) of M is said to be positive (negative)
semidefinite for the set M in the open neighborhood N(M) if
ip(x) .. 0 for aU xEM
t(x) ~ 0 (~(x) ~ 0) fop all xEN(M) \ M
If N(M)" E" then the soalar' fu:notion V" ip(x) is said to be positive (negative)
semidefinite for the set M. If M" {O} and N (M) =E" then the soalar' funotion
v .. ~(x) is oalled positive (negative) semidefinite. If fop the set M" a
funotion V" Hx) defined in a neighborhood N(M) with cp(x) .. 0 fop x t-M
is not semidefinite" '/.lie shaU oaU it indefinite.
0.2.2 Remazak
The definition (0.2.1) as well as the following definitions (0.2.4)
appl.t.s to a slightly larger class of sets than the compact sets, namely for the class
of closed sets with a compact vicinity; viz closed sets M, such that for some
a a) \
> 0 the set S(M, M is compact.
o • 2 • 3 E:x:amp le
°
If X is locally compact, then for sufficiently small > 0, the set
C(S[x,o) is a set with a compact vicinity.
o .2.4 DEFINITION
Given a oompaot set Me E" a oontinuous soalar' funetion V" Hx)"
defined in an open neighboPhood N(M) of M is said to be positive (negative)
definite fop the set M in the neighbophood N(M) if it is
