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Lecture 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. Szeg5 Universit& degli Studi di Milano. Milano, Italy
1967
Dynamical Systems'. Stability Theory and Applications
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 9 Heidelberg 1967. Library of Congress Catalog Card NumbeI 67 - 25757 Printed in Germany. Title No.7355.
PREFACE
This book given by N . P .
began as a series of lecture notes of the course
Bhatia at the W e s t e r n R e s e r v e 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 C a s e Institute of Technology during the s u m m e r of 1965. These courses w e r e meant for different audiences, on one side graduate students in mathematics,
and
on the other graduate students in systems theory and physics. ~owever
in the process of developing these notes w e have found
a
n u m b e r of other results of interest which w e decided to include ( See 1.9, 2.7, 2.8, 2.11, 2.14,
3.3, 3.$, 3.5, 3.7, 3.8, 3.9 ). Therefore, this
m o n o g r a p h is of a dual nature involving both a systematic compilation of k n o w n results in dynamical systems and differential equations and presentation of n e w T h e o r e m s
a
and points of view. As a result, a certain
lack of organizational unity and overlapping are evident. The reader should consider this m o n o g r a p h not as a polished, finished product, but rather as a complete survey of the present state of the art including m a n y n e w open
areas and n e w problems.
Thus, w e feel that
these notes fit the special aims of this Springer-Verlag
series. W e
do
hope that this m o n o g r a p h will be appropriate for a one year graduate course in Dynamical Systems. This m o n o g r a p h is still devoted to a mixed audience so w e have tried to m a k e the presentation of Chapter I (Dynamical Systems in Euclidean Space) as simple as possible, using the most simple mathematical techniques and proving in detail all statements, even those which m a y be obvious to m o r e mature readers. Chapter 2 (Dynamical Systems in Metric Spaces) is m o r e advanced. Chapter 3 has a mixed composition : Sec~ons 3. i, 3.2, 3.6, 3.7 a~d 3.8 are quite elementary, while the remaining part of the chapter
is a d v a n c e d . In t h i s l a t t e r p a r t we m e n t i o n m a n y p r o b l e m s w h i c h a r e s t i l l in an e a r l y d e v e l o p m e n t a l s t a g e . A s i z e a b l e n u m b e r of the r e s u l t s c o n t a i n e d in t h i s m o n o g r a p h h a v e n e v e r b e e n p u b l i s h e d in book form b e f o r e . We would l i k e to t h a k P r o f . W a l t e r L e i g h t o n of W e s t e r n R e s e r v e U n i v e r s i t y , P r o f . M i h a i l o M e s a r o v i d of C a s e I n s t i t u t e of T e c h n o l o g y , and P r o f . M o n r o e M a r t i n , D i r e c t o r of I n s t i t u t e for F l u i d D y n a m i c s and A p p l i e d M a t h e m a t i c s of the U n i v e r s i t y of M a r y l a n d , u n d e r w h o s e s p o n s o r s h i p the a u t h o r s h a d the c h a n c e of w r i t i n g t h i s m o n o g r a p h . We w i s h to t h a n k s e v e r a l s t u d e n t s at o u r universities,
in p a r t i c u l a r , A. C e l l i n a , P . F a l l o n e , C. S u t t i and G . K r a m e r i c h
f o r c h e c k i n g p a r t s of the m a n u s c r i p t . We a r e a l s o i n d e b t e d to P r o f . A. S t r a u s s and P r o f . O. H a j e k f o r m a n y h e l p f u l s u g g e s t i o n s and i n s p i r i n g d i s c u s s i o n s and to P r o f . J. Y o r k e f o r a l l o w i n g to p r e s e n t h i s new r e s u l t s in S e c . 3 . 4 . We w i s h a l s o to e x p r e s s o u r a p p r e c i a t i o n to M r s . C a r o l Smith of T E C H - T Y P E C o r p . , who t y p e d most of the m a n u s c r i p t . T h e w o r k of the f i r s t a u t h o r h a s b e e n s u p p o r t e d b y the N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r G r a n t s N S F - G P - 4 9 2 1 and N S F - G P - 7 0 5 7 , w h i l e the w o r k of the s e c o n d a u t h o r h a s b e e n s p o n s o r e d b y the CNR, Comitato p e r l a M a t e m a t i c a , Gruppo N 0 11, and by the N a t i o n a l F o u n d a t i o n u n d e r G r a n t N S F - G P 6114.
The a u t h o r s
M a r c h 1967
TABLE OF CONTENTS Page
0
Notation , Terminology and Preliminary Lemmas
0.i
Notation
i
0.2
Terminology
4
0.3
Preliminary Lemmas
6
1
Dynamical Systems in a
i.i
Definition of a continuous dynamical system
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 a n d Liapunov functions
85
1.8
Topological methods
96
1.9
Topological properties of attractors
99
Euclidean Space 9
i. i0 From periodic motions to Poisson stability
105
i.ii Stability of motions
108
2
Dynamical Systems in Metric 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
Liapunov functions and asymptotic stability of compact sets
143
2.8
Topological properties of
157
2.9
Minimal sets and recurrent motions
2.7
A (M)
A(M) and P (M)
163
2.10 Stability of a motion and almost periodic motions
171
2.11 Parallelizable dynamical systems
182
2.12 Stability and asymptotic stability of closed sets
201
2.13 Higher prolongations and stability
220
2.14 Higher prolongational limit sets and generalized recurrence
235
2.15 Relative stability and relative prolongations
242
3
The second method of Liapunov 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
407
CHAPTER 0
Notation , Terminology and Preliminary Lemmas
O. i
Notation T:
topological space
X:
metric space with metric
E:
real euclidean space of n-dimensions
E2:
p
the real euclidean plane
G:
group
R:
set of real numbers.
R+:
non-negative real numbers
R-:
non-positive real numbers
I:
set of integers
I+:
set of non-negative integers
I-:
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 which have been used to denote real numbers), (notable exception If
z,
t,s,k,v
and
small greek letters real numbers
which denotes a mapping).
x = (Xl, ...,Xn) EE,
llxll
will denote the euclidean norm of
x
i.e.,
n
0.1.1
2 1/2
l[xll
while
0.1.2
Ixl --~= (Ix~l, i-- l,...,n~ Given two points
between
x
and
y,
w
i.e.,
x,y ~ E
p(x,y)
will denote the euclidean distance
2
n
0.1.3
p(x,y) =[i i [x i -
If
M
yi)2] 1/2
is a non-empty subset of
X,
xs
and
~ > 0,
then we
write
0.1.4
p(x,M) = inf{p(x,y):y([M},
0.1.5
S(M,a)
-- { x E X :
~(x,M)
< a},
0.1.6
s[8,a]
-- { x ~ X :
~(x,M)
.< ~},
0.1.7
H(M,a)
= {x~X:
O(x,M)
= a}.
S~,a),
S[M,a],
and
HfM,a)
will sometimes be referred to as the
open sphere, the closed sphere, and the spherical hypersurface
(of radius
a
about
M). The closure, boundary, denoted respectively by If
{x n}
complement,
M, 8M, C(M), and
and interior of any set
x
is
I (M).
is any sequence such that
llm n-~
denoted by
M~X
x n = x,
then this fact is simply
~
§ x. n
We shall frequently be concerned with transformations (the set of all subsets of
0.1.8
X).
Given
Q:X § 2X,
and
M cr_ X,
Q
from
X
to
2X
we write
Q(M) = O f Q ( x ) : x E M } .
where
O{Q(x):x6M} =
0.1.9
If as an index set,
{Qi }, then
i 6 I,
U
x4M
{Q(x)}
is a family of transformations
from
x
to
2X
with
I
3
o. i . lo
q = U{qi: i e 1}
denotes the transformation from
0.1.11
X
to
2x
defined by
q(x) = U{qi(x) :• { I}. G i v e n two s e t s
Zt~ and Trz w i t h
M , N c X , t h e i r d i f f e r e n c e is d e n o t e d by M'~N. Given two map~
1T~ o "U"2 w e w i l l d e n o t e the c o m p o s i t i o n map.
S o m e t i m e s w e w i l l u s e the l o g i c s y m b o l s ~ , E, ~ , V and - - ~ m e a n i n g " t h e r e e x i s t s " , "belonging to",
"such that", "for all"
and " i m p l i e s " .
S o m e t i m e s the f o l l o w i n g s i m p l i f i e d s y m b o l s w i l l be u s e d :
and
", U(~(x)
2~-I _ : xeM)
-~ tJ~(x) x'~M
9
4 O. 2
Terminology
0.2.1
DEFINITION Given a co,pact set
M~E,
defined in an open neighborhood 8emidefinite for the set
~(x)
M
of
(~(x)
semidefinite for the set
M.
If
.< o)
if
N(M)
for all
xs
for all
x~N(M) \ M
is said to be positive (negative)
v = ~(x)
M = {0}
v = #(x),
is said to be positive (negative)
in the open neighborhood
then the scalar function
N(M) = E,
M
= 0
~,(x) >. o
If
N(M)
a continuous scalar function
and
then the scalar function
N(M) =E,
v = ~(x)
is called positive (negative) semidefinite.
function
v = ~(x) defined in a neighborhood
N(M)
If for the set with
~(x) = 0
M, for
a x~M
is not semidefinite, we shall call it indefinite. 0.2.2
Remark The definition ~.2.1)
as well as the following definitions
~ .2.4)
a~pL,~s ~o a slightly larger class of sets than the compact sets, namely for the class of closed sets with a compact Vicinity; viz closed sets B > 0
the set
0.2.3
Example If
C(s[x,~])
0 .2.4
S(M ,8) \ M
X
M ,
such that for some
is compact.
is locally compact, then for sufficiently small
~ > 0,
the set
is a set with a compact vicinity.
DEFINITION Given a co~pact set
defined in an open neighborhood definite for the set
M
M~E, N(M)
a continuous scalar function of
M
v = ~(x),
is said to be positive (negative)
in the neighborhood N(M~
if it i8
5
If
r
= 0
r
> 0
x EM
(r
then the real-valued function
N(M) = E,
(negative) definite for the set function O. 2.5
r
M.
~1 > ~2"
~ = ~(~)
N(M) =E,
and
e(Vl ) >~ c,(~2) whenever
>~(~2 )
~i > ~2"
v = #(x),
if there exists an increasing function
such that
0.2.7
~(~)
and such that
++
~
as
satisfies in
r
~(oCM, x ) )
0 .2.8
.< Ir
then the real-valued function
~ §
|
the inequality
E
M
l
a compaot set
is called radially unbounded for the set
v = r
M.
DEFINITION If
M
is closed set (not necessarily compact) and the function
satisfies the requirements of definition (0.2.1) ~or 0.2.4) weakly 8emidefinite (or weakly definite) for the set r
i8 defined in
increasing function
0.2.10
~(~1 )
DEFINITION
= ~(~)
further
then the scalar
is called strictly increasing if
and it is called increasing if
Given a scalar function
.2 . 9
i8 said to be positive
v = r
M = {0}
\ M.
DEFINITION
whenever
0
If
x(:N(M)
is called positive (negative) definite.
A scalar function
0.2.6
for all
< 0)
S(M,~)
e(E)~(O) = 0
for some ,
M
~ > 0,
such that
aCPCx,M)) .< $(x),
x6SCM,6),
then
r
in the open set
v = r is called
N(M).
If
and if there i8 a strictly
6
holds, then
is called (positive) definite for the set
~(x)
M
in the neighborhood
s (M, ~). 0.2. ii
DEFINITION If
M
is a closed set and in the neighborhood
real-valued function
8 = 8(~)
v = ~(x) satisfies the condition
is an increasing function, then the function
uniformly bounded for the set
0.2.13
M
in
v = ~(x) i8 called
N(M).
DEFINITION If
is a closed set and there does not exist an
MCE
the real-valued function M
the
l~(x)i ~ ~(o(M,x))
0.2.12
where
N(M)~S(M,=))
in the set If
v = ~(x) then
S(M,n),
is
~(x) will be called indefinite for the set
i8 a closed set, a continuous real-valued function
M~E
will be called indefinite for the set
The properties different spaces:
the (n + i)
O.2. I&
of the sets
in
M
~(x) = k
~(x) which
in an open neighborhood
T > 0
D
v = ~(x)
can be investigated in two (v,x)
and the
In this latter case one actually considers
(- = < k < + =).
of real numbers is called relatively dense if there
such that
D~
N(M),
N(M).
dimensional Euclidean space (x).
M.
DEFINITION. A set
is a
M
of the scalar function
n-dimensional Euclidean space
such that
at least weakly 8emidefinite for the set
is not at least weakly semidefinite for the set
properties
n > 0
(t - T, t + T) ~ ~
for all
t E R.
the
0.3
Preliminary Lemma8 We shall now state a few obvious properties of definite (or semidefinite)
functions both in the space
(v,x)
and in the space
(x).
We shall define in the
following corollaries properties of real-valued functions with respect to a compact set. The statements are identical in the case of sets with a compact vicinity and weaker when, instead of considering compact sets, one considers closed, non compact sets.
In particular,
the statements concerning definite functions become statements
on weakly definite functions, as it must be obvious to the reader by comparing definitions
O. 3.1
(0.2.1) and (0.2.4) with the definition
LEMMA A continuous scalar function
for a co,pact set O. 3.2
M
if
M
v -- r
is positive (negative) definite
is the absolute minimum (maximum) of the function.
LE~4 A continuous scalar function
semidefinite for the co.pact set any hy'persurface on which exist any point O. 3.3
(0.2.9).
= 0
for
x ~M,
is at least
if and only if there does not exist in
E
@(x) changes its sign and it is definite if there does not
such that
y~M
M
v = ~(x), r
~(y) = 0.
LEMMA Necessary and sufficient condition for the continuous real-valued
function
v = ~(x)
neighborhood = a(~)
0.3.4
and
N(M)
to be positive definite for the compact set
M
in some open
is that there exists two strictly increasing, continuous functions
6 =8(~)
a(p(M,x))
such that $ ~(x)
$ 8(p(M,x)),
e(O) = 8(0)
= 0
8
Proof:
The condition
(0.3.~) is clearly sufficient.
e (y) = inf{~(x): 7 .< ~(x,M)
To see the necessity,
define
.< ~},
and
S (y) = s u p { ~ ( x ) : 0 ( x , M ) where
6 > 0
is such that
N(M)~
(0(x,M))
and
~ (y)
and
B (7)
Then indeed
.< ~(x) .< 8 (p(x,M))
a (y) > 0, 8 (y) > 0
~ (y), 8 (y)
increasing functions
S(M,6).
are continuous.
N o t i c e that and the functions
.< 7},
a(7)
for
are increasing.
and
8(7)
y ~ 0
and
~ (0) -- 8 (0) = 0,
Now, there exists strictly
defined over an interval
0 ~ 7 ~ 6' < 6,
such that
~(~') .< a (Y) .< ~ (y) and
~(0) = 8(0) = 0.
(6') = q, q > 0. Yn + 0
as
For example,
.< 8 ( u
~(Y)
may be chosen as follows.
Then there is a sequence of points
n § ~,
such that
~ (Yn) ~
~n '
and
q(Yn ~(y)
=
these
8(7).
a(y)
and
Yl = 5'
N o w define
Y)
(n + l)(n + 2)(7 n - 7n_l )
for
8(7)
71 > Y2 > 73 > .... > 0,
n
n + "i -
The existence of
Let
Y n ~ 7 ~ 7n+ I, n = 1,2,
...
may be d e m o n s t r a t e d in the same w a y and ( O . ~ . ~ The t h e o r e m is proved.
holds with
CHAPTER i
DYNAMICAL SYSTEMS IN A
EUCLIDEAN SPACE
Definition of a continuous dyneonical system.
i.i
1.1.1
DEFINITION A transformation
system
~:E • R § E i8 said to define a dynamical
(or continuous f l o w ~ )
(E,R,~)
on
E
if it has the following
properties: i) 1.1.2
ii)
~(x,0)
for all
= x
~(~(x,t),s)
x~E for all
= ~(x,t + s)
x ~E
and all
t,s 6 R . iii)
~
i8 continuous
For every x
: R § E
of
is called
R
x E E into
the m o t i o n For every
t
: E § E
the m a p p i n g E
such that
through tE R
such that
~
induces
a continuous
~ (t) = ~(x,t). x
map
This m a p p i n g
~x
x.
the m a p p i n g
~t(x)
~
= ~(x,t).
induces The map
a continuous ~
t
is called
map transition
(or action).
1.1.3
THEOREM The mapping
-t
defined by -t
(X) = ~ (x,-t)
~t .
is the inverse of the mapping
Proof. applying
It m u s t be proved to the point
x:y = ~(x,t)
that
x EE
the m a p p i n g
[- t)-i = ~
the m a p p i n g ~
-t
.
-t
.
t ~ ,
This can be e a s i l y
s h o w n by
then to the image point of
The image point of
y
under
this mapping:
10
z = -t(y)
must coincide with
x.
In fact, using axioms (i) and (ii)
we have
z
=
~
-t
(~(x,t))
=
~(~(x,t),-t)
=
~(x,t-t)
=
~(x,0)
=
x,
which proves the theorem.
I. i. 4
THEOREM t
The mapping
is a topological transformation of
E
onto
itself.
Proof.
The map
~
t
is an onto mapping.
image points of points to one.
~(x,-t) E E.
In fact, all points
x 6E
For the same reasons the map
are t
is one
In fact the statement
~(x,t)
= ~(y,t)
= z
x,y,z6. E
t~R
implies, by application of the inverse map
x
which shows that
=
t
y
=
-t,
fixed
that
~(z,-t)
is one to one. -t
Since, by the definition 1,1.1,
is obviously continuous the
theorem is proved.
As a consequence of this fact, it follows that the dynamical system ~is
a one-parameter group of topological transformations, meaning by this
that for each value of
t ~R
a topological transformation is defined and,
furthermore, the transformation { t}, t E R
1.1.5
t
forms a group.
We claim that the set
is a Kroup with the group operation defined by ~ t ~ s -- ~ t+s
.
II
~o
Hereby the inverse.
i) ii)
~t,
is the identity element and for any
-t
is
In fact t o
~ ~ t
iii)
= ~
-t
t+o
= z
t
= ~t-t = n~
t( s q) = t
s+q
and furthermore = t+(s+q)
= ~(t+s)+q = ~t+s ~q = ( t s)~q
so that all axioms of a group are satisfied.
Notice also that
we have in fact a commutative group as: iv)
~t
s
t+s
s+t
s
t
A simplified notation
1.1.6
In most of the following work it will be inessential to distinguish a particular mapping
~.
When its use will not be misleading, we shall,
therefore, introduce the notation xt
xt
instead of
is, therefore, the image point of a point
~(x,t).
x ~ E
For a fixed
under the mapping
t, t
bye.
induced
In this simplified notation the first two axioms of (1.1.2) take the following very simple form:
x0 = x
and
In line with the above notation
1.1.7
MS = { x t : x s
Whenever xS
and
M Mt
or for
S
(xt)s = x(t+s). if M C E
and
M{t}
S CR
we define
t s
is a singleton, namely, {x}S
and
M = {x}
or
S = {t},
we write
respectively.
Remark
1.1.8
One can define dynamical systems in a more general framework as the triplet and
~
(T,G,~),
where
T
is a topological space,
G
the map which satisfies axioms similar to 1.1.2.
a topological group In this chapter
12
besidem (E,R,n) where
I
we shall once in a while discuss properties of
is the group of integers.
The dynamical system
called a discrete dynamical system or continuous cascade.
(E,I,~),
(E,I,~)
is
In the advanced
Chapter 2 we shall discuss the more general case of the dynamical system (X,R,n),
where
X
is a metric space and mention
related to the dynamical system G
(T,G,~) , where
more general problems T
is a topological space and
is any topological group.
1.1.9
Notes and references. The introduction of the definition of a
cannot be attributed to any one person.
dynamical system
Some historical remarks on the
generation of such concepts can be found in a paper by V. V. Nemytskii
[i0]
and in a paper by G. D. Birkhoff [i, Vol. 2 pg. 710]. The first abstract definitions of a dynamical system can be found in the works of A. A. Markov [i] and of H. Whitney [i,II].
Most concepts have
been introduced by Poincar4 and his successor, G. D. Birkhoff~ in the framework of the theory of dynamical systems defined by ordinary differential equations. The theory of dynamical systems received new impetu~ by the publication of the books by Nemytskii and Stepanov, G. T. Whyburn, Gottschalk and Hedlund [4] and Montgomery and Zippin.
13
i. 2
Elementary concepts.
1.2.1
DEFINITION For any fixed
xKE
and
a ~ hER,
the trajector~ segment is the
set x[a,b] = {xt:t ([a,b]}
1.2.2
For every fixed
x EE
the trajectory or orbit (1) through
x
is the
set
1.2.3
xR = { x t : t ~ R ]
The sets
xR +
and
xR-
semi-tr~ectory through
are respectively called positive and negative x.
By the axioms defining a dynamical system, it follows that:
1.2.4
For all
t ER
xR = (xt)R
it follows that the trajectory segment
From the properties of X
is a closed and bounded set.
1.2.5
Remarks on trajectories and motions. The trajectory
xR
is a set, a curve through the point
x.
Therefore, a trajectory is a purely geometrical concept in which the dependence upon the time does not show.
On the whole trajectory
direction of the motion appears. in which the point
yt:y E xR
xR
not even the
By direction of motion we mean the direction
moves with increasing
t
on
xR.
In some cases
it may be possible to recognize on a trajectory a positive and a negative direction of motion, that is, the case if one maps
xR +
and
xR-
separately
(1)Throughout this book the word trajectory will be preferentially used.
14
for any trajectory which is not closed and bounded.
It can be seen in many
drawings
are represented
showing various
flows that the trajectories
sets of points with arrows. on
xR.
These arrows show the direction of the motion
In the case of discrete dynamical
in many cases a disjoint
as lines or
set of points.
systems,
the trajectory
xl
is
For this reason in the literature
a
the set
xl
is very often called/punctual
In some parts of these notes, following notation for trajectories
1.2.6
in particular
in Chapter 2,the
and semi-trajectories
x R + =A 7+ (x),
xR ~ y(x),
trajectory.
will be adopted.
xR - =A X - (x).
+ The symbols
u
and
Thus the notation
7(x)
that the trajectory xR
X-
denote the maps from
E
to
2E
defined by 1.2.6.
etc. will be adopted when it is desired to emphasize
is an element of the maps
will be used when the simple geometrical
E § 2 E,
while the notation
concept of trajectory
is
predominant. The motion
~
through the point
x ~ E
is a mapping which maps
R
x into
E
motion
or to be more exact maps ~
through a point
x
parametrized point
xt
by
t.
xEE
onto
xR.
xR.
One can also say that the
is the locus of
A motion can be visualized
moves on
tion between
R
XR,
for all
tER,
as the law w i t h which the
In order to be absolutely
the concept of trajectory
xt
clear in this basic distinc-
the concept of motion
we ~X ~
may think of the law
1.2.7
xR
as the rail on which a material point moves according
~ . X
DEFINITION A point
x EE
having the property that
xR =
is called critical
or ~ t a t i o ~
{x} or equilibrium or rest point.
to
15
1.2.8
REMARK Critical points are the fixed points of the mapping
~t:E §
The definition 1.2.7 has defined rest points as a particular type of trajectories.
It must be remarked that a critical point can be defined also
from the properties of the corresponding motion
1.2.8
x"
DEFINITION A point
xEE
to which there correspond8 a motion
~x"
h~ving
the property that
x
(t) = x ( t )
=x
for all
t~ R
is called a critical point.
Some basic properties of critical points shall now be proved.
1.2.9
THEOREM If for
1.2.10
x
a < b,a,bER, xEE
x[a,b] = {x}
is a critical point.
Proof.
We shall give the proof for the case of the discrete system
(E,I,~).
For the case of the theorem a very simple proof shall be given as Corollary 1.2.24.
For the case of discrete systems the statement of the theorem could
be rephrased as follows:
1.2.11
x(h I + i) = xh I
then for all
hs
x(h
and
x
If for an
+ 1) -- x h
is a critical point.
hl~ I,
,
16
In fact, by the axiom 1.1.2
x(h + i) -- x(h I + 1 + h-h I ) ~ (x(h I + 1)) ( h - h I)
and because of 1.2.11 it follows that
x ( h + 1) = (x(h I ) ) ( h - h I ) ~- x ( h I + h - h i ) ~ x ( h ) .
Exercise.
1.2.12
Prove the analogue of theorem 1.2.9 for
(E,R,~).
THEOREM
1.2.13
The set of critical points is closed.
Proof.
It must be show~ that the limit of a sequence
{xn}
of critical
points is a critical point. From the definition 1.2.7, for all x t= x . n
t ER,
it follows that
On the other hand, from the continuity of the mapping d e f i n i n g ~
n
we have that if
x
§ x,
then
x t § xt.
n
1.2.14
xt = x
for all
t~R.
THEOREM
~> 0
If for every either
y R+ C
Proof. for
Thus,
n
S (x, ~)
or
there exists at least one y R- r
S (x, E), then
x
y s S(x,z)
is a rest point.
If x i s n o t a c r i t i c a l p o i n t , t h e n t h e r e i s a Tm o
o~ 09 x n
n
+
Xk'
and
xktk § Yk for each fixed n
n
P(Xk,X ) .< ~,
for
n >. k.
we notice that
and
and
8
Then considering the sequences
p(x,x n) .< p(X,Xn) + P(Xn,X n) .< p(X,Xn) + in,
nn nn i O(Y'Xntn) "< 0(Y'Yn) + P(Yn'Xntn) ~< 0(Y'Yn) +--n' which shows xn § x n
We shall
This is so, because there are sequences
We may assume, without loss of generality that
kk i p(yk,Xntn) < ~
be a sequence with
{xn} 9
is compact, we may assume that
y @D+(x). 9
D+(M)
Then there is a sequence
{x },{t } k = 1,2 ..., k.
has stronger properties than
THEOREM If
Proof.
D+(M)
xnt n § y, i.e., also n n
the same is closed.
y ~ D+(x ) Hence also
and
that
y ~ D+(M)
and
Q.E.D.
Additional properties of prolongations will be presented in Sections 1.5 and 2.4 to which the reader is referred. 1,4.14
Notes and References The concept of prolongation is due to T. Ura [2].
in
E2
The example (1.4.7)
can be found in the work by H. Poincar4 [i, Vol. i, pp. 44] and in the
work of I. Bendixson9 but without a formal definition of prolongation.
46
1.5
Lagrange and Liapunov Stability for Compact Sets In the last sections we have performed what can be called the
anatomy of dynamical systems.
In fact, we have been concerned with the
definitions and the essential properties of the elements which constitute a dynamical system:
trajectories, motions, invariant and minimal sets,
prolongations, etc.
The limit sets of trajectories and prolongations have
been defined.
We have proved (1.3 and 1.4) that these limit sets are
closed and invariant sets.
In the next sections we shall be concerned
with what can be called the "physiology" of dynamical systems, i.e.~the study of the behavior, the relationsTand the relstive properties of its elements.
Our analysis will start from the most simple properties.
Consider a point
x ~ E ;
a dynamical system a limit set.
~
the first properties of such a point within
can have is that the associated trajectory
In fact, if
xR
tends to infinity both for trajectory
xR
t + +~
xR
does not have such a limit set, and
t + -~
has
xR
in a certain way.
If a
does not have any limit set this fact classifies this
trajectory in the dynamical system in a certain way which will be clarified in Chapter 2.
Consider then the case of a trajectory which has a limit set
(either positive or negative). mean that the trajectory
xR
If the limit set is not compact this would will cover a non compact region of the space
(Example: an infinite strip), but not the whole space. interesting case, is when a trajectory
1.5.1
has a compact limit set.
DEFINITION A point
L+-stable) if xR~ if xR"
xR
The next, and more
x ~ E is said to be positively Lagrange stable (or is compactj negatively Lagrange stable (or L--stable)
is compact and Lagrange stable if xR is compact.
4?
In the space to the fact that
E
the property of Lagrange stability is equivalent
xR + is positively bounded.
The property of a point
L+-stable can be generalized to a whole set
i. 5.2
DEFINITION A set
points
B C E .
x ~ B
BC are
i8 called
E
is called
L+-stable (L--stable, L-stable) if all
L+-stable (L--stable, L-stable) . A dynamical system
L+-stable (L--stable, L-stable)
L§
(L--st~le, L-stable) .
1.5.3
D~INITION If a point
called
L-unstable.
points
x E E
are
x ~ E
i8 neither
nor
L+ ,
A dyn~nical system ~
if all points
x ~ E
are
L--stable it will be
is called unstable if all
L-unstable.
Lagrange stability is both a property of the trajectory and the motion associated with a given point system ~ .
In the space
E
x ~ E
the statement that
equivalent to the concept that the motion
~
within a dynamical
x ~ E
through
is
L+-stable is
x ~ E
is posl-
X
tively bounded. The properties of Lagrange stable points are essentially characterizable by the properties of their compact limit sets. have been extensively investigated in Section 1.3. stable points in the space
These properties
Thus for Lagrange
E , from Theorem 1.2.35
and 1.3.13
it trivially
follows that:
i. 5.4
COROLLARY If a point
x { E
is
L+-stable,
connected and contains a minimal set. L+-stable also
~B
i8
L§
then
^+(x)
If a c~o.~ed set
is co~pact,
BC E
is
48
Figure
1.5.5
BI y////.y// / / / / / / / z z ,-,-
t
i
Q.
f
It must be pointed out that the second part of the Theorem 1.5.4 holds only if
B
is &Jos~d .
of a non-~lo~ed
set, which is
It is, in fact, easy to produce an example L+-stable, but
example, the flow represented in Figure
~B
1.5.5
is not.
Consider, for
whose trajectories are a
family of parallel straight lines having their positive limit set of the curve
~
each element of which is a critical point.
asympototically to the straight line bounded by the two straight lines not belong to all
x e B ,
B . ~-~
B I.
BI
Consider the open set
and B 2
This set is non-cLosed is compact.
The set
The curve
such that
and it is
~B = B I U
B2
B1
~
tends B
and B 2
do
L+-stable since for is on the other
hand not L ~ s b l e . The concept of Lagrange stability, if applied to certain sets, fails completely to provide us with any additional information about the properties of such sets.
For example, the expression "a compact positively
invariant
B "
B
L+-stable set
is clearly
redvndbnt
since, if the set
is compact and positively invarlant, this implies that for all points
x e B , true for
xR +
is compact and therefore
~B 9
B
is
Clearly the property of being
L+-stable.
The same is
L-stable has a non
trivial meaning if applied to non compact sets or, in particular, to the whole dynamical system ~ .
Also, the
case
of a
L-stable dynamical
49
system may be a rather pathological one like the case of a system in which all trajectories are compact (for example when all motions are periodic). From these remarks it must be obvious that Lagrange stability is a rather weak concept which does not provide us with much information regarding the qualitative behavior of dynamical systems.
In particular, it is strictly
a property of the trajectories within a set, not related with the properties of the system outside the set.
The concept which will be introduced next
will provide us with a much more precise characterization of the qualitative behavior of the dynamical systems. the points in a neighborhood M .
A
This will essentially be a property of
of a set
M CE
with respect to the set
In the case of sets with compact vicinity, these properties will be
characterizable in terms of the properties of the limit sets of the points in a set
N ~
M ,
and it is therefore closely related with the idea of
Lagrange stability.
1.5.6
DEFINITION Let
M CE
x ~ B , A+(x) # ~
to
be a compact set. and
A+(x)_CM .
Let
Then
B C'E
M
be such that for all
is called ~
attractor relative
B 9 The largest set
B
attraction of
M 9
1.5.7
A(M) = {x ~ E:
A(M)
and called
the region of
Thus
If there exists a called/attractor
will be denoted
A+(x) # @ , and 6 > 0
such that
If in addition
A+(x) C M}
S ( M , 6 ) C A(M) ,
A(M) = E ,
then
M
then M
is
is called
global attractor. * Instead of (positive) attractor, a set M , which satisfies the conditions of definition 1.5.6 is sometimes called quasi asymptotically stable set.
50
If there exists a then
M
~ > O,
such that
implies
x~S(M,~)
A+(x)~M
# ~,
is called weak attractor. The set
A (M) -- {x ~E:
1.5.8
A+(x) # ~ and
~0
A+(x)~ M # @}
is then called region of weak attraction. If
M
for all compact sets for all
Kt C S ( M , ~ )
KC
A(M) there exists a
t > T , > O,
If for all implies that
x # S(M, 6)
6>0
is an attractor and it is such that for all
then
M
such that
T =. T(K, 6) >, 0
is called a uniform attractor.
there exists a
xR+C
and
S(M,z) ,
M
~(~) > 0
,
such that
is called stable . Ifa
compact set is not stable, then it is called unstable .
From the definition it obviously follows that if
M CE
is an
attractor, then
1.5.9
~A(M) N 8M =
Notice that
i. 5. l 0
A~(M) ~_ A(M) .
DEFINITION Let
if there is a ^-(x)~M
.
MC
E
6 >
be a compact set. such that
Then
x ~ S(M, 6)
M
is a negative attractor
implies that
A-(x) # ~
The region of negative attraction or region of repulsion
~s defined similarly to what was done in
(1.5.7)
A-(M)
1.5 .ii
A(M);
has similar properties.
THEOREM If
MC
E
A- (M)
9
We shall now study the basic properties of the set set
and
i8 a compact attractor, the set
A(M)\His open.
the
51
Proof.
We have to show that for all
such that
S(x, e) C A(M)\M. The set
in the definition of attractor. as in Theorem that
xT ~ S ( M , ~ ) \ M
such that
x .
hence
.
Since
S(M, d ) \ M
Now let
By the definition of
Remark.
S(M,~)\M .
A(M)
g > 0 ~
is as
By the same argument ~ > 0
is open we can find an
such e > 0
Because of continuity of the map x
~ ,
and is thus a neighborhood
it also follows that
S(xT, ~ ) ( - ~ ) C A(M)~M~
Q.E.D.
Theorem
1.5.]1
is false if
Consider for that the flow shown in Figure
Figure.
1.5.13
is open, where
x ~ A(M)\M.
iS open, it further contains
A(M)~H is open.
1.5.12
S(M, ~ ) ~ M
it can be shown that there exists a
S(x~, ~) C
S(XT, e)(--T) of
1.3.13
x ~ A(M)\Mthere exists ~n
M
is not an attractor.
1.5.13 which has the
following properties: x ~ C(A{x}) ~
A-(x) C A({x})
x E' A({x}) =.+ A+(x)
= A-(x)
= {x}
Clearly, {x}i~,ot an attractor an~ its region of attraction A({x})
THEOREM
1.5.14
Let sets
is a compact set.
A(M)
Proof.
and
M CE
be a compact set.
~(A(M))
Notice that
Since, however for any result follows, from
Then if
M
is an attractor, the
are invariant.
x E A(M) t E R 1.2.36 .
if and only if and
x ~ E ,
A+(x) # ~
we have
and
A+(x) C_CM .
A+(x) = A+(x t) , the
Q.E.D.
In what follows additional properties of attractors will be defined.
52
1.5.15
DEFINITION Let
such that
be a compact attractor.If there exists a point
M CE
then the set
A-(x) ~ M # 0 ,
M
will be called
x ~ M
an unstable
attractor.
DEFINITION
1.5.16
Let set
M
will be called ~
there is a The set
be a positively invariant compact attractor.
M CE
n > 0
A(M)
asymptotic
stable attractor or ~n asymptotically stable set if
such that
defined in stability of
The
x~ S ( M , n ) \ M
A-(x) ~ M
= ~ 9
will in this case be called region of
1.5.7
M .
implies
If
A(M) = E ,
the set
M
will be called
, globally asymptotically stable .
1.5.17
DEFINITION Let
if there exists a M
be a negative attractor as in definition
MCE
n > 0
such that
x ~ S(M, 9 ) \ M
implies that
1.5.10 , A+(x) A M
will be said to be completely unstable (or negatively asymptotically
stable#.
1.5.18 tories,
Remark.
By reversing the direction of motion along the trajec-
sets which are completely unstable will become asymptotically
stable and vice versa.
1.5.19
THEOREM.
If
M CE
the condition that
is a compact attractor and it is positively invariant x C S(M,n)\M
lent to the condition that
Proof.
Let
n > 0
A-(x) ~ M = ~
is equiva-
i8 stable.
be such that
Now assume if possible, * or asymptotically
M
implies that
x ~ S(M,n)\ M
that there is an
stable in the large.
E > 0
implies
A-(x) ~ M ~ # .
such that for every
= ~ ,
53
6 > 0 there is a n < n 9
S(M, 6)
such that
C l e a r l y there is a s e q u e n c e tn > 0 ,
It n} , that
x&
x
.
.
{x } , x § M n n
p(x n tn, M) = e .
and
§ x ~ M
xR+~S(M,e)
W e w i l l show that
As
M
{t }
n
We may assume
and a s e q u e n c e
is compact,
w e can a s s u m e
is not bounded.
For o t h e r w i s e
n
w e can find a c o n v e r g e n t
subsequence,
and so a s s u m e that
t
+ t ) 0 . n
Since n o w and
xn § x ,
t ~ 0 ,
the s e q u e n c e Setting n o w
tn + t ,
this c o n t r a d i c t s
w e have
X n tn § xt.
positive
invariance
is not bounded.
{t n}
x n tn = Y n '
Since
xt ~ M ,
M.
Therefore,
of
W e may assume t h e r e f o r e
we notice
that
x n = yn( -tn)
sequence with
P(Yn' M) = e .
Since the set {y: p(y,M)
w e can a s s u m e
Yn § y '
p(y,M)
however, x ~ A-(y)
as .
with
x n = yn(-tn) Thus
of the a r g u m e n t
§ x ~M
A-(y)~M
~ ~ ,
= e
.
, whereas which
shows that the c o n v e r s e
Then
that
= e}
is compact, Then,
w e see that
is a c o n t r a d i c t i o n . is also true,
tn § + ~
Thus w e h a v e a
y E S(M,~)
-tn § - ~'
x E M
The last part
and the t h e o r e m is
proved.
1.5.20
Remark.
eSSential
in T h e o r e m
The c o n d i t i o n 1.5.19.
of a compact attractor, not satisfy T h e o r e m i. 5.21
Figure.
which
1.5.19
.
that
M
be p o s i t i v e l y
invariant
It is in fact easy to p r o d u c e is not p o s i 6 v e l y
is
the e x a m p l e
invariant and w h i c h d o e s
54
Consider in fact the flow represented in Figure
1.5.21 .
The
trajectories are a family of straight lines through the critical point On each trajectory the motion moves the point toward A-Cp) = @
and
A+(p) = {x}
the compact set for all
.
M ={x}U{y}.
p ~ E ,
Consider a point
x .
y E E ,
Thus all y ~ x
p G E
and
M obviously has the property that
and it is a compact attractor, however,
x .
A-(p)~
Theorem
M =
1.5.19
is obviously not satisfied.
Remark.
1.5.22 that
M
Theorem
be an attractor.
1.5.19
is also incorrect without the assumption
In fact then it is not true that if
positively invariant and compact and is stable.
1.5.23
A-(x) ~ M = @
for
Figure
i
;..~.-I,,-
~ ....
X
i, 89 %, 1/8,
~
"-~ X
then
M
Consider
Clearly for all
with abscisses ...
O,
and fill in the
rest of the flow as shown in Figure 1.5.23.
{0}
is
the sequence of equilibrium points
r"
point
x ~ M ,
This can be shown by the following counterexample.
on the axis
T
M
x E C({0})
,
Consider the equilibrium
A-(x) N
{0}
= @ ,
but
{0}
is not a stable set.
1.5.24
THEOREM. If
Proof. Here
M C E
Notice that xR + C
i8 closed and 8tablej then it is positively invariant.
x ~ M
implies
~ S(M,E) = M , e>O
as
xR+C M
S(M,c)
is closed.
for every ~e,c~
M
~ > 0 .
SS
is positively
invariant.
Remark.
1.5.25 Definition
Theorem
1.5.16
is positively
1.5.24
and in Theorem
invariant
shows that the fact that both in 1.5.19
it has been assumed that
is not a restriction.
is a necessary condition for stability.
In fact, positive
M
invariance
Thence
COROLLARY
1.5.26
If a compact set and if in addition
M
is an attractor it is asymptotically stable.
We now investigate and asymptotic
1.5.27
is stable, it is positively invariant
M CE
the relationship
between uniform attraction
stability.
THEOREM Let
M
be a compact asymptotically stable set. Then
M is
uniformly attracting.
Proof. valent
Notice first that the definition of a uniform attractor to the following:
if given
~ > 0
and a compact set
that K t C S ( M , ~ ) Let
K CA(M)
there is a define and
T
x
However, is a
for
6 > 0
K CA(M)
Let now
And let
such that
Set
T = sup{Tx:XEK}
there will be a sequence K
stable.
be given.
M
S(M,e)
N
of
yT
such that
As
For any
We claim that
{x } n
y~ ~ S(M, 6)
Since
is defined as
x
.
is a uniform attractor
be asymptotically
in
K
is compact we may assume that
such that
open neighborhood
e > 0
9
M
there is a T = T(K, 6)
M
y+(S[M, 6 ] ) C
= inf{t > O: xt~S(M, 6)}
since
T > 0
t > T .
be compact.
K C A(M)
otherwise,
a compact attractor
is equi-
T
such that x
S(M, 6) N C
M
S(M,~)
n
is stable,
x ~ K , is an attractor,
is finite. T
x
+ y E K .
is open,
such
For
§ +~ n Then there
there is an
The inverse image
56
N
*
= N(-T)
y .
of
N
Further,
Since
x
by the transition
N*(~) = N C
E N*
S(M,d)
for large
n ,
,
~
T
is open and a neighborhood
so that
we have
N*tC
T
n
< T
S(M,e)
for
for large
of
t > 9 .
n .
This
x n
contradicts
T
§ + ~. x
Hence
T < += .
Notice now that
x ~ K
implies
n
xT ~ S[M,~]
,
and so
Kt C S ( M , E )
for
attracting,
and the theorem is proved.
t > T ,
i.e.,
M
is uniformly
THEOREM
i. 5.28
A compact positively invariant set
is asymptotically
M~E
stable if and only if it is uniformly attracting. Proof.
Let
M
be positively
shall prove that
M
is stable.
Then there is a sequence t
n=
> 0 ,
such that
x
n
By uniform attraction Thus
Assume if possible that
{x n} ' t
Xn § x E M ,
E H(M,E)
n
be chosen small to ensure that
t > T 9
invariant and uniformly attracting.
S[M,E]
there is a
t n =< T .
for some
E > 0
such that
subsequences
{t
}
and
t
§ t ,
and
x
nk invariant, as
t nk
and also
M ~ H(M,e)
§ y .
Hence
M
it is asymptotically
stable.
proved
(the previous
theorem).
1.5.29
Remark.
necessary.
as
{x
y = xt E M ,
consider
x
t
nk %
is stable,
of t
. for
{x n } , }
%
"k
as
M
converge. is positively
~ H(M,e)
and since
.
This is impossible
M
is an attractor,
The converse of the theorem has already been The theorem is therefore proved.
The assumption
In fact,
S[M,e] C A ( M )
{xnk }
and
may
nk
y $ H(M,e)
= ~ .
Then
e > 0
S[M,e]t C S ( M , E )
% Let
{t n } ,
Indeed
There is then a subsequence
such that the corresponding
is not stable.
and a sequence
is compact,
T > 0
M
We
that
M
is positively
the following example
invariant
is
(Figure 1.5.30)
.
57
i. 5.3 0
The shaded region represents the
Figure.
set
M .
The point
0
is an
unstable attractor, (Example 1.4.9 iii) and
M
is uniformly attracting with
a suitable time-parametrisation,but it is not stable.
The dependence of various concepts is illustrated below in a chart
M Asymptotically Stable + positively invariant
+M stable
M stable
M attractor
M weak attractor
Remark.
1.5.31
M uniform attractor
The definitions given and the theorems proved so far
for compact sets, are meaningful and true under the slightly weaker hypothesis
that 1.5.32
M CE
is not a compact set,
but a closed set with a compact vicinity.
Exc~p les. i)
in Example
1.3.9 (Figure 1.3.10) choose any point
p
on the
58
periodic trajectory
weak attractor. ii)
y
.
The set consisting of the point
This set has no other property listed in
p
is a
1.5.6.
Consider a planar dynamical system defined by the following diffential equations in polar coordinates.
= r (l-r) = sin2 (e/2) The phase portrait consists of two rest points P2 = (i,0)
(Figure 1.5.33),a trajectory
together with the rest point a trajectory
y
and a rest point
be generally called a path circle have
P2
P2
1.5.33
Figure.
p
such that
All trajectories
Pl
A+(y) = A-(y) = {p} will
All orbits outside the unit
Pl ) have
in the interior of the unit P2
as their sole positive
as their sole negative limit point.
is an attractor with
attractor,
circle which
forms a path monogon (the union of
monogon).
circle (except the rest point
P2
on the unit
as their only positive limit point and their negative
limit sets are empty.
limit point and
y
Pl = (0,0), and
A(P2) = E 2 \{Pl } .
and is not stable.
The point
It is not a uniform
59
iii)
In the above example
(ii) ,
the set
M
the unit disc is asymptotically stable. attractor.
consisting of points on This set is also a uniform
However, if we consider a set
M*
consisting of points
on the unit disc and another point not on the unit disc, then is a uniform attractor, but it is not stable.
M*
A similar example
can be built out of example 1.3.9 ~i).
iv)
Consider again a planar dynamical system defined by the following differential equations in cartesian coordinates.
Xl = x2'
x2 = -xl
"
The phase portrait consists of a rest point coordinates a ~ circles with
P
P -- the origin of
periodic trajectories which coincide with concentric as center.
~y
compact invariant set in this
example is stable, but has none of the attractor properties. for example, the point 1.5.34
Figure
P
Thus,
is stable.
~X~ X~
v)
Cons~er
finally a planar dynamical system given by the differential
system in cartesian coordinates
Xl = x2'
x2 = sin2
2 2 xI + x2
x2 - Xl "
60
The phase portrait (Figure 1.5.35) consists of the rest point the origin of coordinates, a sequence which are circles with center
of periodic trajectories
P - Yn = {(Xl' x2): x~ + x~ =
All other trajectories are spirals. no attractor property.
{yn }
P --
The point
P
~}n
"
is stable, but has
No compact set except the point
P
is either
stable or a weak attractor.
Figure.
1.5.35
r
We shall now present further properties of stable and. asymptotically stable compact sets.
THEOREM
1.5.36
A compact set M C E
is stable if and only if each component of M
is stable. The proof is given in 2.6.8.
61
1.5.37
Remark.
Theorem
1.5.36
not been in any way restricted
implies that our theory would have
if instead of considering compact setswe
would have limited ourselves to the case of continua. We are now in the position of discussing the relative properties of and
A(M)
and
C(A(M))
This will be done in the next two theorems.
Similar theorems in a much stronger form will be proved in Chapter 2.
1.5.38
THEOREM Let
M
be an asymptotically stable closed invariant set with a
compact vicinity, then, if the set is completely unstable and Proof.
The set
open (by Theorem
C(M)
A(M)\Mis compact, the set
is its region of repulsion.
C (A(M)) is closed and invariant as the set
A(M~Mis
1.5.11)
.
and invariant
Notice now that for all A-(x) ~ ~ . of r~pui~ion
1.5.39
Hence
C(A(M))
since
(by Theorem
x E A(M),
1.5.14)
A-(x) C C(A(M))
is completely unstable and
x s M
implies
A-(x) C
M
as
M
M
its region
is invariant.
be a closed set with a compact vicinity.
completely unstable and invariant~ then, if the set vicinit~ it is asymptotically stable and
C (M)
If
C(A-[M))
M
is
has a compact
is its region of attraction.
THEOREM Let
M
be a positively invariant compact set, and let
be the largest invariant set contained in attractor, relative to Proof.
C (M)
and
COROLLARY of Theorem (1.5.38) Let
1.5.40
C(A(M))
For any
M .
Then,
M*
M*CM
is a stable
M .
x ~ M ,
A+(x) # ~
and compact, because
xR+CM
and so
62
xR +
is compact.
Again
A+(x) C M
, because otherwise
be a larger compact invariant subset of
M .
Hence
To see that it is stable, we must show that for A-(x) ~ M
Assume the contrary, i.e. t n § -~
,
with
invariant.
Xtn E M , so t h a t
Since
Notice now that larger
than
M
M
x ~ M~M
xR =
U(xt
~ xR =
, which is
and s o
M
is
n
~
# ~ .
M
x e M \M
(Xtn)R+~g
, as 2,
...)
will
is an attractor.
Then t h e r e
R+; n = 1,
[J xR
A+(x) U M
,
is
A-(x) N M
a sequence
M
is
,
we h a v e
= ~ .
{t
n
}
'
positively xRCM
.
is a compact invariant set which is
a contradiction.
Hence
A-(x) ~N ~=
~
for
each
stabie.
The next theorem will further clarify the structure of asymptotically stable sets and of their regions of attraction.
THEORem4
1.5.41
If
M ~ E
is a compact minimal set which is asymptotically stable,
then for all
x ~ A(M)
Proof.
M
As
A+(x) = M. M
the compact set
is compact and minimal, we have for each
Otherwise, since
will not be minimal if
= xR+U
M
of it.
implies that an attractor.
A(M)
y ~ A(M) Again if
asymptotically stable. Thus
~+
and
A+(x)
A(M)
is a neighborhood of
y ~
A(M)
But then
Now
xR +
1.5.6
is a compact subset
xR + .
Now
y ~
and hence
A(M)
A+(y) C M C x R
+ ,
\ x R + , then
A-(y) ~ M = ~ , as
\xR +
xR ~
is M
is
A-(y) ~ xR + = ~ , as A-(Y) ~ A(M)\M=~ .
is asymptotically stable.
The theorem is proved.
The property of stability of a (compact) set 1.5.19 ,
x R + = xR + U A+(x)
Therefore, definition
is open, and
and therefore
x E A(M) ,
is closed and invariant,
is a proper subset.
A+(x) is compact.
Notice that
Therefore
A+(x) C M ,
A+(x)
is compact, as
is applicable.
i8 asymptotically stable.
xR + ~ E
M , defined in Theorem
is a rather weak property which cannot be characterized by the
63
positive and negative limit sets of the points in a neighborhood of
M .
Such property can be characterized as a property of the first positive prolongation of
M ,
as shown by Theorems 2.6.5 and 2.6.6.
We shall close this section by stating some important theorems on the stability properties of the first positive prolongation of compact attractors. 1.5.42
THEOREM Let
M
be a compact weak attractor.
asymptotically stable set.
The region of attraction
coincides with the region of weak attraction D+(M)
Then
A (M)
D+(M)
is a compact
A(D+(M))
of
of
D+(M)
M . Moreover,
i8 the smallest asymptotically stable set containing
M .
The proof of this theorem is given in 2.6.17. 1.5.43
Notes and References Stability
theory for dynamical systems was essentially developed
by T. Ura [2] in the context of theory of prolongations.
Early results
and definitions can also be found in the book by Zubov [6]. The original defintions of stability and asymptotic stability for the case of differential equations are due to Liapunov.
In his work, however,
only local properties of equilibrium points are investigated.
The concept
of orbital stability (usually defined for limit cycles) found in many earlier works is a particular case of stability of sets (see, for instance, the book by L. Cesari Ill). The concept of attraction seems to have been used by many authors, but a syst~mstic study seems to have originated with the example of Mendelson Ill. The definition of weak attractor (1.5.6) is due to N. P. Bhatia [3].
64
Definition 1.5.16 is independent from stability.
Our whole
presentation of stability theory is motivated by this idea. us to prove Theorem 1.5.19 proving that asymptotic
This forces
stability implies
stability; while usually asymptotic stability is defined as stability plus attraction.
We have chosen this way of presenting asymptotic
stability to clearly point out how this is a property of the positive and negative limit sets
A+(S(M,6))
and A-(S(M,~))
only.
On the other hand, stability without attraction is not characterlzable in terms of the properties of the limit sets above. Theorem 1.5.27 is due to S. Lefschetz [2I. Theorem 1.5.28 is due to N. P. Bhatia, A. C. Lazer and G. P. Szeg~ [I].
65
1.6
Liapunov Stability for Sets. In what follows
be extended
and theorems developed
to the general case of a set
no means trivial.
M C E .
One of the major difficulties
erties of the neighboring longer characterisable
trajectories
of
M
These extensions is the fact
with respect
trajectories
tend to
M .
are by
that the prop-
to
M
are no
In addition to this
for non compact sets we are confronted with a very large number
of possible stability properties which degenerate
into
for the case of closed sets with a compact vicinity. of these different
study of these properties erties of time-varying
stability behaviors.
This case is contained
system:
~t
E x R = E n+l
Thus ~ t
an illustration
of the above remark,
alent to these of the invariant with components
set x.. 1
~: E x R • R-->E x R.
defined dynamical
is defined by the mapping
system by letting
~:E n+l x R-->E n+l
assume that the ~ t
M = {0} x R .
case of the
To clear this point we shall define
through a mapping
in the previously
state)
that
The main reason for the
systems will be treated as a particular
a tlme-varying dynamical
E x R = E n+l
We shall present some
of non compact sets is that the stability prop-
stability of non compact sets.
set (equilibrium
a few basic properties
types of stability and instability without claiming
we shall exhaust all possible
Liapunov
so far will
in terms of their limit sets which may now be empty,
even if the neighboring difficulty
the concepts
.
As
has the invariant
Its stability properties
{x:x I = x 2 = ... = x n = 0}
are equivin the space
These concepts will be fully explained
and used in Section 3.4. We shall now proceed with the definitions properties
of sets in the space
E .
of the Liapunov-stability
66
1.6.1
DEFINITION A set
M C E
given any
a > 0 ,
S(x,n)R+C
S(M,E) .
for each
M C E
stable (*) if, given any S(M,n)R + C S(M,e)
x ~ M
such that
O(M) D M
n(c,x)
such that
O(M)R+C
S(M,a)
a > 0
.
is said to be (positively Liapunov) uniformly ~ > 0 ,
there exists a
n(~)
such that
.
From these definitions
1.6.2
there exists a
This is equivalent to saying that given any
there exists an open set A set
i8 said to be (positively Liapunov) stable, if,
it obviously follows that
THEORY74 If a set
M CE
is uniformly stable, it is stable.
On the other hand, it is easy to construct examples of sets which are stable but not uniformly stable.
i. 6.3
Ezamp le. Consider, for instance,
flow has the property that for all
the flow shown in Figure 1.6.4. x E s~
llm
On the other hand the positive semitrajectory x = (Xl, x2) to the axis
with xI .
xI ~ D
and x 2
y+(x)
This
(7(x),{,:x 2 = 0}) § 0
though all points
arbitrary is a straight line parallel
Clearly then the set
[x:x 2 = 0}
is stable, but not
uniformly stable.
Notice that Zubov [ 6] calls this propertv stability. We prefer to call it uniform stability to be consistent with the established terminology in the case of time-varying differential equations.
6?
Figure
1.6.4
X2
!
I l
X l
L
For the case of a compact set the property of stability and that of uniform stability coincide:
1.6.5
THEOREM If a closed set
M CE
has a compact vicinity, then stability
is equivalent to uniform stability. Proof. that
Given
~ > 0 , for each
y E S(x, ~(x))
family of open sets
implies
x ~ M , there exists an yR+CS(M,E)
{S(x, n(x)} ,
x ~ ~M
.
Now
~M
q(x) > 0
such
is compact and the
covers the compact set
~M .
Hence there is a finite subcoverlng
S(Xl, ~(Xl)) , ... , S(Xn, n(Xn))
which covers
y ~ MU
= S(M)
BM .
implies
S (M, n) C S(M)
Notice now that
yR+CS(M,~)
.
S(x I, n(Xl)) U
Since there is a
~ > 0
1.5.2 4.
THEOREM If the closed set
v~i~t.
such that
the theorem follows.
The next theorem is an extension of Theorem
1.6.6
... U S(x n,
M~
E
is stable, then it is positively in-
n(Xn))
68
Proof.
Stability of
M
implies
MR+~
MR+c N since
M
is a closed set.
S(M,e)
c
But
M C MR +
> 0}
for all
e > 0 .
Hence
= M
always holds, so that ~e h~ve
~V
MR + = M
1.6.7
and
M
Remark.
is positely invariant.
It is to be noted that the property of stability may be
trivially satisfied if the set
1.6.8
Example.
M
is not closed.
Let the b o u n d a r y of the circle be a limit cycle
and let the orbits in the interior of the disc be not a rest point. T h e n the set D k { x } 1.6. I.
This is shown by:
Note however
D
(Figure 1.6.9),
a p p r o a c h it spirally. Let
is still stable according to our definition
that it has a c o m p a c t vicinity, but it is not positively
invariant.
i. 6.9.
xED
Figure
\
It is also noteworthy that the property of stability is not preserved for the closure of a set
M ,
as shown by Example
although the property of uniform stability is preserved.
1.6.8
In fact
69
THEOREM
1.6.10
If a set
M
is uniformly stable, then
M
is also uniformly
stable.
Proof.
The theorem is clear when one notices, that for any set
M ,
1
S(M,n) ~ S(M,n)
The above theorems and examples indicate the role played by closedness in connection with stability properties.
The various nice properties which compact attractors always have, are not necessarily all present in the case of non compact sets. "attracting" property is constant on all points of
M
When the
we can define weak
attraction, attraction and uniform attraction, while in the case in which the attracting property of
M
varies from point to point, we shall call
the same properties semi weak attraction and semi attraction.
Essentially
these properties are special forms of attraction relative to a set such that for any
~ > 0
S(M,6)~A(M)
.
A(M)
In the case of a set with com-
p a c t vicinity all these properties are equivalent and coincide with those
given in the Definition
1.5.6.
DEFINITION
1.6.11
If for a set that for each
lim
there exists an open set
y ~ O(M)there is a sequence
p(Ytn, M) § O, M
If
M~E
{tn} : t L §
O(M)~M
+~
such
such that
is called semi weak attractor. i8 such that for each
O(M)~M
o(yt, M) = 0 ,
M
Y 4"0(M)
it i8
is called semi attractor.
t § +~ If for a set
y ~ S(M,~)
M~E
there i8 a sequence
there exists an {tn} : t n § + ~
~ > 0
such that for all
such that
70
p(Ytn,M) + 0 ,
M
is called
If for a set lim
If set
M C E
and a
x ~ S(M,X) ,
P(yt,M) = 0 ,
such that
1.6.12
> 0
such that for all
is called an attractor.
such that for all
~ > 0
there exists a
t ~ 3, p(xt, M) < E
for
i8 called ~ uniform attractor.
M
i8 finally called equiattracting C*) if it is
M~E
attracting and there exists a T > 0
M
is such that for all
T(X,c)
A set
and
weak attractor. there exists an
M CE
y ~ S(M,~) it is
X(c) > 0
~
there exists a
X > 0
9 > 0
such that for each
~, 0 < ~ < X
with the property that for each
E ~ p(x,M) ~ X , x[0, T] ~
x,
S(M, 6) = ~ .
DEFINITION The set
1.6.13
A
(M)
= {x ~ E :
{t
} , t n
+ +~
is called the region of weak attraction of the set 1.6.14
A(M) = {x ~ E : p(xt, M) § 0
is called the region of attraction of the set
Notice that if 1.6.15
M
i. 6.16
The set
M . as
t++'}
is an attractor, then
A (M) = A ( ~ ) D S ( M , T )
for some
T >
A(M)
0 .
is a generalization
1.5.14.
THEOREM For any set
attractor,
§ M} n
M .
The next theorem on the properties of of Theorem
, xt
n
then
A(M)
M ,
A(M)
i8 always invariant.
If
M
is an
i8 also open.
Notice that this property is equivalent to what Zubov attraction.
[ 6 ] calls uniform
71
Proof.
If
= p(x~', A(M)
and T ~ R
x E A(M)
M) § 0
T' § ~
as
,
,
then
where
p((xT)t,
M) = @(x(T + t), M)
T' = 9 + t .
Thus
and
xT 6 A(M)
is invariant. As
M
is an attractor,
.
N o w let
S(M, 6) C A ( M ) exists a
~ > 0 T > 0
that
e) C S ( M ,
S(xT,
t § ~ .
x ~ A(M)\S(M,
such that
there is a
Consider
.
yT ~ S(xT, A(M)
~) .
p(xT,M)
Then
now the set
This set is a neighborhood Thus
is open w h i c h
if
attractor.
~
~
N = S(xT,
of
x .
y ~ N ,
completes
implication
. e)
implies
e)(-T)
e > 0
that, such
p(yt, M) § 0
= {y(-T) y { N
that there ther
observe
C h o o s e now
as
: y s S(xT, e)}
if and only if
and a w e a k attractor
as
t § ~ ,
and
is an attractor, is a semi-weak
need not and does not hold
1.6.17
Ez~le.
1.6.18
To see this,
a uniform attractor
This is shown by the f o l l o w i n g
Semi W e a k Attractor
such that
to show,
p(yt, M) + 0
general.
i)
We need
Note that then
6 > 0
the proof.
is a semi-attractor, Any other
.
y ~ S(xT,
It is easy to see that: an attractor
6) .
S(x,~)~A(M)
such that 6)
there exists a
Consider
examples.
the flow shown in F i g u r e
Figure.
in
In the strip
1.6.18.
-i .< x? .< + i
this flow has the property X2 ~
-I
~ ~
1
~
~
~
~
~
i i
i
I
I
~
_
_ >
k+I
that the positive
i
trajectory
A+(x)
-1 with Xl ~ ~22
~ "l
j
XI
x2
arbitrary
semithrough
and
and the
?2 negative s~_mitrajectory Y-(x) through all points x = (Xl,X 2) with i - x2 Xl ~ 2 and x 2 arbitrary are straight lines parallel to the axis x2 x x2-1 l-x 2 The flow for x I ~ ( ~ , ~ ) is completed as shown in Figure 1.6.18. x2 x2 the separatrix x E E 2, x~ Q
~
is the trajectory
implies
yg(X)
p(x,{x:x2= 0}) # 0,
xI9 Where
with the property that for all
while
lira
p(yg(X),)•
0}) § 0.
xI §
Notice that then the positive limit set of all trajectories bounded by
~
set ~mx 2 = 0}
ii)
and the axis
is the set
A+(G) = ~ U , { x : ~ =
0}.
G
Thus the
is a semi-weak attractor, but not a semi-attractor.
Semi-attractor.
1.6.19
xI
in the region
Consider the flow shown in
Figure
Figure
1.6.19.
The trajectory
~
has the same
properties as the one in
Xz example i).
The trajectories
inside the region and the axis that
y~ G
G xI
bounded by are such
implies that
lira p(yt,~:~Z= 0})=O. The set ~Xl
{X;X2 = 0}
is a semi-attractor,
but not a weak attractor.
iii) 1.6.20
Weak attractor.
Consider the flow shown in
Figure
Figure Let
, lira p(Qi,Y(x)) t§ M =
~
U
§ 0
Q2 and
lira p(Q2,Y(x)) t++~
7(x)
shown in the
Figure 1.6.21, i.e., § 0
uniformly.
Then the set
0.2 is an attractor, but not a uniform attractor. In the case of compact sets, or closed sets with a compact vicinity
one can prove that a semi-weak attractor is a weak attractor, and that a semiattractor is an attractor.
The proof rests on the fact that if
or is a closed set with a compact vicinit~ then~if M,
one has for a sufficiently small
Theorem 1.6.5.
1.6.22
0(M)
M
is compact,
is any open set containing
E > 0, 0 ( M ) ~ S ( M , E ) ,
as is shown in
The proofs of these assertions are, therefore, omitted.
Then:
THEOREM If
M C _ E is a closed set with a compact vicinity then semi weak
attraction implies weak attraction and semi attraction implies attraction. It remains to be proven that in the case of compact sets the definitions 1.5.6 and 1.6.11 of an attractor are equivalent.
Analytical examples can be found in a paper by Bhatia [ i ].
74
THEOREM
1.6.23
If M C E
is compact the definition 1.5.6
of attractor is equivalent
to definition 1.6.11. Proof~
If (1.6.11) holds,
is in a compact set.
then any sequence
Thus we may assume that it converges.
x ~ S(M,6) , A+(x) # ~.
Notice further that
if
p(y,M) = O,
xt n + y,
A+(x) C
we have
{Xtn},tn ++~o,
also
p(Xtn,M) § 0 y 6M
as
M
and
x ~ S(M, 6)
Hence, for each
as
tn § +oo. ThUS
is closed,
i.e., also
S. Now assume that (1.5.6) holds.
p(xt,M) + O,
as
t ++~.
p(X~n,M ) ~ e > 0. {y:P(y,M) so t h a t
= e}
Assume,
Then there is a sequence
We may assume that
p(X~n,M ) = e
is compact, we can assume that
y~M.
But
if possible,
y 6 A+(x) C M,
xT
n
that
[Tn}' ~n § +~ for all + y.
n.
Then
which i s a c o n t r a d i c t i o n ,
and As the set
O(y,M) = e,
and proves t h e
theorem.
If
M
is not compact, we can prove the following weaker version of
Theorem 1.5.24.
1.6.24
THEOREM If
M
is/positively invariant closed set which is uniformly attractingj
then it is stable. The proof follows from that of Theorem 1.5.24, when we notice that for any
e > 0
and
x~M,
there is a
~
> 0
such that
yR+C
S(M,e)
for
x
yfS(X,6x). since
Thus for
0(M)
y~0(M)
=
U S(X,6x), we have x~M is open, this implies stability of M.
yR+~
S(M,~),
and
By combining the five possible attracting properties with the two possible forms of stability we shall now define six different forms of asymptotic stability of sets.
It is, in fact, easy to prove that
75
1.6.25
THEOREM is (uniformly) stable and semi-weakly attracting,
If a set M C E
If a set M C / E
then it is semi-attracting.
is Cuniformly) stable and weakly
attracting, then it is attracting. Proof:
We shall give the detailed proof only of the first statement; the proof
of the second is similar. If the assertion is not true there exists at least one sequence {tn}:t n § ~ , Tn § +~ tn ~ ~n"
such that
such that
p(Ytn,M) ~ 0
g(yTn,X) + 0,
Then the fact that
whereas there is a sequence
for some
x E M.
yt n = y~n[tn - ~n)
{~n }'
We may assume that shows that definition 1.6.1
is contradicted and proves the theorem.
1.6.26
DEFINITION If a set M C E
is [uniformly) stable and semi-attracting it is
called/(uniformly) stable semi-attractor. If a set M C E
is Cuniformly) stable and attracting it is called Q~
(uniformly# stable attractor or/(asymptotically stable set). If a set
M CE
is (uniformly) stable and uniformly attracting it
is called/(uniformly) stable uniform attractor or/Cuniformly asymptotically stable set) . We shall now give some examples of the various properties presented in definition 1.6.26.
1.6.27 i)
Examples
Stable semi-attractor.
Consider the flow shown in Figure 1.6.28.
This flow
76
Figure
1.6.28
X2
is essentially a variation of the flow shown in
Figure
1.6.19.
The
only difference is that while them ;, 1.6.19 was not stable ;,l.6.2g~t (positively Liapunov) stable. Stability is achieved by the
S
property that now for
'
9
jectory
ii)
y+(x)
~Xi
xI ~
x2
arbitrary, the
x I.
Figure
Consider the flow shown in
Figure
1.6.29
and
corresponding positive seml-tra-
is a straight line parallel to the axis
Uniformly stable semi-attractor.
~
x ~G,
1.6.29.
This flow has the property that fr xI ~
~
and
x2
arbitrary the
>
corresponding negative semi-trajer
>4
tory
y-(x)
is a straight line
parallel to the axis the region xI
G
xI .
Thus
bounded by the axi~
and the separatrix
Q is an
infinite strip in the direction
• {x 2 = 0}
is uniformly (positively Liapunov) stable.
1.6.29 has also the property that in the region the trajectories {x 2 = 0}
iii)
x I § -=.
y(x)
Clearly the set
The flow shown in
C(G),
for
F~ure
x I ~ (i - x 2) / x~
are straight lines parallel to the axis
x I.
Hence the set
is a uniformly stable semi-attractor.
Stable attractor.
Consider the flow shown in
Figure
1.6.30.
This flow is
77
1.6.30
F/gure
essentially a variation of the
X2
flows shown in
Figure
1.6.4.
Now the flow has the additional property that for all
li=
pCYCx),{~= o } ) § 0.
Hence t h e s e t {x=x 2
XI
=
iv)
= 0}
stable attractor.
Uniformly stable attractor. In the euclidean plane, consider the system
Xl = i,
i2 = 0
R1 = i,
i2 -
(i+ 2)
The solution through any point
O
O
CXl, x 2)
for
x I .< 0,
for
x I >.0.
has the form
+ Cx~) 2
1 O
xI = t + xI ,
O
x2 =
o 2
x2
i + Ct + x I) for
O
t >. -Xl,
and 0
x I = t + xl,
O
x2 = x2
O
for
t .< -x I
.
i
The
x ~E 2
xl-axis is a uniformly stable attractor, but is not a uniform attractor.
is
a
78
In the proof of Theorems of the compactness of general case.
M.
(1.5.11) and (1.5.14) no use has been made
We may assume that these two theorems are true in the
The proof is left as exercise to the reader.
THEOREM
1.6.31
Let
be a closed attractor. Then
M CE
A(M)
is open.
THEOREM
1.6.32
Let
be a closed attractor.
M CE
Then the set
is invariant.
A(M)
THEOREM
1.6.33
If a closed attractor (1.6.26)
M CE
has a compact neighborhood the definition
i8 equivalent to the definitions (1.5.15)
The proof follows ~mmediately from Theorems
and (1.5.16).
(1.6.5),
(1.6.22) and
(1.6.23). We shall
now define and investigate a certain n,-~ber of other
properties of set; the instability properties.
We shall first define two types of
instability as the opposite of the two forms of stability defined in 1.6.1 then define various forms of negative attraction and complete instability. The classification that we give for these properties may not exhaust all possible behaviors.
i. 6.3 4 DEFINITION A set i) point
~nstable if it is not stable, i.e., if there exists an
x E M , a sequence
such that
O(Xntn,M)
ii) an
M C~E is called
and a sequence
{tn};tns I%+
e.
weakly unstable if it is not uniformly stable, i.e., if there exists
~ > O, a sequence
0(Xn,M) § 0
>.
(Xn}tXnE C(M);xn § x
e > O, a
implies
{Xn}:Xn EC(M); 0(Xntn,M) >- e.
and a sequence
{tn}:tn E R +
such that
79 1.6.35
Remark It is important to point out that a set
M~E
may be both stable
and weakly unstable if it is stable, but not uniformly stable. properties is, for instance, the set
M
A set with these
in the flow of Figure 1.6.4.
Again
1.6.36
THEOREM Ira
compact set
M CE
i8 weakly unstable it is unstable.
The difference between an unstable and a weakly unstable set lies in the different way with which a trajectory or a sequence of points leave the set
M.
again.
If
M
is unstable, such a trajectory may possibly approach the set
M
We can then define a stronger form of instability when this does not
happen, that is, if either the trajectory or the sequence of points will ultimately be bounded away from
1.6.37
M.
DEFINITION A set i#
is called
M CE
ultimately unstable if there exists an
{Xn}:Xn~ C(M),x n § x E M
for all
Thus
and a sequence
a sequence
{ t n } ~ t n E R + such that
P(Xn(t n + ~),M) ~ r
T e N +. ii)
ultimately weakly unstable if there exists an
{Xn}:XngC(M),x n + M and a sequence
for a l l
e > O,
{tn}:tnER +
such that
E > O,
a sequence
P(Xn(t n + ~),M) ~ E
~ R +. Again it is easy to prove that
1.6.38
THEOREM If
M CE
is a set with a compact vicinity then ultimate weak instability
is equivalent to ultimate instability.
We shall now introduce still stronger forms of instability and define
80
properties of sets for which all trajectories and sequences in a certain neighborhood of it tend to leave. that all points of
M
These definitions are made by requiring
have the property 1.6.34 i) or li).
It is, however,
very important to point out that in this case the stronger form of the property 1,6.34 i) defines a weaker property than the stronger form of the property 1.6.34 li).
DEFINITION
i. 6.39
A set iJ any sequence such that addition
is called
weakly completel~ unstable if there exists an
> 0
{Xn}~Xn~ C(M)~xn § xEM,
{tn}~tnE R+
O(Xntn, M) >~ ~
there exist a sequence
for all ~ ~ R +.
completely unstable if there exists an
{Xn}:Xn~C(M) Xn+ M,
~(Xntn, M) >~ E
such that for
and ultimately weakly completely u~stable if, in
~(Xn(t n + T),M) >~ c
ii) sequence
M~E
there is a sequence
~ > 0
such that for any
{tn}ttnE R +
such that
and ultimately completel~ unstable if in addition
O(Xn(t n + ~),M) >~ ~
for all 9 E R + .
All the instability properties lised until now are the analogue of the "semi" properties for stability since they are essentially defined on open sets of
M
1.6.40
and not on spherical neighborhoods.
It is ~mmedlate to prove that
THEOREM A set
M CE
exists an
E > 0
such that
0(xt,M) > ~
is ultimately completely u~stable if and only if there
such that for all for
x~S[M,e] \ M ,
t ~ ~(x).
Again it is easy to prove that
there exists a
T(x)ER +
81
1.6.41
THEOREM If M C E
is a set with a compact vicinity then weak complete
instability implies complete instability and ultimate weak complete instability implies ultimate complete instability. 1.6.42
Remark Obviously, by reversing the direction of motion on the trajectories
all forms of stability and asymptotic stability will lead to some form of instability.
It may happen that those negative asymptotic stability properties
have even stronger instability properties than the one listed above since they characterize and classify the behavior of the flow also outside
M.
For
practical reasons, however, these classifications are not very interesting in the case of instability.
i. 6.43 Examples i)
Consider the flow shown in Figure 1.6.44.
Weak instability.
i. 6.44 Figure
This flow has the
property that for all
•
xEE 2
p(~(x),{x:x~= 0})
lira Xl §
Thus the set ~:x 2 = 0} unstable.
§
O.
is weakly
On the other hand, it
is neither unstable, nor ultimately weakly unstable.
=X I ii) Instabilit5.
1.6.&5
Consider the flow shown in
Figure
Figure
1.6.45. point
2
This flow has an equilibrium P
on the axis
x I.
consider the separatrixes Q2"
Let's denote by
GI
QI
and
and
G2
the region bounded by
~i
and
Q2
respec-
tively.
and the axis
xI
This flow has the property
82 that for all
x~E 2
it is
lira Xl§
p(y(x),{•
it is lira p(y(x),{x'~2ffi0}) § 0, Xl-~ -~ x6G I
implies
A-(x) = {p}.
while
x6G 2
The set {x~x2 ffi 0}
0}) + 0
and for all
x tC(GIU
implies
h+(x)
and
=
{p}
G 2)
is then unstable, but neither
ultimately unstable nor completely unstable.
iii) Ultimate Weak Instability.
i. 6.46
Consider the flow shown in Figure 1.6.46.
Figure
This flow
is essentially a variation of the flow shown in F~ure i. 6.44.
X2
For
xI < 0
the flow is the
same as the one in Figure 1.6.44 . Now for all x I >. ~
x ~ E 2,
and
x2
with
arbitrary
the corresponding positive semi-trajectory -
y+(x)
is a
X I straight line parallel to the
axis
iv)
x I.
Hence the set |x;x2 = 0}
Ultimate Instability.
is ultimately weakly unstable.
Consider the flow shown in Figure 1.6.47.
i. 6.47 Figure
This flow is a
variation of the flow shown in
Figume 1.6.45. properties as I
|L L
It has the same xI + + ~
as the
flow shown in Figure i. 6.45. The set ~x:x2 = 0}
is ultimately
unstable, but not completely ultimately unstable.
83 v)
Weak
i. 6.48
Complete
Instability.
Consider the flow shown in Fi~ume 1.6.48.
Figure
This
flow has an equilibrium point
•
on the axis separatrix
x 1. 4.
Consider the Denote with
the region bounded by Q axis
x I.
and the
x C E 2,
~(~x(t),fx:~= 0}) + 0,
lira
t § + ~
P
vi)
G
This flow has the
property that for all
lira p(~x(t),{x'~= 0}) + 0 t+ -~ weakly completely unstable.
P
for all and for
x 6 C ( G ) , A-(x) = P.
Ultimate Weak Complete Instability.
x ~G,
The set
it is
{~x~ O}
is
Consider the flow shown in Figure 1.6.49.
i. 6.49 Figm~e
This flow is a variation of the
X2
flow shown in F ~ u m e
i. 6.48.
It has the same property for
I I
0
as the flow shown in
Figure 1.6.48.
For
x I .
X
r
84
vii)
Complete Instability.
Consider the flow shown in
Figure
1.6.50
Figure 1.6.50.
This flow has
the equilibrium point
P
and in
addition all points of the semi-
•
axis
x I > 0 are equilibriom points.
We shall denote the semi-axis xI > 0
with
M.
separatrix Q.
Xl
Consider the Denote with
(shaded) region bounded by the set
M.
G the Q and
This flow has the
following properties. the trajectories
7(x)
For
x EG,
are closed
bounded curves clustering around P
and fllling the set I(G).
The set
viii) 1.6.51
M
For
x E ~ ' ~ \ M, A-(x) = (0}
and
lira
§ o.
is completely unstable.
Ultimate Complete Instability.
Figure
Consider the flow shown in set
B(
..
BI ~ B 2
Figure 1.6.51.
pletely unstable.
r
B2 Notes and References Some of the above given definitions have been presented under slightly different forms by Zubov [6] and by Bhatia [i].
The
is ultimately complete-
>
1.6.52
p (.x(t),~)
8~
1.7
Stability and Liapunov functions. In this section we shall formulate some necessary and sufficient
conditions for a closed set semi-attracting
M CE
to be stable, uniformly stable, stable
and asymptotically
certain scalar function
v = ~(x)
stable in terms of the existence of defined on a suitable neighborhood of
M.
In this section we are interested in deriving necessary and sufficient conditions for the above mentioned properties which require weak properties of the scalar function
v = #(x).
In Section 2.
we shall, on the other hand, be
interested in giving necessary and sufficient conditions for the above mentioned properties of a very sharp type~i.e., by using scalar functions
v = ~(x)
of a very special type.
THEOREM
1.7.1
Necessary and sufficient for stability of a closed set is the existence of a function open set
W
such that
w ~
~(x)
M~E
defined on a positively invariant
S (x, 6(x)) for all
x ~
M
and some
6 = ~(x) > 0
and having the following properties: i)
For every implies
ii)
Proof.
~ > 0
6 > 0
and a compact set
such that
~(xt) ~ ~(x),
such that
@(x) ~ ~
for
x~W
for
and
K CE,
xeW\S(M,c)
x~W
there exists a
N S(M,6) N K,
t > 0.
The conditions are sufficient, because for any given
W \ S(M,e) # ~,
and any compact set
sup{$(x):x~WN
and
there exists a
~(x) ~ 6,
For every 6 > 0
iii)
~ > O,
K CE,
we can choose
S(M,6) N K} < i n f { ~ ( x ) : x s
e > 0 6 > 0
such that such that
S(M,e)},
S(M,6) ~ K C W . We can then assert that
For by hypothesis
(iii)
x E S(M,~) N K
implies
y+(x)C
S(M,e).
86
#(xt) .< r
This implies
xt s S(M,e)
< inf{#(x):xEW~S(M,e)}.
for
t > 0,i.e., that
To prove the necessity,
w -- {x:y+(x) C
where
~ > 0
such that
is arbitrary.
MCW.
and such that
M C
~
is compact.
S(x,p),
there exists S(M,9).
Let
S(M,~)}.
The set
x(M
W
is positively invariant, open and
~ > 0
for
such that
~ = rain (~,6).
S(x,q)~W,
S (x, ~(x)) c W .
9 > 0
x~W
~ > 0
Then as
W
is open
such that
and the compact set
y E S(x,p) ~ S ( M , ~ )
i.e., for all For
We prove that
then there exists a
By hypothesis,
Now let
and consequently such that
set
The first assertion is obvious.
S(x,B)
Y+(Y) C
M is stable.
implies
x %M, S(x,n) C
x ~M
S(x,p) (~ S(M,6)
there exists
n(x) > 0
define now
~(x) = sup{ O(x~,M):T >. 0}.
1.7.2
Then
~(Xt) = sup { p ( x ( t
+ T),M): T >. 0}
= sup {p(xT,M):T >. t} .< ~(X),
i.e.,
~(x)
has the property
Since has property
implies
#(x) ~ e,
so that
~(x)
(1).
S(M,E).
x E S(M,6)(~ W.
t > 0)
(iii).
#(x) ~ p(x,M),x ~ W ~ S ~ , e )
Lastly, for any y+(x) C
if
Then
e > 0,
~(x) ~ e
This is property
choose for (il).
6 > 0
x ~S(M,~) Q.E.D.
such that
x ~ S(M,6)
and in particular for
implies
87
i. 7.3
COROLLARY A sufficient condition for stability of a closed set
is the existence of a continuous function S(M,~),
6 > 0,
defined for
M C E
defined in some
~(x)
and a continuous monotonic increasing function
0 0
for
and
xEM.
x~S(M,6).
We now give a similar theorem for uniform stability
THEOREM
1.7.4
Necessary and sufficient for the uniform stability of a closed set MCE
,
is the existence of a function
invariant set
6 > O,
W~S(M,S),
i) For every
~ > O,
x e w\
;
ii) For every
~ > O,
defined on a positively
~(x),
and having the following properties.
there exists a
~ > 0
such that
~(x) ~ ~
for
there exists a
6 > 0
such that
~(x) ~ e
for
x~W~S(M,~); iii) ~(xt) ~ ~(x)
Proof.
for
x~W
and
t > O.
The conditions are clearly sufficient.
For a fixed
e0 > 0
define
W = {x:y+(x)C
positively invariant and such that
60 > 0
y+(x)C
and
and
S(M,co).
t > O,
then
Thus
W
Now, for properties
Thus
S(M,~o)C
xt~W,
for
S(M, e0)}.
W OS(M,6),6
of uniform stability there exists a W
We prove the necessity.
> 0.
W
is
In fact, by the definition
such that
W~-'pS(M,6),
This set
x ~ S(M,60) 6 > 0
.
implies Again if
y+(xt) C y + ( x ) C W .
is positively invariant.
xeW
define
(i),(ii),(iii),
~(x)
as in 1.7.2.
This
~(x)
has all the
which can be verified as in the Theorem 1.7.1.
x~ W
88
COROLLARY
1.7.5
A sufficient condition for the uniform stability of a closed set MC
X
is the existence of a continuous function for
S(M,6)
and
6 > 0,
defined in some
and two continuous monotonic increasing functions defined for
6(~), ~(0) = 8(0) = O,
0 ~ p < 6,
Ci) ~(O(x,M)) ~ ~(x) ~ 8~(x,M)) Cii) ~(xt) ~ ~(x) 1.7.6
~(x)
for
for
~(~)
such that
xES(M,~)
and
t > 0.
Remark It is to be noted that the theorem 1.7.4 does not predict the
existence of a continuous function in case of uniform stability. function
~(x)
~(x)
Notice,
possessing
the properties mentioned
for instance, that no continuous
satisfying the conditions of Theorem 1.7.4 can exist for the
stable rest point
p
in Example 1.5.32
(v).
The situation that even for a compact set
M
which is stable
(and hence uniformly stable), a continuous function satisfying conditions of Theorem 1.7.4 need not exist, has led to the introduction of a host of stronger concepts of stability, asymptotic stability. prolongations.
each lying somewhere between stability and
This has been made possible by the general theory of
These we shall discuss in later sections, but let it be
mentioned that the concept of stability, ~(x)
for which the existence of a continuous
satisfying conditions of Theorem 1.7.4 is guaranteed,
absolute stability. stability,
is called
We may, however, note that even in the case of ordinary
any function
~(x)
continuous at all points of
satisfying conditions of Theorem 1.7.4, is M,
and that, in general, a function
~(x),
continuous along the trajectories of the dynamical system in a neighborhood N
of
M,
does always exist. We now prove a theorem on stable semi-attractors.
89
THEOREM
1.7.7
A necessary and sufficient condition, that a closed set a stable semi-attractor
is the existence of a continuous function
defined in an open invariant set and some
W
such that
~ ~ 0,
ii)
for every
~ > 0
such that
#(x) .< ~
~(xt) < #(x)
iii)
~(xt) + 0
iv)
as
We set M~A(M).
for
t § =.
are sufficient
(i),
For this purpose,
is open, positively
A(M).
is clear, we prove the necessity.
This is an open invariant
(il) and (iii). we define
invariant,
W
c
has the property
that
xT~W e .
= I.
xTfW
,
S(xT,a)
We can choose, if
y s S(x,~),
Let now
x ~W
~ = i/4.
#(x,M)
We can choose
is compact. then a
and
n > 0
Then
a > 0
S(xT,g)(-T)
such that
We prove that it is also
= {x6W:y+(x)~
x ~ W,
and
set with
~ 0}.
has the important property that for each
where
~ > 0
x %W
has properties
continuous.
there i8 a
> 0,
(x) = sup ~ (x~,M):~
~(x)
K,
xEWf]KOS(M,6),
x&W~M,t
W = interior of
Let now for
~(~), ~(0) = 0,
for x ~ w ,
and a compact set
for
That the conditions
WE
x ~ M
such that
~(x) >. ~(o(x,~))
This
for all
there is a continuous monotonic increasing function defined for
Then
W _~ S (x, 6(x))
r
6 = 6(x) > 0, which satisfies the following conditions~
i)
Proof.
be
M CE
S(M ,e)}. M~W
there exists a There exists a T such that
= N
n < (~/4)
e
and further T > 0
> 0
such
such that
S(xT,g) C
WM
is a neighborhood
of
and
S(x,n)~
N.
x.
Then
90
p(xT,M)-sup p(yT,M){
l#(x) - #(y) l ffi Isup 9 ~0
z~O
{p (xT,M) - ~ (yT,M) [
sup 0~T
sup
p(xz,yz)
.
0~T~T
This shows, however, that continuous.
To get a
~(x)
(x)
#(y) § ~(x)
with property
(xT)dT
=
as
y + x.
Thus
is
~(x)
(iii) we can set
,
0 which has all the properties
(i-iv).
Q.E.D.
We now prove the following theorem on asymptotic stability.
1.7.8
THEOREM
A closed set
is asymptotically stable if and only if
M s-E
exists a continuous scalar function 6 > 0
W~S(M,6),
i) and
8(~,
iii) Proof.
defined on an open invariant set
having the propertiesz
There exist two continuous monotonic increasing functions defined for
~ ~ O, ~(0) = 8(0) ffi O,
~(~(x,M))
ii)
r
~(xt)
there
< ~(x)
~(xt) + 0
as
.< ~(x) .< S(O(x,M))
for t
xEW\M,t §
~
for each
open and invariant by Theorem 1.6.16.
(x)
=
such that
for x ( W ,
> O, xs
The conditions are obviously sufficient.
hypothesis, the region of attraction
We prove the necessity.
A(M) ;s such that FoE
~(~
x ( W ffiA(M)
s u p b (x~,M): T >. 0}.
A(M) ~ S ( M , 6 ) , define
By
6 > 0 and it is
91
Clearly
#(x) ~ p(x,M),
the existence of any
e > O,
Then
6(s
8~),
define
so that we can set
we note first that
e > 0,
can choose a continuous monotonic 6 (0) = 0 (e),
and
6(e) ~ 6*(e).
then we have
(iii) for all
T > 0
x t W k M.
Let
Let now
8~)
x ~ M.
Now for
y+(x) C
S(M,e)}.
6(0) = 0.
6 (e),
We
such that
be the inverse function of
Lastly note that
is also continuous.
6 > 0
of stability,
~(x)
has the property
~ > 0,
compact
(this is possible as
zT ~ S(xT,~)
x 6 S(M,6)
p(xT,M)
such that S(M,6)
= {y(-T):ys
Let
p(x,M) = I > 0
be a number which corresponds
i.e.,
be chosen such that
Choose now
implies
increasing function
~(x)
t ~ T.
S(xT,~)(-T)
and
To see
x ~ W.
by the hypothesis Let
for
implies
is nondecreasing
#(x) ~ 8(p(x,M)).
We now prove that for a given
~(x) ~ 0
6(e) = sup {6 > 0:x % S(M,6)
is positive for
e(~) = ~.
< 6.
Then
p(xt,M) < (~/4) and
for
S(xT,~)
is
Then the set
is a neighborhood
and consequently
(~/4)
y+(x) ~ S ( M , I / 4 ) .
S(xT,~) ~" S(M,6), is open).
= N
implies
to
zt ~ S(M,I/4)
of
for
x.
Further
t B T.
ze N
Thus
J~(x) - #(y) l = Isup p(x~,M)-sup p(y~,M~[ T >.0 r >~0 sup IO(xT,M) - O(Yr,M) I 0. 0.
(ii), although
which also has the
x~W.
(ii), note that,
implying
This
~(x)
~(xT) + 0
as
92
Now examine the difference
F
O(x) - O(xT) =
e-T[0(x~)
- 0(x(T + ~))]dT,
T > 0.
0
By t h e p r o p e r t i e s
of
O(x~) - l ( x ( T + ~))
O(x), > 0
there
for
i s an i n t e r v a l
~ ~ (tl,t2).
ft2 %(x) - ~(xT) ~ I J t1
e-X[~(x~)
( t l , t 2)
such that
Then - ,(x(T + T))]dT > 0.
This proves the theorem completely.
1.7.9
Remarks For compact sets
proved next.
M,
That condition
we do not need condition (iii) as will be (iii) is essential for noncompact sets can be seen
from the following example.
1.7.10
Example Consider the dynamical system defined by the differential equations
= i,~ = -2xy/(l + x2)(2 + x 2) = 2xy/(1 + x 2)
in the euclidean plane. the x-axis is denoted by
1.7.11
x ~ 0
for
x ~ 0,
The x-axis is stable, but not asymptotically M,
then we may define
This function has the properties but not the property
for
stable.
~ (x,y) = IYl = P~x,y),M).
(i) and (ii) required in the above theorem,
(iii).
THEOREM A necessary and sufficient condition for the compact set
MC E
to be asymptotically stable is that there exists a continuous scalar function ~(x)
defined in a positively invariant neighborhood of
M
and such that
If
93
there exists two continuous strictly increasing functions defined for
~ >. 0, ~(0) = 8(0) = 0,
i)~(p(x,M)) ii)r
Proof.
.< r
.< 8(p(x,M))
Let
part follows from 1.7.4.
1.7.12
If
Choose
8(~),
The set 6 > 0
for
xgN
for
x6N
S[M,e]
•M,
t > 0
is compact.
The stability
such that
8(6) ~ a(e)
x E S ( M , 6),
that
e > 0.
and
such that
r
Sufficiency.
a(~)
then
XT ~ S ( M , e ) .
xR+-- S(M,e).
For3if not, then there is a
a(p(x ,M)) < r
=
< r
.< 8(6).
which contradicts 1.7.12.
This proves stability of
that
If
is an attractor.
xCS(M,6)
sequence
such
Hence
~(e)
M
~ > 0
such that
M
M.
It must now be proved
is not an attractor then there exists an
lim p(xt,M) # 0.
Then there exists a
~ > 0
and a
s u c h that
{tn}: tn § + ~
n .< p(Xtn,M) < r
Since
S[M,e] ~S(M,n)
sequence. a
z > 0
Thus there is a point
y ~ A+(x)
with
{Xtn} has a convergent sub y ~ M.
There exists therefore
such that by condition (ii)
1.7.13
However
is compact, the sequence
r
since
< r
y~ EA+(x)
and
yEA+(x)
there are sequences
{t } and
n
{t n}
94
such that that
xt
§ y
n
t > t' n n
and
for each
xt' § yT. n n.
We might assume, without loss of generality
Then
(xt n) < ~ (xt~) and proceeding to the limit, since
is continuous we obtain that
(y) .< ~(yz)
which contradicts 1.7.13. x E S(M,~).
lim p(xt,M) = 0 t§ This completes the proof of sufficiency.
Necessity:
The set
x~A(M)
Thus it must be
A(M)
is an open and invariant neighborhood of
M.
For
set.
~(x) = sup {p(x~,M): 9
1.7.14
Clearly
for each
~(x) % p(x,M),
we need only prove that
~ O}
so that we may take #(x)
u(~) = ~.
Since
M
is compact
is continuous, which can be done exactly as in the
previous theorem, then the existence of
8~)
will follow from Theorem 0.3.2
The scalar function 1.7.14 may not have the property (ii), although #(xt) ~ ~(x)
for
t > 0
To have a
~(x)
is satisfied, and
~(x~) § 0
as
~ ~.
satisfying all the properties set
f~ ~(x) = Jn e-X~(xr)dx,
This scalar function
for
x~A(M)
~ ( x ) has the properties (i) and (il).
.
The proof is the
same as in the previous theorem. This proves the theorem completely. 1.7.15
Remark Notice that any
that
~ (xt) § 0
hypothesis.
as
~(x)
t § ~,
satisfying the above theorem has the property
although this is not explicitly assumed in the
9
95
1.7.16
Notes and References
~Imost all results presented in this section are due to N. P. Bhatia Few similar results can be found in the book by Zubov by Roxin [3].
The use of the function
~(x)
[i].
[6] and in a paper
for characterizing
stability properties was introduced by Liapunov[l].See 1.12.13 and 3.6.32.
96
1.8
Topological methods.
DEFINITION
1.8.1
Let
be open sets with
N,M
invariant for a flow
~on
(or an ingress point) of x(- c,O) C M (or (or
M
A point
then
x
be positively
is called an egress point such that
x(O,n) ~ M = 0
i8 called a strict egress point (or strict x ~ ~M ~ N
point, if it is not an egress point. M
N
~ > 0
If in addition,
Sometimes a point
egress points of
xs ~M~N
if there exists an
x(O,e) c M).
x(-n,O) ~ M = ~),
ingress point).
E.
Let
MCNCE.
may be called a non-egress
The sets of egress points and strict
will be denoted respectively by
M
and
M
e
LEMIMA
1.8.2
If then
M
If
t
with
R+
~,x(0,T) C
M
is not positively invarlant, then there is an Xt ~M.
M,
invariant). y
and
T = inf{t ~R+;xt ~M}.
xT ~ ~M 0 N,
Setting
y = xT,
because
we note that
is an egress point of
Let f:U § V
U
M,
and a
Then, by continuity of
xR+~
N
(note
N
is positively
x(0,T) = xT(-Y,0) = y(-T,0),
which is a contradiction.
be a topological space and
is called a retraction of for all
f(v) = v
the set
1.8.4
Let
x~M,
showing
Q.E.D.
DEFINITION
1.8.3
and
is either empty or consists only of non-egress points,
~M~N
is positively invariant.
Proof.
that
. se
V
U
onto
V~U.
V
if
A continuous mapping f(u)~V
for all
When there exists a retraction of
v s
is called a retract of
U
u ~U onto
V,
U.
THEOREM Let
be open sets such that
M,N,M~N~__E,
for a flow - ~ o n
N
is positively invariant
E.
Let
Me = Mse" i.e., all egress points of
strict egress points.
Let
S
be a non-empty subset of
M U Mo
M
are
such that
97
is a retract of
S ~ Me
at least one point
Proof.
Me,
x ~S ~ M
but is not a retract of such that
S.
Then there exists
xR+CM.
Suppose that the theorem is false.
Then for each
x E S~M
there
e
N
is a
t
such that
t
X
map
> 0
and
x[O,tx) ~ M
f:S § M
M
e
defined by:
e
= M
se
.
f(x) = xt
To see this let
smal~ but arbitrary. for is a
tx < t .< tx + e. $ > 0
Then Set
such that for
. tx + e, i.e. 0.< t < tx - e .< ~y -< tx + e. also
xt
f:S § M
also
if
x z
x6 S ~M
xt 6 S ~ M
xES~M
and
e
e
.
Let
for
e
Consider the
e
defining the flow
f(x) = x ~ S ~is
Y CS.
Let
0 .< t .< t
- e,
x
p(x,y)
< 6,
we have
y(t x + e ) ~ M
Therefore
tx
is continuous for
and
p(xt,yt)
e > 0 and
y(t x - e ) ~ M .
< ~/2
.
e
.
be sufficiently xt ~ M Then there
for
Hence
is a continuous function of
x 6S~M
~M
continuous
n = min{p(x(t x + e),M), p(x(t x - G),~M)}.
e
x~SOM
sM . X
This map is continuous since the map and
and
X
xES.
Hence
A similar argument holds for
e
.
e
g:M e § S O M e
If now composite map
gf
is a retract of
is a retraction of
S
onto
Me S ~
onto M
e
S ~ Me,
then the
.
The existence of such a retraction contradicts the hypothesis,
so that
the theorem is proved.
I,8.5
Remark If
Me = M
se .
M
is compact the only non-trivial condition of the theorem is that
In this case the result of the theorem is rather weak as will be shown
in the next section.
98
i. 8.6 Notes and References Theorem 1.8.r is due to Wa~ewskl [3]. This theorem is the cornerstone of the so-called topological methods for the study of properties of flows. Additional results alone these lines are due to F. Albrecht [~]andto A. Pliss[a].
99
1.9
1.9.1
Topological properties of attractors. DEFINITION We shall say that a compact set
M~_~E
has strong stability properties
if it is either asymptotically stable or completely unstable. with
AS(M)
A(M)
or
We shall denote
the open invariant set in which these properties hold, no, ely either
A-(M). In this section and in the next we shall discuss some relationships
which exist between the topological properties of closed sets having certain stability properties and the topological properties of the regions of the space in which such stability properties hold.
E
In particular we shall devote our
attention to the case of closed sets with strong stability properties and discuss the relationship between the topological properties of a closed set and those of
AS(M).
MCE
The case of strong stability properties is not only the
most interestin~ but also the easiest to solve since in this case the existing continuous Liapunov functions have many properties which are very useful in the proof of the various results.
In this section we shall limit ourselves to the
discussion of global properties with respect to equilibrium points.
In the
next we shall present the very few results available for the case of local properties, whil~ we postpone the more general discussion of the properties of sets to Section 2.8.
Since for the proof of these results a more involved mathematical
machinery is required, we urge the reader of this section to read at least the statement of the theorems presented in the advanced section. Most of the theorems that we shall present are given for the case of weak attractor.Obviously they hold for asymptotically stable sets and, with the due changes
(by inverting the direction of motion on the trajectories)~for
completely unstable sets.
100
All the results that we shall present are at a very early development stage; they are incomplete and
~m[t
further improvement.
It is
only because we think that those problems right now are a m o n g the most important problems in stability theory that we expose the reader to these preliminary results and incomplete theories.
THEOREM
1.9.2
Let attractor.
MC Then
E
be a compact minimal set, and let
By Theorem i. 5.42,
x0~ E
be arbitrary but fixed.
x~S[x0,~ ] T
and
(x) = ~(x,T)
"
D+(M)
Choose
such that
t >. T.
For each ~
T
i.e., there is an
D+(M)
is continuous and S[X0,~ ]
and
whenever
~T:X § X
by
~ ($[x0,e])~-S[x0,~ ] .
Thus by
T
contains a fixed point of the map
such that
~ (x) = x = ~(x,T).
Hence
T
~(x,t) = ~(~(x,T),t) trajectory. M
we have i.e.,
is uniformly attracting,
define the map
T
for
Let
sufficiently large so that
~(x,t)E S ( D + ( M ) , e ) ~ S[x0,e], T >. T
x~5[x0,e ]
stable.
sufficiently small such that
By Theorem i. 5.27,
T > 0
Then
~ > 0
e > 0
the Brouwer fixed point theorem ,
is globally asymptotically
Choose further
S (D+(M), ~) ~ S [x0,~] . hence there is a
be a global weak
M is a rest point.
Proof.
D+(M)CS(x0,~).
M
= ~(x,t + ~)
Notice that
is invariant. y(x) --- A+(x),
Y(x) ~ M ~ ~.
y(x) = y(x),
least period say
TO,
t&R,
for otherwise, if
On the other hand, since and as
x~A
(M),
and so x~M, y(x)
r.
is a periodic y(x) ~ M = 0,
is a periodic trajectory, A+(x) 0 M # @,
Y(x) ~ M.
is minimal.
Thus
Since
y(x) =- M,
as
If now
is not a rest point, then it will have a
M
M
y(x) then
we must have
This contradiction proves that
we must have
trajectory with period
is an integer.
x eM,
for all
and all other periods must be the numbers
However, we have in fact shown that all numbers
M
is a periodic
m r0,
where
m
T >, T are periods
101
of
M.
This is a contradiction and so
M
is a rest point, and the theorem
is proved.
1.9.3
Remark An important implication of the above theorem is that if
compact minimal set, and is not a rest point,
M
M
is a
cannot be globally weakly
attracting, or in particular, globally asymptotically stable.
Thus the
trajectory of a periodic motion, or the closure of the trajectory of an almost periodic or recurrent motion cannot be globally weakly attracting.
1.9.4
COROLLARY Let
be a compact minimal set with global strong stability
M~E
properties, then
M
is a rest point.
The following theorem is a generalization of one of the principal results of the Poincare-Bendi~n Theory of planar dynamical systems described by differential equations viz., every periodic trajectory contains in its interior a rest point.
This is clear when we notice that a periodic trajectory
and its interior form an invariant set homeomorphic to the unit disc. The proof of this theorem is an elementary application of the Brou~e~ fixed point theorem and of the following lemma.
1.9.5
LEMMA Let
be a compact positively invariant set.
M~X
sequence of periodic trajectories with periods Tn + 0.
Proof.
Then
M
Tn,
Let
such that
{yn }
be a
YnCM,
and
contains a rest point.
Consider any sequence of points
{Xn} ,
We may assume without loss of generality that
with x
Xns
§ x s M,
n = 1,2,... as
M
is compact.
We
n
will demonstrate that
x
is a rest point.
For suppose that this is not the case.
102
Then there is a The spheres
S(x,4),
T,O < T < T ,
such that
and
such that
there is a for
9 > 0,
6 > 0
0 .< t .< ~.
x r ~(x,T).
S(~(x,T),~) P(x,~r(x,t))
such that
Let
d(x,~(x,~)) = a(> 0).
are disjoint.
.< ~
for
0(x,y)
Now choose
0 ~ t .< T.
< 6
implies
Notice in particular that if
By c o n t i n u i t y
of
p(~(x,t),~(y,t))
0
there
called displacements, such that n
p(xt,x(t + ~n) ) < ~
for all
t ER
and n
Notice that the set
{T }
of the definition 1.10.2 does not depend
n on
x.
Obviously, periodicity implies almost periodicity, while the converse is not true.
I. I0.3
DEFINITION A motion
a
9 = ~(t) > 0
~X
is said to be recurrent if for every
such that for all
ts
e ~ 0
there exists
xR C S (x [t-T,t+~ ], ~).
106
The property of recurrence can be expressed as almost periodicity if the set
{T n } is made to depend on
x.
It can he proved that almost
periodicity implies recurrence and that there exist recurrent motions which are not almost periodic.
Many theorems of the relative properties of compact
minimal sets and recurrent motions are given in Section 2.9.
i. 10. 4 DEFINITION A motion
~x
is called positively Poisson-stable
(P+-stable) if
negatively Poisson-stable (P--stable) if x E A-(x)
x E A+(x),
stable (P-stable) if both
x EA+(x)
and
xEA-(x)
and Poisson-
ho/ds.
Again, the property of Poisson stability may be defined as a weak form of the property 1.10.2 where the set
{~n}
may depend upon
x
and does
not need to be relatively dense. Obviously, P'-stable.
if
~x
is
P'-stable,
is
P+-stable then
xR 0 A+(x) r r
1.10.5
and that if
x
is
P--stable,
then
xROA-(x) r r
1.10.6
We shall then prove that
I. ] O.7 THEOREM If
x
1.10.8
i8
P+-stable
A-(x) C
If
t eR
also
~xt is
Poisson stability can, therefore, also be defined as a property
of trajectories and their limit sets. x
then for all
x
is
A + (x) = xR
P--stable
In fact, it is easy to show that if
i07
i.i0.9
and if
A+(x) C
x is
P-stable, then
A+(x)
i.i0.i0
Proof.
A-(x) =
=
A-(x)
We shall prove 1.10.8,
The proof of 1.10.9 and i.I0.I0 is analogous.
Because of 1.10.5, from the closedness of ~C
A+(x).
A-(x)C: A§
A+(x), if
From the definition of limit sets, and
A + ( x ) C x"R+~___ x--R. Thus
x
is
P+-stable, then
A-(x)C~'C
xR,
hence,
A+(x) = ~ .
The following theorem on Poisson-stable motion is very simple and its proof is left as an exercise.
i.i0.II
THEOREM A motion
is
P+-stable if and only if the trajectory xR
is not
x
positively asymptotic. It can be proved that recurrence implies Poisson stability and there exist Poisson-stable motions which are not recurrent. 1.10.12
Notes and References The definition of a recurrent motion given in this section is due to
G. D. Birkhoff [i, Vol. i, pg. 660).
See also
2.10.17.
108
Stability.of motions
i.ii
Liapunov stability and asymptotic
stability of sets are properties
of a given set with respect to the neighboring stability and asymptotic oriented trajectory:
Thus Liapunov
stability are purely geometrical properties of the
the set
to the case of motions. stability of a motion
trajectories.
xR + U xR-.
We shall now extend these concepts
It is important to point out that the concept of ~
defined by ~
and
x ~E
is completely different
xR.
The stability of a motion
x
from the concept of stability of the set
x
can be defined as follows.
1.11 .i
DEFINITION A motion
~x
there exists a
c>O
for all
is said to be (positivel~ Lia~unov) stable if for every n(c) > 0 yEE
such that
with
p(x,y)
0
p(x~,y) < n
is said to be (~ositivel~ Lia~unov) unifo~ml~ stable, if,
there exists
n(e) > 0
p(x(~ + t),yt) < E
such that for all for all
tER +
and
y ~E
with
TER.
Similarly one can define negatively Liapunov uniformly stable motions and Liapunov uniformly stable motions.
From this definition
follows a rather
109 interesting result which is presented in the next theorem.
Similar results
can be given for the case of negatively stable and stable motions.
1.]1.4
THEOREM A motion
motion
~
Y
~x
with
is positively (uniformly) stable if and only if every
y ~xR
i8 positively (uniformly) stable.
From the definitions i. Ii.i and 1.11.3 it clearly follows that
1.11.5
THEOREM If a motion
trajectory
x~
~ x
is (uniformly) stable then the corresponding
is also (uniformly) stable.
It is on the other handeasy to show that if
~
is stable the closure
X
of the positive semi-trajectory example the continuous flow ~ segment
M,
xR +
shown in Figure 1.11.6.
limited by the equilibrium points
the other hand, one can define motions uniformly stable).
i.ii.6
need not be stable.
Figure
~x,X ~ M
z
and
Consider for Clearly the closed
y
is unstable.
On
which are stable (but not
110 The converse of Theorem 1.11.5 does not hold~ in fact (uniform) stability of a trajectory does not imply stability of the motions defined on it.
This fact can be shown by many examples.
flow ~
For instance, consider the
on the plane represented by Figure 1.11.?.
The trajectory through
each point is a circle with its center in the origin of the plane.
1.11.7
Figure
) Obviously the origin as well as all circular trajectory are uniformly stable. Assume that the tangential velocity of rotation defined by the motion on each trajectory is a constant,
it follows that the angular velocity
is decreasing as the radius of the circles is increasing.
Thus the periodic
motions are not stable. In the case that for
x HE,
the corresponding
set
xR
is compact,
or even in the case of almost periodic motion some stronger connections between the stability properties of motions and those of the corresponding will be shown
trajectories
(Section 2.9).
In the particular case of an equilibrium point the two concept: stability of motion and stability of trajectoryj
coincide.
This is the reason
for the not clear distinction between stabil~ty of sets and stability of motion in the classical literature.
i. 1 i.8
THEOREM The equilibrium motion
and only if the set
{x} ~ xR
~x
[~x(t) = x
is stable.
for all
t ~ R], is stable if
111
Proof.
Let the equlllbriummotlon
w
be stable.
Then, given any
c > 0,
x
there is a t 6 R +.
6 > 0
Since
y~S({x},6)
such that
xt = x implies
p(x,y) < 6
for all yR+C
t~R,
such that
there is a
~ > 0
particular
yt ~ S(x,e),t#R +.
yt 6 S(xt,e),l.e.,
we get
S({x},~),
Now let the invariant set
implies
{x}
{x}
be stable. implies
Since, however, for
t ~ R +,
< r
p(x,yt) < e.
i.e., the set
p(x,y) < ~
P(xt,yt) < e
p(xt,yt)
for all
Clearly then
is stable.
Then, given any yR+C
xt = x
S(x,e). for all
the motion
e > 0,
Also in t ~ R,
~ is stable.
we have This
X
completes the proof.
In the literature,
it is also given, for the case of motions, a stability
property stronger than the one given in definition 1.11.1, namely stability wlth respect to a set.
i. ii. 9
DEFINITION A motion ~
the set every
BC y~ B
E
is called (~ositivel~ Liomunov) stable with respect to
if for every
with
~ > 0
p(x,y~< n
there exists
n(E) > 0
p(xt,yt) < c
for all
such that for t ~,
(positively Liapunov) uniformly stable with respect to the set every
r > 0
there exists
p(x(~ + t),yt) < ~
n(~) > 0 for all
such that for every t&R +
and
y~B
and B c E
if for
with
p(xT,y)
< 6
T~R.
Similarly one can define negatively Liapunov stable and Llapunov stable motion wlth respect to of stability of however, X
that if
B.
If
B
is a neighborhood of
~x and stability of~ x B ffixR +
xR +
B
then the concept coincide.
Notice,
this need not be true since there may exist motions
which are not stable with respect to
Figure i.ii.i0 where
with respect to
x,
xR +.
For instance,
is a straight half line
in the case of
112
i. i i.i0
FiQure
x R+
and the motions is accelerating on it. It will be proved in Chapter 2 that for almost periodic motion this situation cannot arise. Similarly to that done for the case of stability one can define attracting motions as:
I.ii.Ii
DEFINITION is said to be attracting if there exists a
A motion such that
6>0
x
p(x,y) < 6
implies
p(xt,yt) + 0
for
t § + |
Clearly the property of attraction of a motion
p r o p e r t y of a t t r a c t i o n
of t h e c o r r e s p o n d i n g t r a j e c t o r y
x
implies the
x R§
As in the case of trajectories one can also define asymptotic stability.
1.11.12
DEFINITION A motion
is said to be as~toticall~ stable if it is both stable x
and attracting. Notice that for the case of motions having noncompact trajectory closures one can define at least as many properties as the ones defined in Section 1.6.
Since we shall not use these properties it is pointless to define
them in detail.
Their definition is very simple.
113
1.11.13. Notes and Re~erenoes The original definition of stability of motions is due to A. M. Liapunov [1]. The presentation given here is adopted from Nemytskii and Stepanov.
It must be emphasized that the stability of a given motion
was the only form of stability considered by Liapunov [1] as well as from many other authors like Chetaev [5], Malkin [8] ~ Hahn j[2] etc. 3.6.32.
See also
114
DYNAMICAL SYSTEMS IN METRIC SPACES
2.1
Definition of a dynamical system and related notation. A dynamical system or continuous flow on
where
~:X x R § X
X
is a map from the product space
is the triplet
(X,R,~),
X x R
satisfying
into
X
the following axioms:
2.1.1
2.1.2
~(x,0) = x
xEX,
~(~(X,tl),t 2) = ~(x,t I + t 2)
and
2. i. 3
for every
~
for every
x(X,
tl, t 2 ~ R ,
is continuous.
The above three axioms are usually referred to as the Identity,
Homumorphlsm
and Continuity Axioms, respectively. In the sequel we shall generally delete the symbol ~(x,t)
of a point
(xpt) ( X x R
the hom~umorphismaxioms
2.1.i'
2.1.2'
x0
=
will be written simply as
~. xt.
Thus the 4m,ge The identity and
then read
x
for every
Xtl(t 2) ffix(t I + t 2)
x ~ X,
for all xt X
and
and
tl,t2~ R.
Notice also that the continuity axiom is equivalent to:
2.1.3'
If {xn} , {tn} are sequences in x n + x, tn § t,
then
{Xnt n}
In line with the above notation,
X
and
R
is a sequence in
if
M~X
and
respectively such that X
A~R,
such that
we set
Xntn § xt.
115
MA = { x t : x % M
and
t~A}.
If either
M,
containing exactly one element) i.e., xA,
or
Mt
for
{x}A~
or
M{t},
The phase map or
t
is fixed.
~t(x) = xt X
~
w
is a singleton, or
A = {t},
we slmply write
respectively. X~
the space
X
is generally called
determines two other maps when one of the variables
Thus for a fixed
Again for a fixed
t~R
the map
For each x~ X
wt:x § X
t ~ R,
the map
t
is called a motion
(through
x).
d e t e m l n e d by
is a homeemorphism of
~ :R + X X
~x(t) = xt
(a set
as the phase map (of the dynamical system).
is called a transition.
onto itself.
A,
M = {x},
For a given dynamical system on the phase space~ and the map
or
determined by
x
116
Elementary Concepts:
2.2
For any trajectory,
Trajectories and their Limit Sets.
x ~ X ,
the trajectory
(or orbit), the positive semi-
and the negative semi-trajectory
are the sets given respectively
by
~,(x)
2.2.1
= {xt: t f R}
,
2.2.2
~+(x) -- {xt: t G R +} ,
2.2.3
~,-(x)
= {xt: t ( R-}
and
. B
We shall reserve in the sequel the symbols for the maps from
X
to 2 x
defined respectively by
Y,
Y
, Y
2.2.1, 2.2.2, and
2.2.3. A subset
M C X
will be called i n v ~ a n t ,
or negatively invariant if the condition
y(M) -- M ,
positively
invariant,
y+(M) = M
or
m
y (M) = M
is satisfied,
respectively.
DEFINITION
2.2.4
A subset
M C X
is called minimal~
and invariant, and no proper subset of
M
if it is non-e~ty, closed,
has these properties.
DEFINITION
2.2.5
For any
x ~ X ,
the positive or omega limit set~
and the
negative or alpha limit set are the sets given respectively by A+(x) = {y ~ X:
there is a sequence
A-(x) = {y I X xt
+ y}
:
{tn}, tn § + ~ ,
there is a sequence
such that
{t n} , t n § -~
@
n
Examples of limit sets are given in Section
1.3.
,
xt n § y} ;
such that
117
Exercises
2.2.6
i)
Show that
il)
~(~(x))
y(x) = T(Xt)
= ~(x)
for every
, ~+(~+(x))
y(x) , y+(x) ,
and
t ~ R .
= y+Cx)
y-(x)
,
and
~-(~-(x))
are respectively,
= ~-(x)
.
Thus
invariant, positively
invariant, and negatively invarlant. iii)
iv)
Show that A+(x) = ~ { y + ( x t ) :
t (R}
,
A-(x) = ~ {
t ~ R}
.
y-(xt):
and
Show that A+(x) = A+(xt) ,
2.2.7
Note.
The relations
and
A-(x) = A-(xt) ,
(lii)
in
2.2.6
t ~R
.
are frequently used to
define the positive and negative limit sets. 2.2.8
Exercises
i)
Show that
ii)
A+(x)
2.2.9
7+(x) = 7+(x) U A+(x) .
is closed and invariant.
THEOREM If the space
A+(x)
X
is locally compac~ then a positive limit set
is connected whenever it is compact.
Further~ whenever a positive
limit set is not compact, then none of its components is compact. Proof.
Let
A+(x)
A+(x) = P U Q , A+(x)
be compact, and let it be not connected.
where
P,Q
is compact, so are
compact, there is an disjoint.
Now let
E > O y s P
Then
are non-empty, closed, disjoint sets. P
and
Q .
such that and
z ~ Q .
Further, since S[P,~] ,
S[Q,r
X
Since
is locally are compact and
Then there are sequences
{tn} ,
118
{ T n}_ ,
t
§ + ~,
T
n
such that
§ +~, n
xt
we may assume without loss of generallty, and
T
-
n
t
n = i, 2, ... , and
H(Q,r
for all
> 0
n
9
n
§ y
and
,
n
that
xT
§
z
xt n E S ( P 9
,
XTn
Since the trajectory segments
.
And
.
n
E
X[tn, Tn|.-
are compact connected sets, they clearly intersect
Thus, in particular,
such that
xT n ~ H(P,E)
xT n § ~ ,
and as
there is a sequence
which is compact.
Tn § + = ,
which is a contradiction.
we have
S(Q,e)
{Tn } , tn
H(P,~) < Tn
0},
D-(x) --N {y-(S(x,=))
: = > 0},
D+(x) ~
y+(x),
and
u
iv)
2.3.4
D (x)~y
(x).
THEOREM
For any
xEX,
D+(x)
is closed and positively invariant.
The proof is left as an exercise.
2.3.5
THEOREM
Let
X
be locally compact.
Then for any
x E X, D + ( x )
is connected
122
whenever it is compact.
Further, if D+(x)
is not compact, then none of its
components is compact. The proof follows exactly the same lines as that of Theorem 2.2.9
and
is, therefore, omitted. 2.3.6
DEFINITION The (first) positive prolongational limit set, and the (first) negative
prolongational limit set of any
xE x
are the sets given respectively by
J+(x) = (y~X: ~(x n } C X, {tn}CR , such that
xn §
tn §
{tn}CR , such that
xn §
tn §
~
xl-axis,
J+(p)
is the
J-(x) = ( y ~ X : ~ { X n } ~ X , 2.3.7
x n tn §
and
x nt n §
Ex~wp/e In Example 1.4.7 (i) for any
In Example
2.3.8
and
1.4.7. (ii) for any
p
in the
P (Y-I'
x2-axis.
J+(P) = Y0 ~j Y-2"
Exercises Show that for any i)
x~X
J+(x) = [~ {D+(xt) :tf R},
and
J-(x) = N {D-(xt):t6R}. ii) iii) 2.3.9
J+(x) = J+(xt), J-(x) = J-(xt) y(J+(x)) = J+(x),
i.e.,
and
J+(x), J-(x) 2.3.10
is invariant.
Note The relations (i), (ii)
D+(x)
J+(x)
t ~ R.
D-(x). once
in Exercises 2.3.3 are frequently used to define
The relations (1) in Exercise 2.3.8 are frequently used to define D+(x), D-(x)
have been defined.
Exercises Show that i)
if)
D+(x) = y+(x) U J+(x)
J+(x),
J-(x)
and
D - ( x ) = y - ( x ) tJ J - ( x ) .
are closed and i n v a r i a n t .
123
2.3.11
THEOREM Let
compact.
X
be locally compact.
Further, if
J+(x)
Then
J+(x)
is connected whenever it is
is not compact, then none of its components is
compact.
The proof of the first part although similar to that of Theorem 2.2.9 will be made to depend on the following l ~ a .
2.3.12
Proof of the second part will be omitted.
LEk~4 Pot any
particular, if
X, {t n}
and
x~Xj
J+(x) C
mEA+(x) ,
{~n }
in
D+(~),
and R+ ,
y ~ J+(x),
with
Tn + + ~j Xn
§ ~oj and
Xn
Proof.
~ E A+(x),
and any
Given
Tn' § + ~, x ~
§ ~,
and
whenever
xn
A+(x) @ r
and
y E J+(x),
k, k = 1,2,...,
k,
and
t' - T' > 0
the sequence
i ~ ~
for
have then the required properties.
arbitrary, we have
2.3.13
in
tn §
~,
{T~},
x n t'n + y . We can assume, if
for each
n.
Consider for
n
{XnT~}.
By the continuity axiom
n ~ k.
and
i + ~.
The sequences
Now notice that
t' - ~' > 0. n n
J+(x) C D + ( ~ ) ,
This shows that
Hence
Xn nT' § ~,
because
{T~}, {t'} n' !
Xnt ~ = X n ~ ( t ~ - ~n ),
y~D+(~).
As
y~J+(x)
was
and the lemma is proved.
Proof of the first part of Theorem 2.5.12 Notice that
then
n,
{xn}
We may, therefore, assume without loss of generality that
P(XnT~,x~)
x t' + y, x ~' + ~, nn nn
for each
{xn} ' Xn + x , t'n § + ~ ,
pC~,Xn~ ~) ~ pC~,xT~) + p(x~,Xn~ ~) ~ pC~,x~) {xn}
> 0
there exist sequences
n
for each fixed
n
In
§ y.
{t~}
XnT ~ § xT~, k = 1,2, . . . .
~EA+(x).
then there exist sequences
§ x, tn - 9
necessary by choosing subsequences, that each fixed
and
h+(x) ~ ~ ,
A+(x) C J+(x) whenever
J+(x)
holds always. is compact.
And if
X
is locally compact,
To see this assume that
h+(x) = ~.
124
Since that
X
is locally compact, and
S[J+(x),e]
is compact9
y+(xT) N S[J+(x),e] t
§ + ~,
= 4.
such that
J+~)
If
is compact, we can find an
A+~)
= 4,
This is so for, otherwise,
xt E S[J+~),e],
n
there exists a
and as
e > 0
T > 0
such that
there will be sequence
S[J+(x),e]
{Xtn} ,
is compact, the
n
sequence
{Xtn}
will have a limit point
Notice also that A+(x)
x~J+~),
contradicting
y + ~ ) C J+(x),
y+(x) N S[J+(x),~]
and as
= 49
{Xn}' { tn }' Xn § x, t n + + ~,
X n ~ S[J+(x),e],
and
X n t n 6 S(J+(x),e)
{Tn}, 0 < T n < tn'
sequence
w E A+(x)
such that
for all
claim that
{Yn }
z EJ+~),
whereas
T
z = xts
x r nn
H(J+(x) 'e)
is
§ zE H(J+(x),e).
We
§ t.
= ~
But then by the continuity axiom
J+(x) = P U Q,
where
y+(x) ~ S[J+(x),~]
is non-empty and compact,
Further, choose
S[P,e], S[Q,e]
P
e > 0
and
Q,
x T nn = 4.
§ xt ~ y+(x), Thus
J+(x)
such that
A+(x) C
P.
n
§ x, t
Xn nTE S(P,E), < t' < tn, n
n
- T
and
n
S[F,e] N S[Q,e] = 4, X
+ + ~, x T nn
Xntn~ S(Q,E)
such that
for all
x t'EH(P,e), nn
+ ~, n.
and
x t nn
+ y.
t' + + ~. n
and
and P,Q
{x }, {t }, n
We may assume that
But then there is a sequence
and indeed
As
y ~Q,
n
n
cannot
A+(x)
is locally compact, and
Now, by L~mma 2.3.1~, there are sequences
> 0, T
i.e.,
is not connected.
Choose
9
{Tn} , x
A+(x)
(Theorem 2.2.9) and so say
are compact (this is possible as
are compact and disjoint)
if
are compact, non-empty and disjoint.
it is connected
is a subset of only one of the sets w~A+(x) C P .
P,Q
However,
and we may assume without loss of
To complete the proof of the theorem, assume that
Then we have A+(x)
n
This is impossible as
be empty.
We may assume
Since
J+(x) N H[J+(x),e]
is bounded, it has a convergent subsequence,
generality that
Then
is bounded, because otherwise it will have an unbounded subsequence
so that we will have {T } n
yE J+~).
But then there is a
x n Tn ~ H(J+~),e) "
compact, we may assume without loss of generality that
such that
x n tn + y. n.
= 49
is compact,
~ > 0
Now let and
A+~)
J+(x)
It is thus clear that there is an
is compact, and
there are sequences that
for then
will not be empty9
S[J+(x),u]
n
such
Since
H(P,e)
{t'},n is
125
compact, we may assume that
x t' § z~H(P,e). n
z ~ P U Q.
But then
z~J+(x),
while
n
This contradiction proves the theorem.
The first part of the above proof contains the following lemma which we give below for future reference.
2.3.14
LEM~Z4
If A+(x) # ~
2.3.15
X
i8 locally compact, and if
J+(x) # ~
is compact, then
and i8 indeed compact.
Exercise Give an example to show that Lemma 2.3.14 does not hold in general metric
spaces
X.
2.3.16
Exercises i)
If
M~X
is non-empty and compact, then
ii)
If
M CX
is connected, and
connected if
2.3.17
X
D+(M)
D+(M)
is closed.
is compact, then
D+(M)
is
is locally compact.
Example Consider the Example 1.4.7 i) modified by deleting the origin of the
coordinates from the plane. origin of coordinates)
Then for each
A+(p) = ~.
If
p
p
in
X
(i.e., the plane without the
is a point in the
xl-axis , then
has two components, viz. the positive and negative parts of the
x2-axis.
J+(p) Both are
indeed non-compact. In Example 1.4.7 ii) note that whereas
J+(p) ~ ~
for all
p ~5Y-l"
A+(p) = ~
for all points
p
in the plane,
126
2.3.18
Notes and References The d e f i n i t i o n
(see also 1.4.14)
of p r o l o n g a t i o n i s due to T. Ura [ 2 ] .
He adopts the
relations 2.3.3 (I) and 2.3.3 (il) as definitions. Theorem 2.3.5 is essentially due to N. P. Bhatia [3]. prolonsational
limit sets is due to J. Auslander,
Theorem 2.3.11 is due to N. P. Bhatla [3].
The concept of
N. P. Bhatia and P. Seibert.
127
Self-intersecting trajectories
2.4
2.4.1
DEFINITION A point
x~X
such that
xt
= x
t~R,
for all
is called a rest point
(or a critical point, or an equilibrium point).
2.4.2
DEFINITION For any
x ~ X,
the trajectory
is called periodic with a period
Notice that a rest point T~R
T
(and also the motion
y(x)
whenever
x~X
x(t + T) = xt
~x
for all
through
x)
t ~R.
is a periodic orbit having every number
as a period. However, the following lemm~ holds.
2.4.3
L~I
I f { x } # y(x)~
i.e.,
x
is not a rest point, and if
there exists a least positive number if
9
Proof.
is any period, then
Notice that if
T
T,
such that
T
y(x)
is periodic,
is a period of
y(x)
then
and
~E{kT:k = ! i, ~ 2, ...}.
is a period, then so is
-~,
because if
for all
xt = x(t + ~)
t~R,
then by the homomorphism axiom
x(t
showing that as periods.
-T
-
~)
= xt(-~)
is a period.
k~
trajectory
+
is a period. 7(x).
+k~)
~)(-T)
ffi x ( t
+
T -
~)
ffi x t ,
Thus the periodic trajectory has positive numbers
Notice further that if
x(t
so that
ffi x ( t
T > 0
for all
ffi x t
Now let
is a period, then for any integer
P
k
ts R,
be the set of positive periods of the periodic
If there is no least positive period, then there is a sequence
128
{Tn}, T n ~ P, periods). p(x,xt) < s
and
Hence
It] < 6.
if
As
rl,~ 2
~n § 0,
are perlods,
s > 0
Tn > 0
we notice that > 0} = {x},i.e.,
there is a
then
T 1 _+ T 2
6 > 0
are
such that
is a period, then obviously 7(x) = x [ 0 , T n ] C S ( x , e ) x
for large
is a rest point contrary
This proves the lemms.
DEEINITION For any
if there exist 2.4.5
Further, if
7(x) = ~ { S ( x , e ) ; r
to the hypothesis.
2.4.4
(because, if
By the continuity axiom, given
7(x) = x[0,T n]. n.
Tn + 0
x(X,
the trajectory
tl, t 2 E R , t I # t2,
7(x)
such that
is said to be self-intersecting, xt I = xt 2.
THEOREM If for any
point or
xs
7(x)
is self-intersecting,
then either
x
is a rest
7(x) is periodic.
Notice that rest points and periodic trajectories are self-intersecting. The above theorem shows that these are the only self-intersecting The proof of the above theorem is trivial and is, therefore,
2.4.6
omitted.
Exercise A self-intersecting
trajectories.
trajectory is a compact minimal set.
129
Lagrange and Poisson stability
2.5
2.5.1
DEFINITION For any
if
xEX,
the motion
is compact.
y+(x)
Further, if
called negatively Lagrange stable.
~x
is said to be positively Lagrange stable
y-(x)
is compact, then the motion
It i8 said to be Lagrange stable if
~X
is
y(x)
is compact. 2.5.2
Remark If
X = E,
then the above statements are equivalent to the sets
-~+(x), -~-(x), w(x) 2.5.3
being bounded, respectively.
Exercises (i)
If
X
is locally compact, then a motion
stable if and only if (ii)
If a motion
~x
A+(x)
~x
is positively Lagrange
is a non empty compact set.
is positively Lagrange stable, then
A+(x)
is
compact and connected. (iii)
If a motion
~
is positively Lagrange stable, then
p(xt,A+(x)) § 0
X as
t
§
~ .
It will be useful to compare the statements in the above exercise with Theorem 2.2.9 a~d the Exercise 2.2.16 (i) and (ii).
2.5.4
DEFINITION A motion
~
is said to be positively (negatively) Poisson stable if
X
x ~A+(x)
(x ~ A-(x)). It is said to be Poisson stable if it is both positively and
negatively Poisson stable, i.e., if x CA+(x) N A-(x). 2.5.5
Exercise (i)
A motion
is positively Poisson stable if and only if X
w(x) CA+(x).
130
(ii)
A motion
~x
is positively Poisson stable if and only if
A-(x) C A+(x) = ~(x) (iii)
If
~
is positively Poisson stable then for any
x
motion
~xt
t ~ R,
the
is positively Poisson stable.
Exercise
2.5.6
A self-intersecting
trajectory is Lagrange stable and Poisson stable.
Indeed the following theorem holds.
2.5.7
THEOREM A motion x
is positively Poisson stable if and only i~
~+(x) = A+(x). The proof is trivial (see Exercise 2.5.5) and is left as an exercise. In view of the above theorem it is interesting to inquire about the consequences of the condition
y+(x) = A+(x).
The answer is contained in the
following theorem.
THEOREM
2.5.8
y+(x) = A+(x) if and only if either x is a rest point or
y(x)
is a
periodic trajectory. Proof.
Let
y+(x) = A+(x).
nothing to prove. Suppose A+(x)
x
and, therefore,
homomorphism axiom
trajectory
y(x)
x
is a rest point, the relation holds and there is
is not a rest point.
is invariant we see that
x T~y+(x), the
If
Indeed
y+(x) = A+(x) = y(x).
there is a
T' ~
xt = x(t + T' - T)
is periodic with a period
0
and as
Thus for each
such that
for all
x ~ A+(x)
t ~ R,
T' - z(> 0).
xz
= xT'.
T < 0~ Hence by
showing that the The converse holds trivially
and the theorem is proved.
2.5.9
Remark It is to be noted that if
Y+(x) = A+(x)
then the motion
w
x
is indeed
131
Poisson stable.
It is, therefore, appropriate to inquire whether there exist
motions which are Poisson stable but are not periodic point).
(i.e., also not a rest
We give below an example of a motion which is Poisson stable but is
neither a rest point nor a periodic motion.
Exegnple
2.5 .i0
Consider a dynamical system defined on a torus by means of the planar differential system d ~ = f(~,0), dt where
dO d-~ = ~f(~,0),
f(~,0) E f(~+ i, 0 + i) ~ f(~ + i, 0) E f(~,0 + i),
and
O
are not both zero (mod I),
f(0,0) = 0.
Let
and
~ > 0
f(~,0) > 0
if
be irrational.
It is easily seen that the trajectories of this system on the torus consist of a rest point YI
p
such that
corresponding A-(YI) = (p},
A+(Y2) = {P}.
to the point
There is exactly one trajectory
and exactly one trajectory
For any other trajectory
A+(Yl) = A - (y2) = the torus.
(0,0).
Y2
such that Further
y, A+(y) = A-(y) = the torus.
In this example,
therefore,
the trajectory
Y1
is
positively Poisson stable, but not negatively Poisson stable.
The trajectory
negatively Poisson stable, but not positively Poisson stable.
All other trajectories
are Poisson stable.
Note that no trajectory except the rest point
p
Y2
is
is periodic.
The following theorem sheds some light on a positively Poisson stable motion X
when
2.5 .ii
y+(x) ~ A +(x).
THEOREM Let
X
be a complete metric space.
Let a motion
x
stable, and let it not be a rest point or a periodic motion. A+(x) ~ y(x)
Proof.
Since
is dense in
~
Then the set
A+(x), i.e.,
is positively Poisson stable, we have X
be positively Poisson
A+(x) -- y(x).
To see that
132
A+(x) N y ( x )
= A+(x),
it is sufficient to show that if
y~ y(x)
and
e > 0
is
arbitrary, then there is a point z E A+(x) \y(x)
such that
To see this notice that since
there is a monotone increasing
sequence
(tn} , tn § + ~,
yTl~ S(y,e).
Then
such that
yt n § y.
Y~I ~ Y[-tl'tl]
61 = P(YXl'Y[-tl'tl]) S(Y~l,el)C S(y,e)
y C A+(x) ~ A+(y),
> 0.
and
Set
Choose
(otherwise
~x
choose
where
6n = p(y~n,Y[-tn,tn]).
TI > t I
such that
will be periodic).
eI = rain{2 , ~ - p(y,yTl),~-}.
S(Y~l,el) ~ y[-tl,tl] - @.
en_l,
z E S(y,E)
Also
Then
Having defined
YTn_ 1
and
> t such that yTn~ S(YTn_l,en_l) (possible because of positive n n en-i ~n--i Poisson stability of #x ) . Then define Cn = min{ -2--'en-i - P(YTn-I'YTn ) ' 2 } '
Clearly
T
Note that
S(YTn,en)~S(YTn_l,en_l)
,
~n > 0
and
as the motion is not periodic.
S ( Y T n , e n ) ~ Y [ - t n , t n] = ~. for
n = 1,2 ,...
{yTn }
has the property that
{yT n}
is, therefore, a Cauchy sequence which converges to a point
X
is complete.
p(y,yTn) < e, if
z 6Y(x)
tn > [~I,
Since so that
E y(y), so that
p(YTn,Y~n_ I) < en_ 1 ~ 2n_l
y~n ~ y(x), p(y,z) ~ e.
we will have z~y[-tn,tn].
S(YTn,en) ~ y[-tn,tn] = ~, i.e., z~y(x)
and
Tn § + ~'
we have
Notice further that
But there is an
However,
z6S(YTn,e n),
z~y[-tn,tn].
z
as the space
z ~A+(x).
z ~ y(x).
z = yT.
The sequence
n
Further
For, otherwise,
such that
and by construction
This contradiction proves that
and the theorem is proved.
It is now clear that
2.5.12
THEOREM If
X
i8 complete, then a necessary and sufficient condition that
be periodic i8 that 2.5.13
y(x) -- A+(x)
y(x)
[= A-(x)].
Remark Theorem 2.5.12 is not true if
X
is not complete.
This can be shown for
instance by constructing an almost periodic motion on a torus and then delete from the space all points which do not belong to the trajectory defined by that motion.
133 Obviously, 2.5.14
7(x) ffi A+(x),
~(x)
is not periodic.
THEOREM A motion
r > 0
but
x
there exist a
is positively Poisson-stable if and only if for every t ~ 1
such that
xts S(x,r
The proof is left as an exercise to the reader.
2.5.15
Notes and References This section has essentially been adopted from the book by Nemytskii and
Stepanov.
134
2.6
Attraction, stability, and asymptotic stability of compact sets
2.6.1
DEFINITIONS A compact set
> 0
such that
for all
x~K,
whenever
if there is an
if there is an
x~S(M,E);
~ > 0
such that
A+(x) r 4,
and
x~S(M,E);
a uniform a~ractor, > 0
is said to be a weak attractor,
A+(x) N M # 4
an attractor, A+(x) C M
M~X
and a compact set there exists a
K
if it i8 an attractor and is such that given any with the property that
T ffiT(K,6) ~ 0
stable, if given any
~ > 0
with
KtC
there is a
A+(x)@ 4, A + ( x ) ~ M
S(M,~) ~ > 0
for all such that
for all
t > T; 7+(S(M,~)) C
S(M,~);
asymptoticall~ stable, if it is both stable and an attractor; and finally unstable, if it is not stable.
2.6.2
Remark The concepts of attraction and stability are in general independent of each
other as we shall presently see.
However, under certain circumstances attraction
and uniform attraction do imply stability.
Further, if a stable set is a weak
attractor, then it is an attractor and hence asymptotically stable, and an asymptotically stable set is a uniform attractor.
Thus the combination of stability with any one
of the attractor properties yields asymptotic stability. Section 2.6.3
For details see
1.5 DEFINITION Given any set
MCX,
set
A (M) ffi {x~X:^+(x) n M ~ 4},
A(M) = {x~X:A+(x) ~ r
and
and
^ + ( x ) C M}.
135
The sets
A (M), and
A~)
a~e respectively called the region of ~eak attraction,
and the region of attraction of the set Note that
A (M)~A(M)
M.
holds always.
Exercise
2.6.4
Show that the sets
A (M)
and
ACM)
are invariant.
The implications of the various stability properties defined in 2.6.1 and the elementary properties of compact sets having one of these stability properties have been discussed at length in Section 1.5.
We shall now present some more
results.
THEOREM
2.6.5
If a compact set Proof. Let
M CX
Let, if possible,
0~,M)
= 6 > 0.
hence sequences
is stable, then
D+CM) # M.
Since
~,
0 < u .< 5'
XnCM ,
there is an
§ x, t n
we may assume that
Then there is a point
yfD+(M),
{Xn} , {tn} , x
>. 0, x t n
n
6 X n ~ S ( M , ~),
y+(S(M,e))-- ~ S ~ ,
~)'
D+CM) = M.
xEM
+ y.
yED+(M)~M.
with
y&D+(x),
and
In view of Theorem 1.5.24
n
X n t n ~ S [ M , 26--I.This shows that for every
i.e.,
M
is not stable.
This proves the
theorem.
The converse of the above theorem is not in general true. locally compact metric spaces
2.6.6
in
we do have
THEOREM If
if
X
However,
X
is locally compactj then a compact set
MC
X
is stable if and only
D+(M) = M.
Proof. is an
Let e > O~
D+(M) = M, a sequence
and suppose if possible that {x } n
and a sequence ~
M
{t } n
is not stable. with
~
t
% O, n
Then there
136 P(Xn,M) + 0, c > 0
and
P(Xntn,M)
~ e.
We may assume without loss of generality that
has been chosen so small that
[this is possible as xn § x s
X
S[M, el
is locally compact).
We can now choose a sequence
XnTn~ H(M,~), n = 1,2,... Xn Tn + y ~ H(M,e) shows that theorem
M
and hence
Since
Then clearly
is stable.
H(M,e)
Further, we may assume that
{Tn }, 0 ~ Zn ~ tn,
K(M,e)
y E D+Cx) C
is compact
such that
is compact, we may assume that D+(M),
but
y~M.
This contradiction
The converse has already been proved in the previous
and so the proof is completed.
The following example shows that Theorem 2.6.6 does not hold in general metric spaces.
Example
2.6.7
Consider Example 1.5.32 (ii) (see Figure 1.5.33). the set
X \ Y.
D+(P2 ) = P2'
P2
but
does not have any compact neighborhood). P2
is not stable.
The trajectories
be
Note also that
in the present example
are the same as in Example 1.5.32 ii) except that the trajectory
7
has been deleted.
The following exercise contains yet another characterization compact set
X
This space with the usual euclidean distance is not locally compact
(note that the point now
Let the space
of a stable
M .
2.6.75Erercise8 i)
If
M
is stahle, then
J-(X\M)
N M = ~ .
ii)
If
X
is locally compact, then a compact set
M
is stable if and only if
J-(X \ M) N M = r
iii) Further
Show t h a t i f
xED+(y)
x,
y~X,
i f and only i f
then
x~J+(y)
i f and only i f
y~J-(x).
Y~ D-(x).
We shall now present an interesting property of the components of stable compact sets.
137
THEOREM
2.6.8
Let
be compact and let
M~X
X
be locally compact.
stable if and only if every component of M Proof.
Note that if
Further if Now let
M
Let
Mi
and
is stable.
Mi
2.6.9
M
is compact.
is positively invariant, so is every one of its components. where
I
M.l be stable, i.e.,
Since
is stable.
is an index set, and
D+(Mi) = M i.
To see the converse, let
be a component of
D+(Mi) C M .
is
is stable.
is compact, then every component of
M = U { M i : i ~ [}
Let each and
M
M
Then M
M.
D+(Mi)~M.
Then
D~(Mi)
and
l
M.
l
Then
Mi
are components of
D+(M) = U D + ( M i ) = U M i
D+(M) = M,
i.e.,
M
M.
= M
is stable.
is a compact connected set, and
is a component we have
D+(Mi) = M.
i
The theorem is proved.
Remark Theorem 2.6.8 is not true if
X
is not locally compact.
We shall now prove that in any dynamical system there do not exist compact stable sets which are weak attractors but not attractors.
Before doing so, we shall
prove a number of preliminary le-~as.
2.6 .i0
LF2~I4
For any given set M C Proof. and let n
§ + ~
Since
x~A~(M),
z~A+(x) such that
X, x e
we have
n
+ y,
without loss of generality that
~i)
implies
A+(x) (~ M ~ ~.
be arbitrary. xt
A
h+(x) C_ D+(M).
Choose any
Then there are sequences
and r
xT n
n
§ z
- t > 0 n
(since
yEA
n.
(x) (~ M,
{tn }' {Tn}' tn § + ~'
y,z &A+(x)).
for each
+
Setting
We may assume
138
Xtn E Yn' n = 1,2,... Yn § y 6M,
Tn
,
we notice that
tn > O,
-
z ~ D+(y) C D+(M) 9
yn(Tn - tn) = xT n § z,
and
Thus
xT n = Xtn(T n - tn) = yn(rn - tn).
A+(x) C D+(M)
as
z(A+(x)
Since
we have was arbitrary.
This proves
the lemma.
2.6.11
LEPI~A
Let
be compact.
M GX
open invariant set containing invariant set containing
Proof. same. A (M) ~
If M.
M
is a weak attractor,
If
M
then
is an attractor, then
A (M)
A(M)
is an
is an open
M.
We shall only prove the first statement, as the proof of the second is the By definition of weak attractor, S(M,e).
Now let
x~A
w
(M).
there is an
Then since
e > 0
such that
A+(x) 6~M # @,
there is a
T > 0
w
such that
x T E SCM,~).
S(xT,~)~S(M,e)CA
Since
(M).
S(M,e)
is open, there is a
6 > 0
such that
Consider now the inverse image of the open set
S(xt,~)
w
by means of the transition S(xT,~)(-T}
is open and contains
yTESfxT,6),
and, therefore,
A+(yT)/~ M ~ @ A (M)
T ~ .
because
is open.
A (M)
Since
~
x.
is continuous,
l~ote that for any
A+(y) N M
yT E S(M,~).
T
~ ~,
He~ca
as
9 the inverse ~mage y K S(xT,~)(-T),
A+(y) = A+(yT),
S(xT,~)
(-T)~Am(M).
and This shows that
is indeed always invariant and the L~-,,a is proved.
w
We are now ready to prove our promised theorem.
2.6.12
THEOREM Let
M
be a compact stable set.
If
M
is a weak attractor, then it is an
attractor and hence is an asymptotically stable set.
Proof.
Since
M
is stable, we have
D+(M) = M.
If
x6A
(M),
then we have by
139
Lemma 2.6.10
A+(x) C D + ( M )
A ( M ) ~ A (M) A (M)
= M.
This shows that
holds always9 we have
is a neighborhood of
M.
A (M) C A ( M )
A (M) = A(M).
Thus
M
As
M
and as
is a weak attractor
is an attractor.
The theorem is proved.
We shall next characterize the property of asymptotical stability of a compact set
2.6.13
M
in terms of
J+(x).
THEOHEM s
M
be compact and positively invariant.
stable, if and only if there is a and
J+(x) @ ~
Proof.
Let
M
be asymptotically stable.
J+(x) C D+(z).
Since But as
Now let for Hence
an attractor.
A+(x) ~ @, M = D+(M),
2.6.14
x ES(M 9
is asymptotically
x ~ S(M, 6)
implies
A+(x) # @ Now let
and further
xE M,
then
is stable, we have
and
A+(x) # ~
J+(x)~M.
There is then a
A+(x) C M9 we have
M
implies
J+(x) ~ M .
x ~ S(M, 6)9 J+(x) # ~,
is positively invariant, we have
stable.
Since
x ~ M 9 J+(x) # ~
Let, if possible9
Choose z ~ A+(x).
M
such that
This shows that for
x ES(M,n) \ M.
compact.
such that
M
J+(x)CM.
There is further an n > 0 A+(x) C M .
~ > 0
Then
we have
J+(x) C M.
Clearly
A+(x) C J+(x) C M .
D+(x) = y+(x) L2 J+(x). D~[x)~M.
Hence
and Now let
y 6J+(x), y ~ M .
D+(z) ~ M.
D+(z)~D+(M) = M9
M = D+(M).
By Lemma 2.3.12
also
J+(x) C M.
J+(x)
is
This shows that Since
D+(M) = M9
J+(x) C M i.e.,
M
M
is and
is
The theorem is proved.
Exercise If a compact set
M
is asymptotically stable9 then
J+(A(M))CM.
We shall close this section by proving a very interesting property of compact weak attractors in locally compact metric spaces.
2.6.15
THEOREM Let
X
be locally compact.
Let
K
be a compact weak attractor.
The~z
140
D+(M)
is a compact asymptotically stable set, with
D§
is the smallest asymptotically stable set containing
A(D+(M))
Moreover,
- A (M).
M.
For the proof we shall need the following lemma.
2.6.16
LE~ Let
> 0.
X
be locally compact.
Then there is
T > 0
~
Proof.
M
be a compact weak attractor, and let
such that
D+~) C S[H,=] where
Let
[0,T] - ~(S[M,=], [0,T]),
is the map defining the dynomical system.
Choose
e, 0 < e .< u,
For
x~H(M,e),
define
Set
T = sup{Tx:X~H(M,E)}.
is a sequence
{x }
such that
S[M,E]
is a compact subset of
Tx = inf{t > 0 : x t 6 S ( M , e ) } ;
in
We claim that
H(M,e)
since
T < + ~.
for which
T
n
§
x6%(M),
A 0J (M).
Tx
is defined.
If this is not the case, there
+ ~.
We may assume that
x n
x n + x ~H(M,e).
Let
have then
T,
9
0
such that
which contradicts
x~ T
x
§
{Xn}, {t n}
Then for all sufficiently and
0 < tn - ~n < T.
Then
y E D+(M) \
we
S[M,e].
Then
with n
x
n
§ xEM,
there is a
for
>. 0
such that
Tn, 0 < rn < tn
such that
Tn < t .< tn.
and
t
n
x t n
n
§ y.
By the first part of this proof
Xntn = Xn~n(tn - ~ n ) & S[M,~][0,T].
Therefore,
y6S[M,e][0,T],
The ipn,,a is proved.
Proof of Theorem 2.6.15 Notice that if Thus
D+(M)
D+(M) C
S[M,e]
is compact,
then
S[M,e][0,T]
being a closed subset of the compact set
above lemma) is compact. we have
large
xt~S[M,e]
since this set is closed.
T > 0.
Now let
n,
n
there are sequences
2.6.17
+ ~.
For sufficiently large
x n
Xn~n~H(M'e)'
S(M,E).
Further, as
S[M,e] [0,T] C A
[M).
e > 0 Thus
is compact for any S[M,e] [0,T]
is chosen such that A (M)
(by the
S[M,e] C A w ( M )
is an open invariant set containing
141
D+(M),
and is, therefore, a neighborhood of
x ~ A (M)
implies
[0
attractor.
Notice that
then there is a and since let
A+(x) # 4,
t > 0
Since
J+(x) C D+(w) C D+(M)
A+(x) ~ D+(M).
A (M) = A(D+(M)), such that
A+(x) - A+(xt),
x~D+(M).
and
D+(M).
x~A
xtEA
we have to
(M),
(M).
proved that such- that
D+(D+(M)) = D+(M),
D+(M)
MC
M C
D+(M *) = D+(M).
If
M*
Then
i.e.,
xCA(D+(M)),
D+(M)
w~A+(x) f]M.
D+(M)),
is stable, Then
for
D+(M)
D+(M)
is positively invariant.
is stable (2.6.6).
Finally, let
M*
M* = D+(M *) = D+(M).
smallest stable (also asymptotically stable) set containing
We have thus
by any compact set
D+(M) C D+(M *) C D+(D+(M)) = D+(M),
is stable, then
is an
Thus
is asymptotically stable. D+(M).
To show that
we can choose an
by Le~m~a 2.3.12.
D+(M)
(this being a neighborhood of
D+(x) = ~+(x) 12 J+(x) C D+(M) k] D+(M) = D+(M), This shows that
Therefore,
for if there is an
(M)
x~A
By Lemma 2.6.10
Thus
M.
and so D+(M)
is the
The theorem is
proved.
2.6.18
Exercises i)
X
Let
M
be a compact invariant set.
be locally compact. ii)
Then 2.6.19
M
Let
X
Then
A-(y) ~ M # ~
be locally compact, and
Let
M
for every M
be
a
weak attractor.
Let
y ~D+(M).
a compact invariant weak attractor.
is a negative weak attractor if and only if
D+(M) E A (M).
THEOREM Let
X
be a locally compact and locally connected metric space. Let M C X
compact asymptotically stable set.
Then
M
be a
has a finite n~nber of components, each of
which is asymptotically stable. 2.6.2g
Remark Theorem 2.6.19 is not true if the space does not have the properties listed
above.
Consider for the case of a dynamical system defined only on a compact sequence
of poin~tending to one point. The compact set is asymptotically stable and so are
142
its isolated components.
But the limit point (a component) is not asymptotically
stable and there are an infinite number of components.
2.6.21.
Notes and References The first systematic application of the notion of a prolongation to
attractors seems to have been done by Auslander, Bhatia, and Seibert.
Most
results in this study were shown to be valid for weak attractors by Bhatia who introduced this later notion.
Theorem
2.6.19
[3]
is essentially due to
Desbrow, who proves it for a connected, locally connected, locally compact metrizable space
X .
We observe that connectedness of the space is not required,
but local connectedness is essential. last remark.
A trivial counter example was given in the
143
2.7
Liapunov functions and asymptotic stability of compact sets. The basic feature of the stability theory ~ la Liapunov is that one seeks
to characterize the stability or instability properties of a given set of the phase space in terms of the existence of certain types of scalar functions (i.e., real valued functions) defined in suitable sets (usually neighborhoods of the given set) of the phase space.
Such functions are generally required to be monotone along
the trajectories of the given dynamical system.
Any such function which guarantees
a stability or instability property of a set is termed as a Liapunov function for that set.
In what follows, we shall present some very strong results.
By this we
mean theorems on necessary and sufficient conditions for asymptotic stability of compact sets based upon the existence of continuous functions of very special types.
real-valued
These functions will indeed characterize the
behavior of the dynamical system much better than the functions presented in Section 1.7. The simplest and perhaps the best known result on asymptotic stability is
2.7.1
THEOREM A compact set
MCX
a continuous scalar function
is asymptoticall H stable if and only if there exists v = ~(x)
defined in a neighborhood
N
of M
such
that i) ~ ( x ) = 0
if x ~ M
ii) ~ ( x t ) < ~(x) Remark.
for
and
~(x)>
xcM, t > 0
0 and
if x ~ M ; x[O,t]CN.
This theorem is similar to Theorem I0 in Auslander and Seibert [2].
minor changes being necessitated as we have not assumed invariant as is the case in [ 2].
M
or
N
The
to be positively
The corresponding minor changes in the proof of
sufficiency can be made and so we omit this part of the proof. different proof of necessity we give it below.
Since we have a
The difference lies in the fact,
that in [ 2 ] the authors prove the existence of a suitable function in a relatively
144
compact positively
invariant neighborhood of
method as in [ 2 1
yields a function with desired properties defined on the whole
region of attraction
A(M)
of
M.
Since
M,
A(M)
whereas we show that the same
need not be relatively compact we
need a different proof.
Proof of necessity in Theorem 2.7.1.
2.7.2 A(M)
be its region of attraction.
~(x) = sup(p(xt,M):
Indeed a
T
#(x) >0
x[T, + ~) C
p(xt,M)
is a continuous
properties: t >~ 0.
~(x) = 0
A(M)
it is defined for all continuous in x~ M
A(M).
with
because if
and
t,
~(x)
~(x) > 0
this is possible as
t ~R.
uniform attractor
41.5.28),
t >~ T.
y ~ S[x,e]
Thus for
then there is
M
for
x~M,
and
M
~(x)
is defined for any ~(x)
for
~(X)
as follows.
such that
For
Six,c] A(M)
such that
x~M,
>~ O} - sup{p(yt,M):t
on
M.
set
is a compact subset of
is open. S[x,r
>~ O}
= sup{p(xt,M):O ~< t .< T} - sup{p(yt,M):
x~A(M),
is
~(x)
we have
~(x) - ~(y) = sup{p(xt,M):t
has the
~(xt) .< ~(x)
implies continuity of
T > 0
~(x)
is stable and hence positively
is locally compact and there is a
This
We further claim that this
e, 0 < r < ~, X
p(x,M) = e,
is defined.
So that if
Indeed stability of
and choose
define
0 .< t .< T}.
is invariant.
xt
stable and let
Thus
we can prove the continuity of
p(x,M) = ~(> 0) A[M);
x E A(M),
function of x~M,
be asymptotically
x EA(M)
This is clear when we remember that
invariant and that
For
for
M
t >~ 0}.
S(M,~).
~(x) - sup{p(xt,M):
As
For each
is defined for each
with
Let
0 .< t .< T}.
Since ~)
M
is a
for all
145
So that
I~(x) - ~(Y) I -< sup{Ip(xt,M)
- p(yt,M)[: 0 .< t .< T}
sup{p(xt,yt): 0 ~ t ~ T}.
The continuity axiom implies that the right hand side of the above inequality tends to zero as in
A(M).
y § x,
for
T
is fixed for
y ~ Six,el.
~(x)
is therefore continuous
The above function indeed may not be strictly decreasing along parts of
trajectories in
A(M)
which are not in
M
and so may not satisfy (ii).
Such a
function can be obtained by setting
~(x) =
0 (xt) exp (-~) d~ 0
That
~(x)
is continuous and satisfies (i) in
satisfies (ii), ~(xt) ~ ~(x) have
holds.
x~M
and
M
have and
as
@ [x)
~ Cx) > 0, t > 0.
2.7.3
we get
implies that for
n § ~,
t > 0.
To rule out
~(x(t + T)) ~ ~(x~)
= 0,t,2t,... of
let
T ~ 0.
= ~x(nt)),
x~A(M),
is continuous.
r
To see that
$ r
~(x)
holds, because
observe that in this case we must
Thus, in particular, letting
n = 1,2,3,...
p[xt,M) + 0
as
This shows that
a contradiction.
is clear.
Then indeed
~(xt) = ~(x),
for all
~x)
A(M)
t § ~. @(x) = 0.
We have thus proved that
But asymptotic stability Thus
~(x(nt)) § 0
But as
x~M,
[xt) < ~(x)
as
we must for
x~M
The theorem is proved.
Remark Theorem 2.7.1 says nothing about the extent of the region of attraction of
Thus if a function N
of
M,
~(x)
as in Theorem 2.7.1 is known to exist in a neighborhood
we need not have either
to elucidate this point.
N C A(M)
or
A(M)~N.
We will give an example
(The observation is indeed well k n o w n
are woefully lacking in the literature).
but examples
In particular this means that the above
theorem cannot ~mmediately be stated as a theorem on global asymptotic stability:
M.
146
A compact set
M
asymptotically
stable and
2.7.4
is said to be globally asymptotically
stable if it is
A(M) E X.
EmoTnple Consider a dynamical system defined in the real euclidean plane by the
differential equations 2.7.5
~ = f(x,y),
~ = g(x,y)
,
where
g(x,y) = -y
for all
(x,y),
and
2.7.6 f(x,y) =
x
if
x2y 2 >~ i ;
f(x,y) = 2x3y 2 - x
if
x2y 2 < i.
These equations are integrable by elementary means and the phase portrait is as in
F~u~e
2.7.7
2.7.7 F i b r e
xZy2>l
Iy
!
t
~y~' ~~-L--~I~
,\ | / , ' / "
x~'Y~'I > ~y 0
and
Consider any function
Choose
x s
xs
~(xt) ffi e -t ~(x)
neighborhood
that
for
K CN,
.
P
T(x)
is
P
is a
with
x~P
is defined and T(x)
such that
being compact
A(M)
151
A-(x) ~ @ , A-(x) C P
, but
A-(x)~M
unstable).
A-(x)
is compact and invariant, so that if
y
~
Now
A-(x)
we have
we have
A+(y) O M
A+(x) ~ ~ , and = ~
A+(y) # ~
defined for each
x 6 A(M)\M
and A(M)
(otherwise
A+(Y)CA-(x)
and on the other hand
This contradicts
M
= ~
Thus .
y(x) O
are both invariant.
For
.
will be
Then on one hand
A+(y)CM
8P
Note that
M
as
# ~ , and
A(M)XkM
y
~ A(M)
T(x)
is
is invariant, as
x E A(M)\M
and
T E R
observe
now that
T(xt)
-
~(x)
-
t
.
This follows from the fact that any trajectory ~P
at exactly one point.
Thus
y(x)
in
xt(~(xt)) = x~(x)
A(M) ~ M
i.e.
C~
'
intersects
by the homomorphism '
axiom
2.7.15
As
y(x)
x(t + T(xt)) -- x(T(x))
can neither be periodic nor a rest point, we have
t + r(xt) = ~(x) . function of T(x)
and
T(xt) + + ~
is continuous on
the point hood
t
This shows further that
N
Y
A(M)\M
y ~ x(~(x) + E) of
y such
neighborhood
of
x
as .
~
that and note
t § ~ ~ .
For any
I(P )
N CP Y
a
that
.
for
Then each
neighborhood
N
= N (-T(x) + E)
, and ~ > 0 ,
z
N + = N (-~(x) -~) y ~ ~
Thus there is a Then
We now claim that
There is therefore a neighbor-
z E x(T(x) - ~) E ( A ( M ) ~ P ) of
is a continuous
x E A(M)~M
Again the point
Nz
T(xt)
N+ ,
~(~)
is a
~ T(x) + c
, the last set being open.
such that
is a neighborhood of
x
Nz~(A(M)XkP
and note that for
Z
each
~ ~ N-
neighborhood
we have N
= N+NN
~(~) ~ -
of
T(x) - ~ . x , we have
Thus if
~) .
~
is in the
152
T(x)
This proves
continuity
T(x) + - ~
as
- ~ .< T(~)
of
x + M ,
T(X)
there w i l l be a
T
xn § x s M
and
-T =< T(x n ) =< 0 .
it contains
a convergent
T(x n) § T,
where
x n ~(x n) § xT Xn T(Xn)
.
~ ~Pa
M
is invariant
(x)
=
o
=
e "r
The above observations clearly positive
for
-- e ~ (xt)
Lastly
to see that this
2.7.17
in A ( M ) ~ x M
{X(Xn)}
then
, such that
is a bounded
We may therefore
assume
sequence that
axiom
xT ~ M , on the other hand
Therefore
xT ~ 3P
the function
r
.
on
But
A(M)
for
x ~ M,
for
x ~A(M) ~
~
N
M = ~ ,
as follows
and M.
is continuous
on
A(M).
It is
and
= e 9 (x)-t
~ (x)
invariant.
= r (x) e -t
is u n i f o r m l y
] : n = 1,2,3,
9 (x) > n , so that
{x n}
(x)
x ~M,
(xt)
and p o s i t i v e l y
We now show that
show that this function
2.7.16
A(M) = U { P a [ - n , 0
.
Then by the continuity
We now define
r
e
If this w e r e not true,
Since
subsequence.
w h i c h is compact.
a contradiction.
.
and a sequence
-T =< T =< 0 . As
+
in A(M) \ M
x ~ A(M)~xM
> 0
T(x)
.
e n . This proves
that
, then
the theorem completely.
THEOREM If
MC
X
is any compact asymptotically stable set, then there exists a
continuous uniformly unbounded function
i) ii)
~(x) = o ~(xt)
for x CM,
< ~(x)
for
and
x ~/ M
r
and
~(x) > 0
t > O.
on
A(M)
for x ~ M ,
such that
153
Proof.
Consider any function
~(x)
defined in a neighborhood
satisfying conditions of Theorem 2.7.1. P .
For each
x s A(M)~
and
T(x) + 0
as
x + P ,
(x)
-
r
This
~x)
P
define x~Pa.
r
= a e T(X)
Choose
T(x)
as before.
Let
M
and
as before and consider
This
T(x)
is continuous
Now define
for
x s P
, and
for
x s A(M)~P
.
has the desired properties as may easily be verified.
be compact and let there exist a continuous uniformly
M~X
unbounded function
~(x)
i) ~ ( x ) = 0 ii) ~ ( x t ) < Then
M
for ~(x)
defined on an open neighborhood x~M,
for
and x/M,
~(x) > 0 for
t > 0
i8 asymptotically stable and
condition guaranteeing the invariance of
Proof: ~(x)
N =
N
and
of
M
sueh that
x/M,
xE0,t]CN.
If, in addition, any
N C_ ACM).
holds, then
is uniformly unbounded on
N = A(M).
U{K
: n = 1,2,
N , then for any a > 0 , the set
Then by Theorem 2.7.9
... } ,
and since each
n
N CA(M)
N
The proof follows from the observation that
Ka = {x : ~(x) =< a} is compact. Now
of
THEOREM
2.7.18
if
e, ~ > 0
N
Lastly if
is a neighborhood of
K
,
CA(M)
we have
n
N M.
is invariant we must have
N = A(M)
as
N
The remaining details of the proof will he
the same as in any proof of sufficiency of Theorem 2.7.1. leave to the reader.
K CA(M)
These we
154
For global asymptotic stability we can state the following two theorems as corollaries of the above results.
THEOREM
2.7.19
A compact invariant set exists a continuous function i) # ( x ) = 0
for
MC
~(x)
x~M,
i8 globally asymptotically stable if there
X
defined on
~(x)> 0
x
for
such that
x/M,
ii) ~(xt) = e -t ~(x) (Note that any such
will be necessarily uniformly unbounded on
~(x)
Proof:
The sufficiency follows from Theorem 2.7.18 as
of
The necessity follows from Theorem 2.7.14.
M.
X
x).
is an invariant neighborhood
THEOREM
2.7.20
A compact set
MC X
is globally asymptotically stable if and only if
there exists a continuous uniformly unbounded function i) ~(x) = 0
for
ii) ~(xt) < ~(x)
x~M,
for
~(x) > 0
x~M
and
for
~(x)
defined on
x
such that
x~M,
t > 0.
Proof:
Sufficiency follows from Theorem 2.7.18, the necessity from Theorem 2.7.17.
2.7.21
Remark In dynamical systems defined in locally compact metric spaces, one
may define ultimate boundedness of the dynamical system by the property that there is a compact set A+(x) C K
for each
global attractor if
K CX
in
x~ X X .
KCX
with
A+(x) ~ @ , and
, i.e. whenever there exists a compact It is shown in Theorem 2.6.15 that
is a compact weak attractor, then
D+(K) (the first positive
155
prolongation of
K) is a compact positively invariant set which is asymptotically
stable and has the same region of attraction as show now that the largest invariant set in with the same region of attraction as of
K.
D+(K) K.
Following Ura [ 7], one can is compact and asymptotically
stable
These observations will allow one to
write theorems on ultimate boundedness which are similar to those on global asymptotic stability.
2.7.22
We leave these to the reader.
Remark If
in Bhatia
M~X [i]
is a compact asymptotically
stable set, then following the methods
one can obtain a Liapunov #unction
~(x)
defined in
A(M)
with
the following properties i) ii)
~(x) ffi 0
for
$(xt) ~ e -t ~(x)
x~M,
~(x) > 0
for all
for
xs A(M)
xCM, and
t > 0.
This function, however, need not be uniformly unbounded on
A(M).
To obtain a function
which is uniformly unbounded and has the above two properties, we may use the above function in the construction of
~(x)
of the proof of Theorem 2.7.17.
the following stronger result for a compact (not necessarily invariant)
2.7.23
set
M.
THEOREM If
A(M),
We thus have
is a compact asymptotically stable set with the region of attraction
M CX
then there exists a continuous uniformly unbounded function
(x)
on
A(M)
having the following properties i#
= o
for
and
ii) ~(xt) ~ e -t r 2.7.24
for
> 0 for
xEA(M)
and
x M,
t ~ 0.
Remark In Theorem 2.7.18, 2.7.19 and 2.7.20 the proof of sufficiency can be completed
without the explicit assumption that remaining conditions on
r
r
> 0
for
x~M,
for this follows from the
156
2.7.25
Notes and References This section contains results of Bhatia [6].
Some remarks are in order.
Earlier results in this direction, for example those of Zubov [6], Auslander and Seibert [2], and Bhatia [i], used essentially the same methods as used for the welldeveloped theory in the case of ordinary differential systems.
For results on
ordinary differential equations see, for example, A. M. Liapunov, I. A. Malkin, Barbashin, Krasovskii, Kurzweil, Vrkoch, K. P. Persidskii, S. K. Persidskli, Zubov, Massera, Antosiewicz~ Yoshizawa, W. Hahn.
The basic feature of the results in this
section isthat Liapunov functions are shown to exist on the whole region of attraction as against on a sufficiently small neighborhood in earlier results.
The functions, in
general, have sufficient properties to allow the derivation of theorems on global asymptotic stability and ultimate houndedness as corollaries.
Indeed Auslander, Seibert
established formally the long suspected duality between stability and boundedness in locally compact separable metric spaces.
157
Topological properties of
2.8
and
A (M),A(M)
P (M).
In this section we shall present some additional properties of attractors, region of attractions and the level lines of the corresponding Liapunov functions.
We shall present results for the case of weak attraction and asymptotic
stability. "The latter results are valid with few obvious changes also for the case of complete instability, i.e., in all cases of strong stability properties.
The
results that we shall present are extensions and improvements of the ones presented in Section 1.9 and they are based upon the following two lemmas, the first of which is an obvious restatement of the results proved in Theorems 2.7.9 and 2.7.17.
2.8.1
N
of
LEk~A
M
Let
X
be a locally compact metric space.
Let
M CX
Let
v = r
be a compact asymptotically stable set. be any continuous function
and having the properties
i) ~(x) = 0 ii) r
for
x~M,#(x)
for
< ~(x)
a compact subset of
2.8.2
N.
Let
for
> 0
xCM, t > 0 ,
(such functions can always be defined on %
and
0 < a < re(c)
m(~) = mi~{r
P
= K
K
c~
(~ S[M,e],
= {xEN:r
x[0,t] C N.
where
:xs H(M,~) }.
where
2.8.4
x~M;
A(M)). Let
Then the set
2.8.3
defined on some neighborhood
.< a},
~ > 0
be such that
S[M,~]
i8
158
is a compact positively invariant set, with
2.8.5
P
LE~IA For each sufficiently small
retract of
%
the set
This is so because we can define a map
x~e
and
,
h(x) = xT(x)
Theorem 2.7.14. the fact that A(M)
defined in 2.8.3 is a
P
A(M).
Proof.
of
C A(M).
into
x~P
,
where
T(x)
Because of the continuity of ~(x) = 0
P
A(M) ,
if
h:A(M) § P
for
x ESP
,
T(x)
P .
if
is defined as in the proof of and of the phase map ~,
it follows that
which is an identity on
by h(x) = x
h
is a continuous map of
Thus by definition
and hence also a retract of every subset of
and
A(M)
P
is a retract
which contains
P ~"
We are now in the position to prove the following important result which is a generalization of Theorem 1.9.6.
2.8.6
THEOREM Let
(E ,R,~). M
M C E
be a compact set which is a weak attractor for a dyneonical system
Let the region of attraction
contains a rest point.
weak attractor),then
Proof.
Lemma
Let
D+(M)
~(x)
property, as
B
A(M) = E
(i.e.,
M
E.
Then
is a global
be any function for the asymptotically stable set
we can choose a compact set Then
be homeomorphic to
is an asymptotically stable compact set with
P
positively invariant, and is a retract of
E.
M
contains a rest point.
2.8.1, and consider a set
unit ball in
of
In particular, when
By Theorem 2.6.15
A(D+(M)) = A(M). as in
M
A(M9
B,
P
P C
for A(M).
B CA(M),
is a retract of
~(x). As
Then
A(M)
where
B.
Thus
invariant, the transition
~
T
maps
P
into
P
is compact,
is homeomorphic to
B
E
is homeomorphic to the
P
has,by the Brouwer Fixed-Point Theorem.
P
D+(M)
has the fixed point Since
for each
P
9 ~ 0.
is positively Thus for each
159
fixed > 0
9 > 0,
~
T
has a fixed point in
there is an
xT~_ P~
such that
is closed and has a period
T,
~(x
moreover
We have thus shown that,corresponding is a sequence of closed orbits
P ,
i.e., corresponding ) = ~(x~,~) = x~.
y(x ) C P
,
to any sequence
(Yn }' Yn = y ( x
),
to any
Thus the orbit
+ y(x ) = y ( x ) ~ P ~ .
because
{~n }, ~n > 0,
with
Yn
y(x )
Tn § 0,
having a period
there
Tn.
n
This sequence being in However,
M CD+(M)C
for each
x~A(M).
point.
Hence
P , P
contains a rest point
P CA(M). Thus
x E M.
As
M
A+(x ~) ~ M
2.8.7
(say)
(lemma 1.9.5).
is a weak attractor we have
~ ~.
But
A+(x ~) = {x },
as
A+(x) ~ M ~ x
is a rest
The t h e o r e m i s p r o v e d .
For the following corollaries defined on
x
the dynamical system is assumed to be
E .
COROLLARY If the dyneonical system is ultimately bounded, then it contains a rest point.
This is so, because ultimate boundedness of a compact globally asymptotically
is equivslent
to the existence
stable set (Remark 2.7.21) which by the above
theorem contains a rest point.
2.8.8
COROLLARY The region of attraction of a compact minimal weak attractor
homeomorphic to
E,
unless
Note, however, or attracting,
then
lytic example
1.5.32
M
(ii)
cannot be
is a rest point.
that if a rest point
A(p)
M
p ~ E
need not be homeomorphic shows.
However,
is weakly attracting, to
E ,
as the ana-
if a rest point
totically stable then its region of attraction is homeomorphic
P
is asympto
we shall prove next; its proof depends on the following topological
E
This theorem
160 2.8.9
THEOREM Let
n
Un C U n +
I, n = i=2,...
2.8.10
THEOREM
Then
If a rest point to
p~ E
~]{Un:n = 1,2,... }
is an open
is asymptotically stable, then
i.e., n-cell.
is homeomorphic
A(p)
E.
Proof.
Since
A(p)
is a neighborhood of
closed ball
SiP,el C A ~ ) .
of
E,
E
onto
n-cell.
For each
the image
S~,~)t
S[P,~]t I
tI
Further,
for
e > 0
the transition
~
S~,e)
S[p,e]tl~A(p),
t ~ T.
S [ p , ~ ] t l C S ( p , e ) t 2.
we can choose a sequence sequence of open n-cells. n-cell.
homeomorphic to
2.8.11
Hence
{tn}, tn + - ~, By Theorem 2.8.9
But this last union is
A~),
t
is an open
A(p) T > 0
is open.
p
homeomorphic to
S [ p , ~ ] t l C S(p,e)(t I - T) ~ S [ p , e ] ( t I - T).
such that
The above analysis shows that
{S~,e)tn}
is a monotone
~{S(p,e)tn; n = 1,2,... }
so that
A(p)
is an open
is an open n-cell and hence
E.
is an asymptotically stable rest point, then E \ {0}, where
0
i8 the origin in
A(p) \
{p}
i8
E.
We can now prove the following result.
2.8.12
THEOREM Let
M I'"E
Since
such that
COROLLARY If
such that
In particular,
Setting
we get
~
t2, t 2 < t I
and
there exists a
I.
,
being a homeomorphism
by
there exists a
S[P,e](t I + T) C S ( P , e ) t l C S [ p , e ] t t2 = t I - T
t
such that the
S(p,e)t I being a subset of the compact set
is uniformly attracting (Theorem 1.5.27)
S[p,e] (tI + t) C S ( p , e ) t I
there is an
of the open ball
This is so because
is itself compact.
p,
t~R,
We claim now that for any given
S ( p , e ) t l C S ( p , e ) t 2.
p
E,
be a monotone sequence of open n-cells in
{U }
be a compact globally asymptotically stable set.
Then
161
E\M
= C(M)
Proof.
is homeomorphic to E \ { 0 } .
By Theorem 2.8.6,
of generality that
M
M
contains a rest point.
contains the origin
now the homeemorphism
0
and
We may assume without loss 0
h:E X {0} + E ~ {0} defined by
is a rest point. h(x) =
x
Consider
where
{{xll
l lxll2 ' is the euclidean norm of system on E~M
E,
0
is mapped onto
attraction of E~{0}.
with
0.
x.
h
maps the given dynamical system into a dynamical
becoming a negatively asymptotically A(0) \ {0},
where
A[0)
By the Corollary 2.8.11,
stable rest point, and
is now the region of negative
A(0) ~ { 0 }
is homeomorphic
to
Kence the result follows.
We shall now present one example of application of Theorem 2.8.12.
2.8.13
Example Consider
a
flow ~
Theorem 2.8.12 shows that A ({y}) = E ~
2.8.14
{x}
in y
,
E
with only two rest points
cannot be asymptotically since
C({x} U
{y})
x
and
y, x # y.
stable with
is not homeomorphic
to
E \ {0}.
Notes and references Most of the results presented in this section are contained in the paper by
N. P. Bhatia and G. P. Szeg~[l]. An analytic example showing that if not be homeomorphic
to
E
and P. Seibert at pg. 58) 9
is
1.5.32 (ii)
p ~E
is attracting,
(J. Auslander,
then
N. P. Bhatia
A(p)
need
162
Theorem 2.8.9 is due to M. Brown.
The results contained in this section
and in particular Theorem 2.8.10 and the natural conjecture which generalizes theorem to sets useful.
MCE
such that
E \ M
is homeomorphic
to
E\
{0}
this
are rather
In particular they may have a strong influence on the solution of one of the
most important still open problems in the stability theory of dynamical systems, viz. the problem of local properties and the related theory of separatrices. A separatrix, "a trajectory
according to S. Lefschetz
(i, pg. 223) is, in
(not a critical point) behaving topologically
E 2,
abnormally in comparison
with neighboring paths". A theory of separatrices
in
E2
was formally suggested by Markus
who gives a definition of separatrix and concludes that the union set) of all separatrices
of a differential system in
Each component of the set
C(o)
E2
or homeomorphic
(separating
is closed.
is called by Markus a canonical region.
Markus proves that in each canonical region the flow is "parallel" parallelizable
o
[5]
i.e., either
to a family of concentric cycles.
Clearly since the flow is parallel in each canonical region it admits there a transversal section.
The results presented in this section are helpful in
generalizing some of these results to flows in (after a suitable generalization canonical regions homeomorphic of the flow.
E.
For instance,
of the concept of separatrix)
one can show
that the number of
to balls cannot exceed the number of equilibrium points
If, in addition, one defines the separating set in such a way that in
the corresponding
canonical regions the flow has only strong stability properties
the characterization
of the separating set above (which may have a very complicated
structure) would be enough for the complete global description of the stability properties
then
of the flow.
163
2,9
Minimal Sets and Recurrent Motions. A rest point and a periodic trajectory are examples of com-
pact minimal sets (for definition see 2.2.4). A rest point and a periodic motion are also Poisson stable. Example 2.5,10 indicates that the closure of a Poisson stable trajectory need not be a minimal set (in the example the closure of every Poisson stable trajectory except the rest point the whole torus, which is not minimal as it contains a rest point). G.D. Birkhoff discovered an intrinsic property of motions in a compact minimal set, which is usually called the property of recurrence. The aim of this section is to study this concept of recurrence. We start with some characteristic properties of minimal sets.
THEOREM
2.9.1
Every compact invariant set
K~X
contains a minimal set.
Proof.
Consider the set
set
is partially ordered by the inclusion relation
G
Since
K is compact
it
G
of all closed invariant subsets of
2.9.2
M~G.
Then
This
C.
has the finite intersection property [Dugunji I, pg. 223].
Thus every chain has an upper bound. element
K.
M
Hence by Zorn's lemma there is a maximal
is maximal and the theorem is proved.
OOROLLARZ For any x~
X, if the motion ~
is positivel~ (negatively) x
Lagrange stable, then A+(x)
(A-(x)) contains a minimal set.
An elementary characterization of a minimal set is given by
2.9.3
THEOREM A set
MCx
is minimal if and only if for each
x~H
one
164
has
~(x) = M .
Proof. x E M
Let
M
be minimal, and suppose if possible that there is an
such that
indeed
y(x) C M .
set of
M,
y(x) Thus
# M.
As
y(x)
M
is closed and invariant we have
is closed and invariant and a proper sub-
a contradiction. Hence for each
Conversely, assume that for each
x E M,
x K M
y(x) = M.
we have
y(x) = M.
Let if possible
M
be not minimal. Then there is a non-empty closed and invariant subset
N
of
M,
N # M.
Then for any
x ~ N,
T(x) ~ N
# M, a contradiction.
The theorem is proved. We now introduce the notion of recurrence ~ la Birkhoff. 2.9.4
DEFINITION
(recurrence)
For any for each ~ > 0
x ~ X,
the notion
there exists a
~
is said to be recurrent if
x
T -- T(e) > O, such that
~(x) C S ( x [ t - T , t+ T], r
for all 2 .9.5
t E R. Remark. It is clear that if a motion
~
is recurrent then every x
motion
~ Y
with
of the trajectory 2.9.6
y E y(x) y(x)
is also recurrent. Thus we shall also speak
being recurrent.
Exercise. Show that every recurrent motion is Poisson stable. That the concept of recurrence is basic in the theory of
compact minimal sets is seen from the following theorem of Birkhoff
[2].
165
THEOREM
~9.7
Every trajectory in a compact minimal set is recurrent~ Proof.
Let
M
be a compact minimal set. Suppose that there is an
such that the motion sequences
~
is not recurrent. Then there is an
{T }, {tn} , {~n }, n
with
T
n
> 0, T
n
+ + ~ ,
x E M
e > 0
and
and
XT n # S(x[tn-Tn, tn+Tn] , e) , n=l,2,...
This shows that
p(x~ n, X(tn+t)) ~ e sequences
{xt }, {xT } n
whenever
Itl ~ Tn, n=l,2, . . . .
are contained in the compact set
M
and may
n
without loss of generality be assumed to be convergent. So let and T > 0
x~
n
+ z.
Then
y, z
E y(x) = M.
Consider now the motion
be arbitrary but fixed. Then there is a
p(yt, wt) < ~
whenever
We can now find an integer
n
Itl < T
such that
p(z, XTn) < ~
.
Then we have
E
p(yt, X(tn+t)) < ~
Moreover,
from the choice of
for Itl < T.
x~
we have n
and
6 > 0
such that
p(y,m) < 6.
Tn > T, p(y, xt n) < 6
e
and
The
xt n § y # . y
Let
166
p(x (tn+t), xT ) > ~ 9
The above inequalities
n
=
whenever
It I < T.
for all
t s R.
As
Thus
It I < T < T . =
n
show that
p(yt, z) ~ p(X(tn+t),
-
for
~(XTn,
T
XTn) - p(yt, X(tn+t))
z) ffi > c
.
.
3
.
was arbitrary,
z E Y(y),
i.e.
.
.
3
.
3
'
we conclude that
z ~ M
as
M = Y(y).
E
p(yt, z) >
This contra-
diction proves the theorem.
THEOREM
2-9.8
If a trajectory
~x)
is compact then
u
is also minimal.
~x)
Proof.
Set
y(x) = M.
Let if possible
a non-empty
compact invariant
(otherwise
y{x) C N
~x
is recurrent and
is recurrent,
2.9.9
y ~ A+(x)
there is a
or
y ~ N.
T > 0
t+T], 5)
Since
y E A-(x).
{t n}, t n § + ~ ,
N
such that
N # M.
Now let
for a l l
y C h+(x).
Xtn § y"
M,
Then there exists
Clearly
x ~ N
p(x,N) = e (> 0). As
such that
y ~ M = y(x),
Let
be not minimal.
of
which is impossible).
y(x) C S ( x [ t - T ,
Now choose any
subset
M
t E R.
and
y r y(x),
we have
Then there is a sequence
By the axioms of the dynamical
system,
167
there is a Itl ~ T
~ > 0
such that
p(yt, zt) < ~
This shows that there is an
0(yt, X(tn+t)) < ~ for
n
whenever
0(y,z)
< 6
and
with
Itl ffi< T.
From this it follows that
0(x, X(tn+t))
__> 0(x,yt)
>
~
-This however 2.9.10
2.9.9
contradicts
-
-0(yt,
-- = - 3 3
for
X(tn+t))
]It < T. --
The theorem is proved.
COROLLARY. If the space
recurrent trajectory
X 7(x)
is complete, then the closure
of
y(x),
of any
is a compact minimal set.
The proof follows from the observation imply compactness
y(x)
that the conditions
so that the result follows from theorem 2 .9.8.
The details are left to the reader as an exercise. Another way of defining a Lagrange provided via the concept of a relatively
stable recurrent motion is
dense set of numbers(0.2-14)~
168
THEOREM
2.9. 11
A Lagrange stable motion
is recurrent if and only if for
x
each
~ > 0
the set
K
= {t: 0(x, xt) < e ]
is relatively dense. Proof.
Let for each
e > 0
the set
K
be relatively dense. For any
e > 0
E
there is by definition a
K e ~(t-T,
As
7(x)
T
= T > 0
t+T) # ~
such that
for all
t s R.
is compact, to show that the motion
~
is recurrent we need x
show only that
y(x)
is minimal. Let
is a minimal subset y(x) C M any
of
which will imply
y E M.
0(y,z)
M
< 6
clude that
7(x),
{t n}, tn § + ~ p(yt, X(tn+t))
6 > 0
Itl ~ T = T .
y ~ A+(x) and < ~
or
be not minimal. Clearly
It] < T = T . E
(otherwise
0(yt, zt) < E and
Let
x ~ M
p(x, M) = 3e(> 0). Choose
such that y s
Then there
y ~ A+(x).
But then for
whenever
y~y(x),
we con-
Then there is a sequence
Thus for all sufficiently large
=
we have
As
Set
Y E A-(x).
Xtn § y" for
M + y(x).
y(x) = M).
Then there is a and
y(x)
n
we have
t ( [tn-T, tn+T]
169
p(x, xt) > O(x, M) - 0(xt, M)
>
3e
-
~
=
2~.
This shows that
Ke ~
[tn-Te' tn+Te] = r
which is a contradiction. the motion
~
This shows that
y(x)
is minimal and hence
is recurrent. The converse holds trivially. The theorem
X
is proved.
2.9.12
THEOREM There exist non-compact minimal sets which contain more than one
trajectory. Proof.
Consider a dynamical system defined in a euclidean 3-space, in
which the torus
T
of example
2.5.10
is embedded with the rest point
on the torus coinciding with the origin of the euclidean space. consider the transformation
X
Y = l--~-
'
x # 0 ,
given euclidean space into a euclidean space.
We now
which transforms the
The set
T \{0}
is now
transformed into a closed minimal set which is not compact, since it is not bounded, as is evident from the considerations in Example
2.5.10.
Notice that in the example of the unbounded minimal set given above the motions are not recurrent, showing that Theorem true if the minimal set is not compact.
2.9.7
is not
170
2.9.13
Notes and References G. D. Blrkhoff defined the notions of a compact minimal set and of
recurrent motions and showed the deep connection between them. here is adapted from Nemytskii and Stepanov's book. minimal sets is very scanty.
The presentation
The literature on non-compact
The example in Theorem 2.9.19 is included to give
an idea that these sets do not have many known interesting properties.
171
Stability of a Motion and Almost Periodic Motions.
2.10
In this section we shall assume throughout that the metric space
X
is complete.
The concept of almost periodicity is intermediate between that of periodicity and recurrence, and the concept of stability of motion plays a central role in its study.
We therefore first intro-
duce the concept of stability of a motion.
DEFINITION
2.10.1
A motion in a subset that
N
y ~N
is said to be positively
x
of
X ,
N S(x, 6)
Any motion
if for any
implies
~ x
fied with
by
replaced
of
X ,
t 6R-
If in the above definition
X
"in the subset
N
of
is positively stable if given
y ~ S(x, 6)
2.10.2
implies
for
t s
~ > 0
such
+ .
is called negatively stable, or stable in N
the qualifier
there is a
o(xt, yt) < E
both directions in a subset t ( R+
E > 0 ,
CLiapunov) atable
N
or
for
t ( R
respectively.
is a neighborhood of
X "
~ > 0
p(xt, yt) < c
if the above condition is satis-
will be deleted. there is a
6 > 0
x ,
then
Thus a motion such that
t ( R+ .
Exercise Show that a motion
~
is positively stable if and only if every x
motion
2.10.3
~xt '
t ( R ,
is positively stable.
DEFINITION If
A C B C X
then the motions through
A
(i.e. motions
X
172
x ~ A)
with
will be called uniformly positively 8table~ uniformly
negatively stable, or uniformly stable in both directions in given any
E > 0,
t ~ R- , or
t s R+ .
and
there is a t ~ R
6 > 0
such that
B ,
p(xt. yt) < r
respectively whenever
if for
x s A . y ~ B .
0(x, y) < ~.
Exercise.
2.10.4
Show that if
A
is a compact subset of
through
A
are uniformly positively stable in
through
A
is positively stable in
B ,
B ,
then the motions
whenever each motion
B .
We now introduce the concept of almost periodicity.
DEFINITION
2.10.5
A motion
x
i8 said to be almost periodic if for every
there exists a relatively dense subset of numbers
{z } n
E > 0
called displace,~nts
such that
o(xt, for all
t
~ R
and each
T
n
x ( t + Zn))
0 .
Proof.
> 0
Indeed
for any
there is a set of d i s p l a c e m e n t s
{T } n
such that
p(xt,
x(t + Tn) ) < c
then there is a inequality
T ~ R
for
such that
t ~ R,
y = xT,
together with the h o m o m o r p h i s m
and each Tn.
or
If
x = y(-T).
y & y(x),
The above
axiom then gives
p(y(t - T), y(t - T + T )) < e n
for
t s R. Setting
t - T = s,
we see that
p(ys, fixed.
y(s + ~n )) < E
This proves
s ~ R
and each
rn '
as
r
is
the lemma.
THEOREM
2. i0.7
Let the motion Then
for
(i)
# x
evez,~3 m o t i o n
#
strict i n e q u a l i t y
0 ,
y(x)
with the
but with the
the motion
~
both directions in
be compact.
y(x)
#
is stable x
.
Proof. i)
For any
y ~ y(x)
xn § y .
p(yt,
Given
,
for any
such that: ,
and
T
m
fixed but arbitrary
y(t + Tm)) ~ e
completes li)
2.10.6
{T n}
x n ~ {x n}
m ~ {T n}
there is a sequence
By Lemma
displacements t ~ R ,
,
the proof
e > 0 ,
let
for all of
(i)
{T } n
{x } C y ( x ) n
e > 0
there is a set of
P(Xnt , Xn(t + Tm))
~ {~ } n
Now keeping
and proceeding t # R
and
such that
< e
for all
t ~ R , and
to the limit we get
Tm ~ {~n }
This
. be a set of displacements
corresponding
in
174 E
to
~
for the almost periodic motion
~
,
and let
T > 0
be such that
x
{T } N [t - T, t + T] # ~ n
theorem each
for
p(yt, y(t + ~n )) ~
T
.
~
t ~ R
Then by part
for all
y ~ y(x) , E
By the continuity axiom, for
= > O
and
(i)
of the
t ~ R ,
and
T > 0
as above
n
there is a
~ > 0
It I
for all compact.
such that
& T ,
whenever
Now for any
p(y,
z) < 6
implies
{y,z} C y(x)
y ~ y(x)
and
as t h i s
O(x, y) < ~ ,
p(yt,
zt)
last
set
0
there is a
whenever
{x,y} C
(Theorem 2.9.12)
6 >
y(x)
,
0
such that
and
0(x,y)
< 6 .
there is a r e l a t i v e l y
such that
of displacements
n
O(x, x3 n) < 6
for each
~n
F r o m the above two results we conclude
o(xt,
and e a c h
x(t + 3n)) < ~
The theorem
n
A stronger
for
t ~ R
is proved.
result
is the following:
THEOREM.
2.10.11
If a motion
~
i8 recurrent and positively stable in
x
~(x)
,
then i t i s almost periodic. Proof.
(a)
By positive
stability
of
in
y(x)
,
we have given
~ > 0
X
there
is a
for all
6 > 0
t ~ R+
such that (b)
0(x, x3) < 6
By recurrence
implies ,
of
0(xt,
x(t + 3)) < ! 2
there is a relatively
X
dense set
{3n }
(c) such t h a t Tn
6 p(x, x~ n) < ~
By t h e c o n t i n u i t y
0 ( x , y) < c
be a r b i t r a r y
such that
such that
implies
but fixed.
0 ( x , xT) < min
axi om ,
for each
f o r any
5) .
Then by
n
there is a
o > 0
n
6 p(X3n, y~n ) < ~ .
Then by r e c u r r e n c e (c,
T
T
of (b)
Now let ,
t ~ R
there is a
and 3 < t
X
0(x 3 n, X(T + Tn)) < ~
,
so that
p ( x , x(T + "~n)) < o ( x , x~ n) + p(x3 n, x(T + "~n)) < ~" + Hence
by
(a) ,
since
t - 3 > 0 ,
we get
0(x(t - 3) ,
6 ~" --
~ .
x(t + 3n)) < ~ 2 "
176 Further, p(x,
x~)
p(xt,
x(t
< min
(G,
-
T))
-- p ( x T ( t
6) < 6,
and
-
T)
t -
,
x(t
9 > 0
.
-
T))
E
< ~
by
(a)
~
T h u s we g e t
E
p(xt, x(t + Tn )) =< p(xt, x(t - T)) + p(x(t - T), x(t + Tn)) < ~ + ~
This shows that
~
as
E
= e
9
is almost periodic and the theorem is proved. X
THEOREM
2.10.12
If the motions in
are uniformly positively stable in
y(x)
y(x)
and are negatively Lagrange stablej then they are almost periodic.
Proof.
It is sufficient
to prove that the motion
is recurrent,
X
as the
rest follows from the last theorem. By negative Lagrange stability of
~
9
A (X)
iS compact,
variant,
and indeed
A (x)C
y(x).
there is a compact minimal set
X 9
we conclude that
Since
A-(x)
M, M C A - ( x ).
is also in-
If
~
is not X
recurrent,
then
M#
y(x),
and in particular
x~M.
We will show, that every motion
~ , y ~ A-(x), y
To this end, given ~ > O, there
is a
stability
in
y(x)
P(Xn, x m) < 6
of motions
imply
in
{t }
y(x))
in
R-, t
n
is then an integer p(Xtn , y) < 6, Keeping/in and letting
N
such that for
m § ~, we get
+ - ~,
Now for
such that
p(Xtn(t),
t ~ R
e = ~
xt
and
y ~ A-(x), -~ y.
There
and
and
imply Xtm(t))
n > N for
t -- - t
p(Xtn, xt m) < 6 < e
for
t __> O.
arbitrary but fixed t => 0
whenever
we see that n'
P(y (-tn), x) < ~.
Since
that
The theorem is proved.
p(x, M) = e.
Y(x)
n
p(yt, Xtn(t)) =< e
Choosing now
stable in
{Xn, x m} ~ y ( x )
t __> 0.
n __> N, m __> N
and consequently
this last inequality,
p(y, xt n) < 6.
is positively
n
such that
p(x,M) = ~ > 0.
6 = 6(e) > 0 (by uniform positive
P(Xnt, Xmt) < ~
there is a sequence
Let
y(-t n) ~ M,
this contradicts
the assumption
and
177
The remaining portion of this section will be devoted to finding conditions under which a limit set
A+(x)
is compact and minimal
andgfurther~when such a set consists of almost periodic notions only. For this
the following definition is useful.
DEFINITION
2.10.13
A semi-trajectory limit set that
A+(x) ,
limit set
A+(x)
Proof. The set
A+(x)
uniformly approximate y, z f A+(x) y E A+(x),
and
S(x[t, t + T], ~ ) ~ p(y,w) < 6
there is a point
for each
such
T = T(c) > 0
t E R+ .
if and only if
A+(x).
If
A+(x)
as
implies
that
A+(x)
for
z ~ y(y)
t ~ 0.
(otherwise
y(y) = A+(x)
such t h a t
for
Itl ~ T.
P ( X l , y) < 6.
0 =< t =< T,
then
p(x I t , y t )
E g < g + ~ = e,
0 ( z , y ( y ) ) = ~.
y+(x)
such t h a t
f o r any
Thus
as
A+(x)
for
Let T > 0
Further, there is a
p(yt, wt) < ~
x 2 ~ Xl[0 , T]
y+(x)
is not minimal, then there are
A+(x), . we h a v e i n p a r t i c u l a r
be m i n i m a l , so t h a t
if possible,
y+(x)
the semi-trajectory
is minimal; Theorem 2.9.3).
p ( z , y t ) ~ p ( z , x 2) + P(x2, y t ) contradiction,
Then the
is non-empty, compact,and invariant. Now let
x1 E u
where
be positively Lagrange stable.
By uniform approximation there is a
Thus t h e r e i s a p o i n t
A+(x)
there is a
^+(x) .
A+(x)
E t + TI, ~) ~
x2 = x1 t,
x
such that
p(z, ~(y) ) = E > O.
S(x[t,
~
i s minimal
uniformly approximates
that
> 0 ,
THEOREM Let the motion
each
is said to uniformly approximate its
if given any
A+(x) C S(x[t, t + T] , e)
2.10.14
points
y+(x)
such that
6 > 0
Since
such
y ~ A+(x),
And b e c a u s e
S(x 1 [0, T I , ~)--~ A+(x). e p ( z , x 2) < ~. c
< ~,
If then
so t h a t
x2 = Xlt. is min~al.
y ~ A+(x), y ( y ) = A+(x).
does n o t u n i f o r m l y a p p r o x i m a t e
This is a Now l e t Now assume,
A+(x).
Then
178
there is an
e > 0, a sequence of intervals
{yn}CA+(x)
such that
Yn ~ S(x[tn'
Tn]' e),
Then for arbitrary
E
We may also assume that
P(Yn' y) < 3
for all
and n.
n
Tn]) -- O(Y n, Y)
Consider now the sequence of points
Clearly
and a sequence
tn + + ~' (Tn - tn) + + ~' Yn § y (5 A+(x)),
P(y, X[tn, Tn] > P(Yn' X[tn'
x
{(tn, Tn)} ,
{Xn} ,
e 3
>
2 3
where
t + T = x( n n t' n 2 )= x n"
t' § + ~. n
x t' § z (~ A+(x)).
Since
y+(x)
Since
A+(x)
is compact, we may assume that is minimal,
y(z) = A+(x), so that
n
T ~ R
there is a
such t h a t
E
p(zT, y) < ~.
By t h e c o n t i n u i t y axiom g
we can choose a Now choose
N
o > 0
such that
p(zT, wT) < ~
large enough such that
whenever
p(z, x N) < o,
and
p(z, w) < ~. ~N - tN > ITI. 2
Then
XNT = x(t~ + T) ~ X[tn,
and hence
p(y, XNT) > 23e .
Tn],
On the other hand
P(XNT , zT) < ~,
E
p(y, XNT ) < P(XNT , zT) + p(zT, y) < ~ + ~ -
This contradiction
limit set
No necessary
c
3 "
proves the result.
The following positive
2
so that
theorem gives a sufficient
A+(x)
and sufficient
condition for a
to be a minimal set of almost periodic motions. condition is known as yet.
179
2.10.15
THEOREM. Letthe motion
~
be positively Lagrange stable, and let the
x
motions in y+(x)
be uniformly positively stable in
y+(x)
uniformly approximates
then
A+(x) ,
A+(x)
y+(x) .
If moreover
is a minimal set of
almost periodic motions. Proof.
By Theorem
of Theorem
2.10.14 ,
2.10.11
A+(x)
is a compact minimal set.
we need only prove that every motion through
positively stable in
there is a
x(t 2 + t)) < T > 0
E
~
~ > 0
for
such that
t ~ 0.
Let
p(xtl, xt2) < 6
{y,z}CA+(x)
be arbitrary. We wish to estimate
axiom there is a p(y~,wT) < ~ t2 > 0
^+(x)
is
y+(x) .
By uniform positive stability of motions in e > 0,
In view
o > 0
such that
and
p(yT, ZT).
p(y, w) < o,
y+(x), we have given implies
p(x(t I + t),
6 0(y, z) < 3"
Let
By the continuity
p(z, u) < o
and
p(ZT,UT) < ~.
If
~ = min[o, ~].
such that
p(xtl, y) < ~
and
p(xt2,Y ) < ~.
imply
There are
tl> 0
and
Thus
6 p(xtl, xt 2) < p(xtl, y) + p(y, z) + p(z, xt2) < ~ + 3 + ~ < 6-
Consequently
E p(x(t I + T), x(t 2 + T)) < 3'
and
p(yT, x(t I + ~))
0,
T' ~ 0
i.e., x = x(T' - T).
Since
the trajectory
is closed and has a period
y(x)
J+(x) C
i.e. , y(x) = y+(x).
then there is a
y(x)
D+(x) = y+(x)
J+(x) = ~.
implies that
being closed and invarlant, we conclude if
empty, that
that if
J+(x) = y+(x)
x t X.
x r X, and there are no
is a rest point or
there are no rest points or periodic orbits, then indeed
for each
is and
For y+(x).
is not This shows
such that
xT = xT',
the last equality shows that T' - T. Since we as-
sumed that there are no rest points or periodic orbits, we have arrived at a contradiction.
Thus
system is dispersive.
J+(x) =
@ for each
x G X,
and the dynamical
The theorem is proved.
We now develop a criterion for parallellzahle
dynamical systems.
For this purpose the following definition is needed. 2.11.18
Definition A set
x 6 X
S~iX
~here is a unique
is called a section of ~(x)
such that
(X, R, 7)
if for each
x~(x) r S.
Not every dynamical system has a section.
Indeed any
(X, R, ~)
has a section if and only if it has no rest points or periodic trajectories. The function
T(x)
will be basic in what follows. In general
188
T(x)
is not continuous,
T(x)
implies certain properties
in the following
but the existence
of a section
of the dynamical
S
with continuous
system which we sum up
lemma.
LEI~IA
2. ii. 19
If
S
is a section of the dynamical system
T(x)
continuous on
i)
S
is closed in
ii)
S
is connected, arcwise connected, simply connected if and only
if
x
X
with
(X, R, 7)
then X ,
i8 respectively connected, arcwise connected, simply
connected, iii)
If
KC
every iv)
If
S
is closed in
S ,
then
i8 closed in
Kt
X
for
t ~ R ,
K C S
i8 open in
interval in
R,
S,
then
is open in
where
KI,
I
i8 any open
X 9
Proof. i)
If
{x } n
in
continuity. Thus
and
Since
x
is closed
in
n
§ x r X ,
T(Xn) = 0
xT(x) -- x 0 = x ~ S S
ii)
S ,
then
for each
by definition
T(Xn) § T(x)
n , of
we get T(X)
Then there are disjoint closed
disjoint. {Xn }
in
SI
X -- S I R U
sets
Sl, S 2
S2 R
.
SIR , Xn § x .
Then
S .
H~n~e
is closed,we conclude
that
.
be not connected.
such that
Consider
T(x n) § T(X)
# : XnT(X n) § x T(X)
S
Note that
We prove that they are closed.
the phase map
and
The interested reader can
supply the proofs of the remaining parts. Let
X = SR , we have
T(x) = 0 .
X .
We shall prove only the first part.
As
by
Since
x~(x) e S I .
,
SIR
SI U S 2 = S . and S2R
SIR , and let
and by continuity
{XnT(X n)} Then
are
in
S I and
of
189
x = x T(x)(-~(x)) s xT(x)R C SIR 9
Thus
SIR
we can prove that
Thus
X
S2R
is closed.
is closed.
being the union of two
disjoint non-empty closed sets is not connected. connected, so must iii)
S
be .
iv)
We conclude that if
X
is
The converse follows similarly.
The proof follows by observing that if it is closed in
Similarly
K
is closed in
S ,
then
X .
The simple proof is left to the reader.
The following theorem now gives a criterion for parallelizable dynamical systems.
2.11.20
THEOREM A dynamical system
only if it has a section Proof.
Sufficiency.
~(x)
and the
h-l(x, t) = x t of
X
onto
~(x)
with
Indeed
h(x) = (x T(x), - Y(x)). of
S
(X, R, ~)
SR = X.
Then
phase map w.
h
is
is parallelizable if and continuous on
Define i-i
h: X + S x R
h-l: S x R § X
and is clearly continuous. This
h
where
t
Further for any
is given by h (x) = ~t),t).
from that of 2.11.21
X.
h.
is given by
is thus a homeomorphism
is parallelizable.
note that if the dynamical system is parallelizable, definition is a section of
by
and continuous by the continuity
The inverse
S x R, i.e.~ (X, R, n)
X .
To see necessity, we
then the set
x ~ X
set
Then continuity of
S
in its
~(x) = - t ~(x)
follows
The proof is completed.
Remark The above theorem shows that the dynamical system of example
2.11.13 is not parallelizable. Notice however that the phase space in this example is not locally compact. The following is the most important theorem in this section.
190
THEOREM
2.11.22
A dynamical system metric space
X
on a locally compact separable
(X, R, ~)
is parallelizable
if and only if it is dispersive.
The proof of thls theorem depends on properties
of certain
sections which we now describe.
DEFINITION
2.11.23
An open set a
T > 0
U
and a subset
i)
SI
ii)
for each
T
in
X
SCU
will be called a tube
if there exists
such that
C U , and
x~(x)
x ~ U
~ S.
there is a unique
Here
I
T(x)
such that
, IT(x) I < %
~ (-~, T) . T
It is clear that if
(1)
and
also called a T-tube with section S , tube
and
hold,
S
a
then
U = SI T .
U is
(T- U)=section of the
U . If
tion. T(x)
(ii)
I
T
= R ,
then
U
Is an
In this last case indeed which maps
U
into
I
=-tube ,
U = SR . is
i-i
and
S
an
(~ - U)-sec-
Note also that the function
along each trajectory
in
U .
T
LEnA
2.11.24
Let
U
then the function
Proof.
To show
be a T(x)
: if
T-tube with section is continuous on
{x n}
in
KI
and s
Note that the sequence convergent
as
K
{XnT(Xn) }
is compact.
If
K ES
KI s
for any
x
+ x ~ KI n
is in
Further
S .
K,
is compact,
s , 0 < s < T
,
then
T(Xn) + T(x).
s
and we may assume that it is
{T(Xn) }
is in
I
and hence bounded
s
so that we may also assume that x n~( x n) + x* ~ K,
and
{r (x n) }
T(Xn) -~ T* e I T, .
is convergent. Since
Thus let
x n + x,
we have
191
x* = XT*. lemma i s
Since
IT*I__< T'
there
2.11.25
is
a
t h e o r e m shows t h a t
tube
T* ffi T(X)
by uniqueness.
containing
if
x $ X
is
not
a rest
The
point,
x.
THEOREM
If containing Proof.
we h a v e
proved. The n e x t
then
< T,
x r X
is not a rest point, then there exists a tube
x.
Since
p(x, XTo) > 0.
x
is not a rest point, there is a Consider the function t+T
~(y, t) =
I ~
p(x, yT) dT.
t
It follows that tI + t 2 + T o
r
p(x, y~) dT
~(Y, t I + t 2) ffi J tI + t2
t2+T
o(x, y(T + t l ) ) dT
I
~
t2
t =
i
+ To p(x, Ytl(T)) dT
t2
=
~(ytl,
t 2) 9
T
o
> 0
such that
192 Further the function
~(y, t)
is continuous in
(y, t)
and has the
partial derivative
~t(y, t) ffi p(x, y (t + To)) - 0(x, yt).
Since
~t(x, 0) = 0(x, xT o) > 0,
there is an Define
e > 0
To > 0
such that
such that
St(y, 0) > 0
x[- 3To, 3 T o ] C S ( x
$(x, t o ) > $(x, 0) > $(x,- To).
(S[XTo, ~] U
and such that for y e S(x(-To) , ~)
Now choose
we have
y G S(x, e).
, e). ~ > 0
S[x(-To), ~]) C
y e S(XTo, ~) we have
for
Then, in particular, such that
s(x, ~),
$(y, 0) > $(x, 0),
$(y, 0) < $(x, 0).
and for
Finally determine
6 > 0
such that
S[x,~] T0 C
s[x, ~] (-t o) CSCX&To),O,
S(xTo, ~)
and
S[x,
6] [ - 3To, 3To]
We will show that if IT(y) l < T O that
C
s(x,
~).
y ~ S[x, 6], then there is exactly one
such that
$(y, t) = $(yt, 0)
$(y, T(y)) = ~(x, 0).
T(y),
This follows from the fact
is an increasing function of
t,
and
~(y, t o) > ~(x, 0) > ~(y, - To ). C o n s i d e r now t h e open s e t
U ffi S ( x , 5) I t , o
and s e t
193
s : {y
u: r
0) :
0)}.
claim that
We
S
is a
(23
-section.
- U) 0
For this we need prove that if
y ~ U,
IT(y) l < 2T
C S
such that
yT(y)
O
It'I < T
then there is a unique Indeed for any
y ~ U,
T(y), there is a
t'
"
such that
y' = yt' s S(x, ~),
and for
y' s S(x, 6)
there is
O ~
a
t",
It"l < To,
such that
y't" ~ S.
IT(Y) I -< It ' I +
T(y) = t' + t",
and
be two numbers,
T'(y), T"(y),
y T'(y)
r S
and
It'l $ T .
y T"(y) s S,
Then
~(y',T'(y)
Thus
y(t' + t") = yT(y) ~ S,
It"l < 2T o .
13'(Y) I < and let
Now let if possible
2T o, IT"(Y) I < 2T o, y' = yt' G S(x,
- t') : ~(y,T'(y))
6),
= ~(yT'(y),
where there
such that where
0),
and
O
~(y' T"(y) - t') = ~(y, T"(y)) = ~(yT"(y) ~(y',T'(y) and
increasing
T'(y) = T"(y). 2.11.26
SO that
- t') = ~(y', T"(y) -- t') = ~(X, 0). NOW
IT"(y) - t' I ~3TO,
strictly
0),
and
for
~t(y ', t) > 0
Itl $ 33o.
Hence
IT'(y) - t' I ~ 3To,
for
Itl ~ 33o,
i.e.
~(y', t) is
3'(y) - t' = T"(y) - t',
or
The theorem is proved.
Remark If
X
is locally
the above proof to ensure that
compact,
then we can restrict
S[x, 6]
is compact.
6 > 0
Thus the
in
(2T
- U)-section O
S
constructed
we may S
in the above proof will also be locally compact.
further assume the function
to be continuous
on
T(x)
corresponding
By Lemma 2.11.25
to the section
U.
In fact the following more general theorem can now be proved.
2.11.27
THEOREM Let
x ~ X
restricted only by
be not a rest point. W
T < ~
if the motion
Let
#
T > 0
be given,
is periodic with least X
period
w .
Then there exists a tube
(T - U)-section
S .
Further,
function T(x) corresponding on
U.
if
X
U
containing
x
i8 locally compact,
to the section
X
with a then the
can be assumed continuous
194
The proof of this theorem is left to the reader. For wandering points
one can prove:
THEOREM
2.11.28
If X
x 9 X
x ~ X
is a wandering pointj i.e., x ~ J+(x) ,
is locally compactj then there exists a tube
an
(~ - U)-section
Proof. S,
T(X)
there is a
continuous
~ > O~
U = S* R,
S*
6 > O,
on
W.
containing Since
= S*
x
x,
is an
such that
Yntn + x, i.e.,elther
section of U = S* R. U
~ > 0
that
such that
y(y)
we claim that
with
or x
y ~ S*, inter-
there will be a sequence tn + + ~
R,
x c J+(x),
which are ruled out by the assumption shown that there is a
in
(T - W) -section
x. To see this notice that
such that every trajectory
{t n}
W~U,
This we leave to the reader to verify.
{yn }
(or t n § - ~),
x ~ J-(x),
both of
is wandering.
S* = S(x, ~ ) ~ S
We have thus
is an
is further open, and continuity of
follows from its continuity on
with
(~ - U) -section of the open set
~ - tube containing
and a sequence
x ,
U .
with a
is wandering,
only at the point y. For otherwise,
S, Yn § x,
containing
continuous on
T(x)
W
S(x,6)~S
which is a
there is a
in
and with
Indeed there is a tube
and
sects
S ~
U
and moreover
(- - U)-
T(x)
on
U
and continuity of the phase map #. The theorem is proved.
For further development we need the following definition.
DEFINITION
2.11.29
Given an open on
U ,
in
S
K .
~-tube
U
let there be given sets and
K
with a section N,
K ,
is compact, we shall call
Then indeed
T(x)
restricted to
N CK~S
KR KR
S
and ,
~(x)
where
continuous N
is open
the compactly based tube over is continuous on
KR .
195
2.11.30
Remark. A compactly based tube need not be closed in
X.
As an example,
one may consider a dynamical system defined in the euclidean plane
R 2,
as shown in figure 2.11.31. The x2-axis consists entirely of rest points, all other trajectories are parallel to the point on the
xl-axis , with each having a rest
x2-axis as the only point in its positive limit set, whereas
the negative limit sets are empty. Here, for example the set { (xl, x2): 0 ~ x 2 ~ i, closed in
2.11.31
x I > 0}
is a compactly based tube, which is not
X.
Figure
X2
r
j r
r
We can now prove the following. 2. ii. 32
THEOREM If X is locally compact and separable, and if every
is a wandering point, then there exists a countable covering X,
by compactly based tubes
Proof.
~R
each with
x ~ X
{~R}
Tn(X ) continuous on
of KR. n
The proof is immediate, when we notice that by using theorem
2.11.28, one can find a compactly based tube containing a wandering point of
X.
The rest follows by the assumption of separability of
X.
196
We gave an example above to need not be closed in system
(X, R, ~),
X.
show that a compactly based tube
One may wonder if for a wandering dynamical
a compactly based tube is not closed. Here is a
counter-example.
Example
2.11.33
Consider again example 1.4.7il,referred to in example 2.11.12. Any compactly based tube containing a point because its closure will contain
yo
p ~ T_ 1
is not closed,
which is not in such a tube. This
is an example of a wandering dynamical system which is not dispersive. In the case that 111.34
(X, R, ~) is dispersive one obtains.
LEMMA
A compactly based dynamical system (X, R, ~) Proof.
U = KR
sequences
and if in
{yn }
assume that the sequence
K
with section
{Tn} K
in
K
of a dispersive
X.
is a sequence in
and as
U
i8 closed in
{x } n
Yn + y ~ K {T }
~-tube
R
KR,
then there are
such that
is compact. If now
is bounded, so that
x n = Yn Tn .
we claim that
x n § x~
x ~ y R CKR.
We may
For otherwise if
n
{Tn}
contains an unbounded sebsequence
clearly x r J+(y),
{Tnk} ,
which is absurd, as
say
J+(y) = #
T
+ + ~,
for each
then
y C X
by
Theorem 2.11.15. The lemma is proved. We now prove the last lemma required to prove Theorem 2.11.22. 2.11.35
LE~4A Let
U 1 , U 2 be two compactly-based tubes of a dispersive
dynamical system with sections and
~2(x)
respectively.
If
and continuous functions
K1 , K2
U1 ~U 2 = 0
pactly based tube with a section
K D K1
Moreover, if the time distance between
K1
then
U = UIU
U2
Tl(x)
is a com-
and a continuous function and
K2
along orbits in
~(x)
197
UI ~ U2
is less than
along orbits in
U
Proof.
U2
UI
and
and closed.
2.11.36
T(>
0),
the time distance between
is also less than
and
K2
T .
are invariant and closed.
Further,
K
Therefore
UI N U2
is invariant
K2 ~ UI
Figure
Sz
/
is compact and non-empty. Set UI(~ U 2
intersects
S 2 ffiK 2 ~
UI
and
Tl(X) ffi z2(x) + ~l(X T2(x)). x(~2(x) + ~l(X T2(x))) , Tl(X)
x ~ UI~U
This is so because
can have T(x)
IT(x) l < 9
K ffiK l ! J { x
follows:
then
on
being continuous
set now
T*(X)
for
S2
is continuous on
x r S 2. K 2.
Further if
{x T(X): x ~ K~
for KR
x ~ KIR ,
z(x)
for
K2,
x r S2,
{x T(X): x e S 2} ffi Sl,
is compact as
and
by Tietze's
defined on
[TI(X) I < 9
Notice now that
~*(x)
K2 on
we and
is compact. We
KR = K I R ~ K 2 R
ffi T2(x) + T(x T2(x))
and we need only verify that if
Tl(X) ffi T2(x) + T(x ~2(x)),
is proved.
we have
(which is compact), and
T(X): X ~ K 2}, and define
~*(x) ffi Tl(X)
2
and there are no rest points or periodic orbits. The
is continuous on
T(x) E Zl(X)
Any orbit in
XZl(X) ffixT2(x)(Tl(X ~2(x))) ffi
theorem it can be extended to a contin~us function where
I.
K 2 and hence S 2 in exactly one point, and also intersects K 1
and hence S I in exactly one point. Thus for any
function
SI = K I ~ U
if
as
x ~ K2R.
x ~ UI~U2,
which has already been proved. The lemma
198
Proof of Theorem 2.11.22. sufficient to prove that
Only the sufficiency part needs proof. It is X
has a section
By Theorem 2.11.32 there is a pactly based tubes
U
countable
with sections
n
S
covering K
ning with set
K 2,
U1
and
U2 .
it together with
function X
This leaves
Un+ 1
K 1 = K I,
Thus
un+l
defined by
X
unaltered.
and
.
Now set
S = U
T(x) ffi ~n(x) xT(x) ~ S;
has a section
S
for
K1
by comT (x). n
U n+l K n,
with
then
x r ~
Begin-
to a compact
U2 = U I u u 2
T2(x)
with U n,
we take
Kn+IDKn
X = SR,
,
and
and the
is continuous on
moreover) T(x)
with continuous
system (X, R,w) is thus parallellzable
X
U 1 = U I.
Having found
and construct similarly
continuous on T(x)
K1
of
X.
of compactly based
we use lemma 2.11.35 to enlarge
with the property that
x ~ X.
{U n}
thus obtaining the compactly based tube
continuous on
n+l T (x)
U2
Set
{Un}
continuous on
and continuous functions
n
We replace this covering by a like covering tubes which we construct as follows.
with T(x)
is unique for each
T(x) defined on
X.
The
and the theorem is proved.
We shall now make some applications of the theory of sections to asymptotic stability of compact sets of a dynamical system defined in locally compact metric space Theorem
X. 2.7.11
shows that if
M CX
tically stable, then its region of attraction dense subset in it. Thus if asymptotically
stable set
A(M) M,
separable metric suhspace of 2.11.37
contains a countable
is the region of attraction of a compact
then A ( M ) ~ M X.
A(M)
is compact and asympto-
= A*(M)
is a locally compact
We will show more (see Exercise 2.6.24).
THEOREM If M
is a compact asymptotically stable set of X, then for
199
each
x 6 A(M),
Proof. and
J+(x) C M .
If possible let
A+(x)+ ~,
x e A(M)
we have
and
J + ( x ) + @.
Consequently by lemma 2.3.12 for any But
D+(M) = M
each
x ~ A(M).
2.11.38
J+(x),~M.
Since
J+(x) D A + ( x )
Thus there is aI y ~ w ~ A+(x)CM,
by Theorem 2.6.6 , as
M
J+(x),
wJ have
y~
M.
J+(x) C D + ( w ) C D + ( M ) .
is stable. Therefore
J+(x) C M
for
COROLLARY If
M is a compact,invarian~
asymptotically stable set of a dynamical
system (X, R, ~), then the dynamical system induced by the given one on the invar~ant set
i8 dispersive.
A*(M) = A ( M ) ~ M
If
X
i8 locally compact, then
by the above observation and Theorem 2.11.22 it i8 parallelizable. 2.11.39
Remark It is clear now that if
stable set with region of attraction variant set in dispersive. let
~(x)
If
M, X
M
is a compact asymtotically A(M)
and
M*
then the dynamical system induced on is locally
Lemma
2.8.1.
If
S[M, E] is compact, then let
0 < e < mo,
A(M)~M*
is
compact, then it is parallelizable. Now
be any function defined in a neighborhood
conditions of
is the largest in-
m
set
o
e > 0
N
of
M
is chosen such that
= min{~(x): x E H(M,E)}.
and satisfying S[M,E]CN,
For any
~,
Pe = {x E S[M,e]: ~(x) . 0
and
x~M,
there is a
8 = ~(x,e) > 0
such that
ii)
equi-stable, if for each x ~ S(M, 6 ) R t
iii)
x~M,
there is a
6 = 6(x) > 0
such that
and
uniforml~ stable, if for each
e
> O,
there i8 a
~ = 8(e) > 0
such
that
s(M, )R+C 2.12.2
Proposition If
X
is locally compact and
M
is compact, then
M
is uniformly stable
whenever it is either equi-stable or stable (or both). Proof.
(i)
If
M
is stable, then for a given
number corresponding to U{S(X,6x):X~M}
x~M
such that
i s an open c o v e r o f
U{S(xi,6xi):i -- 1,2,.~n;xi~M} S(M,6) C U { S ( x i , ~ x . ) : i
of
M,
M.
-- 1,2,...,n}.
e > 0,
S(X,~x)R+C
let
~E > 0
S(M,e).
there is a finite
be a
Since
open c o v e r , s a y
But then there is a
6 > 0
such that
Notice now that
1
S(M,6)R +
C
[ U {S(xl,Sx.):i -- 1 , 2 , . , n } ] R + C S(M,e).
Thus
M
is uniformly stable.
1
(ii)
Let
there is a
M be e q u i - s t a b l e .
6 > 0
such that
Since
M i s compact and
X
is locally
S[M,6],
and hence also
H(M,~)
compact,
are compact.
202
Then f o r e a c h But t h e n
xt=H(M,8) ,
x~C(Sx) ,
there
where
H ( M , ~ ) C ~ { C ( S x ) : X t H(M,~)}, Thus there are points
is a
6
> 0
x Sx ~ S(M,Sx)R +.
such that
x~S(M,6x)R+
Since each
C(S x)
i s o p e n , and
we have an open cover of the compact set
Xl,X2,...,Xn
in
H(M,6)
such that
H(M,~).
H(M,6) c U { C ( S x i ) :
i = 1,2,. .. ,n}.
Since U{C(S x ):i = 1,2,...,n} =C ( ~ S x :i = 1,2,...,n}) we 1 1 have ~ { S x i - i = 1,2,...,n}C S(M,r If now 6 = min{~xl,SX2,...,~Xn}, then 9
S(M,6)R+~ O { S
:i = 1,2,...,n}C
~
S(M,E).
Thus
M
is uniformly stable.
xi 2.12.3
Remark Note that part (i) of the above theorem did not use the fact that
X
is
locally compact. Further~uniform stability implies both stability and equi-stability, but it cannot be asserted that a closed set which is both stable and equi-stable is uniformly stable.
2.12.4
THEOREM There exist
closed sets which are both stable and equi-stable but are
not uniformly stable. We leave the proof to the reader.
2.12.5
Proposition If a closed set is either stable, or equi-stable, then it is positively
invariant. The proof is simple and is left as an exercise. We now indicate the connection between various kinds of stability and Liapunov Functions.
2.12.6.
THEOREM A closed set
defined on
X
M is stable if and only if there exists a function
with the following propertiest
v
=
r
203
i)
For every
>~ e,
p(x,M)
e > O,
there is a
and for any sequence
ii)
~(xt) .< ~(x)
for all
Proof. (a)
Sufficiency.
Given
~ > 0
such that
{Xn} , ~(Xn) + 0 x~X,
whenever
~(x) >~ ~
whenever
Xn § x ~ M .
t >. 0.
E > 0, set
m
-- inf{~(x):p(x,M)
>~e}.
By
o
(i)
m
> 0.
Then for
xfM
find
6 > 0
such that
~(y) < m
o
for
y~S(x,6).
o
This is also p o s s i b l e by ( i ) . there is
y ~ S(x,6),
~(yt) .< ~(y) < m
and
We claim t h a t
t ~ 0
such that
on one hand by ( i i ) ,
S(x,g)R+C. S(M,~). p(yt,M)
and also
= e.
For otherwise
But then
~(yt) >~ inf{~(x):p(x,M)
>. g},
o
as
p(yt,M)
= e,
i.e.,
~(yt) ~ m .
This contradiction
proves the result.
o
(b)
Necessity.
Let
M
be stable.
Define
~(x) -- sup{ p(xt'M) i+p(xt,M): This
~(x)
is defined on
The varification
2.12.7
and has all the properties
is left to the reader.
required in the theorem.
The theorem is proved.
Remark Condition
if
X,
t >~ 0}.
{x n}
(i) in the above theorem is equivalent
is any sequence such that
a continuous
strictly increasing
Xn + x E M ,
function
~(p),
then
to the requirement
~(Xn) § 0,
defined for
that
and there is
~ >~ 0,
such that
~(x) >. ~ ( p ( x , M ) ) . 2.12.8
THEOREM A closed set
~(x)
defined on
X
i) ~(x)=
0
ii) for every
MC
X
is equi-stable if and only if there is a function
such that
for
x~M,
e > 0
for
~(x) > 0
there is a
6 > 0
and iii) ~(xt) ~< ~(x)
for
x~X,
t >~ 0.
x~M,
such that
~(x) .< e
if
p(x,M)
.< 6,
204
Proof.
(a) Sgfficienc[.
Let
x I M.
Set
p(x,M) = e.
Then by (ii) there is a
e
6 > 0
such that
by (iii).
E
r
Hence
.< ~
for
Set for each
= sup{6 > o:x
r
= 0
S(M,6)R+C'-~S(M,~),
x ~ S ( M , 6 ) R +.
(b) Necessity.
and
Then indeed
p(x,M) .< 6.
for
x~M.
x~M,
/S(M, O R + }
This
r
verification is left to the reader.
,
has all the desired properties, whose Note that
#(x) ~ p(x,M).
Remark
2 .12.9
Condition (11) in the above theorem is equivalent to the existence of a continuous strictly increasing function
s(r), a(0) = 0,
such that
=(p(x,M)). THEOREM
2.12. i0
A closed set (x)
defined on
X
M
is uniformly stable if and only if there is a function
such that
i) for every
c > 0
there is a
6 > 0
such that
r
~ 6
whenever
r > 0
there is a
6 > 0
such that
r
.< r whenever
p (x,M) >.e,
ii) for every
p(x,M) .< 6, iii) r
Proof.
.< r
for
x E X, t >~ O.
We leave the details to the reader, but r~mork that in the proof of necessity
one may choose either of the functions given in the necessity proofs of the two theorems 2.12.6, and 2.12.8.
2.12.11.
Remark. Our theorems above differ from the usual theorems on stability in that
the existence of the functions is shown in neighborhood of
M.
all
of
X
Theorems 2.12.6 and 2.12.8 are new.
rather than in a small Indeed for sufficiency the
205
functions need be defined on just a neighborhood of
M.
We shall now discuss asymptotic stability of closed sets and its relation with the Lyapunov Functions.
2.12.12
DEFINITION A closed set i)
MCX
a semi-weak attractor, if for each for each
y ~ S(X,6x)
such that ii)
a semi-attractor,
if for each
xEM, § 0
{tn}
in
as
t §
+
~ > 0
R, tn § +
an attractor, if there is a
~ > 0
~,
{t n}
in
~
x
> 0,
and
R, tn
T = T(~) > 0,
6x > O,
such that
t + + ~, and for each such that
y CS04,6),
there
p(Ytn, M) + 0,
such that for each
y ~S04,~),
~,
a uniform attractor, if there is an is a
there is a
there is a as
a weak attractor, if there i8 a
p(yt, M) + 0
v)
there is a sequence
y ~S(x,~x),p(yt,M)
i8 a sequence iv)
x ~ M,
p(Ytn,M) + 0,
for each iii)
will be said to be
such that
~ > O,
x[T,+ ~ ) C
and for each S(M,e)
~ > 0
there
for each
x~ S[M, ~], vi)
an equi-attractor, such that for each
if it is an attractor, and if there i8 an ~, 0 < c < ~,
with the property that vii) viii) ix)
semi-asymptotically
and
T > 0,
x[0,T](] S04,6) = ~
a > 0
there exists a
whenever
6 > 0
~ .< p(x,M) -< l,
stable, if it is stable and a semi-attractor,
asymptotically stable, if it is uniformly stable and is an attractor, uniformly asymptotically stable, if it is uniformly stable and a uniform attractor.
2.12.13
DEFINITION i)
{tn} in
For any set R, tn § +
~,
M CX,
the set
such that
A 04) = {y~ X:
p(Ytn,M) § 0}
there i8 a sequence
is called the region of weak attraction
206
of M,
and ii)
the set
A(M) = {y(X:p(yt,
region of attraction of 2.12.14
M) + 0 as
t §
|
is called the
M.
Proposition If
M
is an attractor then ~ A (M) ---A(M).
The proof is trivial and is left as an exercise.
2.12.15
Proposition If
M
is a weak attractor (attractor),then
invariant set which contains
S(M,6)
for some
A (M) ((A(M))
is an open
~ > O.
The proof is simple and is left as an exercise.
2 .12.16
THEOREM
If a compact set
M
is a semi-weak attractor (semi-attractor), then it
is weak attractor (attractor). Proof is similar to that of Proposition
2.12.2.
We now discuss the existence of Liapunov functions for various kinds of asymptotic stability.
2.12.17
THEOREM
A closed set a function i)
M
is semi-asymptotically stable, if and only if there exists
defined on
@(x)
For each
x
which has the following properties:
y~ M, @(x)
is continuous in some neighborhood
S(y,6y)
Y, ii) iii)
r
-- 0
for
x~M,@(x)
> 0
for
x~M,
there is a strictly increasing function >. O,
such that
$(x) >. a(p(x,M)),
~(~),~(0)
= 0,
defined for
of
207 iv) ~(xt) ~ ~(x) a
6y > 0
t > 0
Proof.
x~X,
for all
such that if
and
(a) Sufficiency.
x#M,
as
~(xt) § 0
t ~ O,
y~M,
and for each ,
x@S(y,6y)
then
there is
~(xt) < ~(x) for
t § + ~.
Stability follows from Theorem 2.12.6.
The semi-attractor
property follows from (iii) and (iv). (b) Necessity.
Consider the function ~ (xt,M)
~(x) = suP~l+0(xt,M ) : t % 0}. This has all the properties along trajectories
(i) to (iv) except that it may not be strictly decreasing
originating
in any neighborhood of points of
this we complete our construction.
(x)
io
-
,
M.
Before proving
We define
(x~) e-~ dT
.
0 This
~(x)
has all the properties
(i) to (iv) except possibly
(iii).
The construction
is now completed by setting
r
=
r
+
,,,(x).
To see for example that for each
y~M,
there is a
6
> 0
such that
~(x)
is
Y continuous in containing invariant,
M.
if
If
%
~(x)
is continuous in an open set
is the region of attraction of
I(A(M))
= {x~ ~(A(M)): y + ( x ) C S ( M , e ) } .
then there is a
p(x,M) = I.
There is a
M,
then
A(M)
is also invariant and open and contains
and contains an open set containing
x ~ I(A(M)),
and let
we need prove that
A(M)
and indeed
define the set invariant~
S(y,6y),
T > 0 T
>0
M,
such that such that
This
We
is M.
We now
is openjpositively
and has the important property thatj xT ~ W e. x T ~ W%/4.
Now let Since
x ~ I(A(M)) W%/4
is open
208
we can find a neighborhood neighborhood of such that
x,
and indeed
n < ~/4,
~(x) - ~(y) =
S(xT,o)C
and s
NC
S(x,n) C
Then
I(A(M)). N.
rp (xt,M) uPil+p(xt,M):t
=
Wl/4.
is a n > 0
y ~ S(x,n),
sup;0 (yt,M) =~l+p(yt,M):
>. 0} -
(-T) = N
We can thus choose an
Then if
0 .< t .< T}
- ptxt,~)
S(xT,o)
t > 0}
- sup(P(yt'M)
l+p(yt,M)
:0 .< t .< T},
and so
,p (xt.M) (yt.M) l~(x) - ~(y) l -< sup{ll+~(xt,M ) - l+p(yt,M) l:. 0 .< t .< T}
I~ (xt,M) - p (yt,M) = sup{~(l+p(xt,M))(l+p(yt,M)
I: 0 .< t .< T}
.< sup([ o(xt,M) - o(yt,M) i: 0 .< t .< T}.
.< sup{p(xt,yt):
0 .< t .< T}.
By the continuity axiom the right hand side tends to zero as is continuous in
I(A(M)).
The rest of the observations
on
y +x, ~(x),~(x)
hence are
~(x) easy
to
verify and are left as an exercise.
2.12.18
THEOREM Let
M
be a closed set.
if there i8 a function i) ~(x)
Then
M
defined in
~(x)
is asymptotically stable if and o~ly x
with the following properties:
is continuous in same neighborhood of
S(M,6)
ii) ~ ( x ) = 0
for same for
which contains the set
~ > O,
xEM , ~(x)>
0
for
x~M,
iii) there exist strictly increasing functions defined for
M
~ >~ 0,
such that
~(p),8(p),=(O) = 8(0) = 0,
209
iv)#(xt)
.< ~(x)
for all
if
x ~ S(M,~),x~M,
as
t§
x ~.X,t > O,
then
and there is a
~(xt) < ~(x)
for
~ > 0 such that
t > O,
and
~(xt) § 0
~.
The proof follows exactly the same lines as that of the previous theorem and is left as an exercise. since
A(M)
functions
We note, however, that in the proof of necessity,
is open and invariant, and ~(x)
and
In the present case
~(x)
A(M)~S(M,6)
for some
6 > 0,
the
can be taken as being defined and continuous on
~(x)
will have the property
(iii),
whereas
m(x)
A(M).
may not
satisfy the left inequality in (iii) although it will satisfy the right inequality. Thus
~(x) = ~(x) + ~(x)
will have all the desired properties.
We shall now prove the following very important theorem, which in the case of asymptotic stability of a closed invariant set flow in the set
2.12.19
the
THEOREM
x~A(M),J+(x)C
Proof:
M
,
M
N - S(x*T,q) xT ~S(x T,q)
~).
x E A(M)
Since
and
(-T)
is a neighborhood of
and consequently
exist sequences
{x } n
in
X
x
y [T, + ~ ) ~ and
{t } n
~ > 0
in
then for each
y ~ M.
Set
T > 0
such that
such that
S(x*T,n)~
such that for each S(M, ~). R,
P~,M)
= e (>
0).
such that
there is a
q > 0
stable,
M ~ J - ( x ) ~ A(M) = ~.
y ~ J+(x*),
is a
x ~A(M),
is open, there is an
is asymptotically
xEA(M)~
is uniformly stable, there
S(M,~)
x ntn § y"
M CX
and for each
Let, if possible,
y+(S(M,~))~S(M, Since
characterizes
A(M) ~ M.
If a closed invariant set
Since
M,
t
n
We may assume without loss of generality,
Now since § + ~, that
S(M, 6).
Now
xEN, y ~J+(x),
such that XnE N,
x*T~S(M,d).
and
x § n
there x ,
tn >~ T.
210
But then
x t ~ S(M, e ) nn ~'
contradiction as fact that if
Thus if
p(y,M) = a.
y&J-(x),
y ~ J-(x) ~ A(M).
x t § nn
Thus
then
y,
J+(x)~M.
xs
Then we have
we
must have
P(y,M) -< ~.
A
The second statement follows from the
Now let
x ~A (M ) ~ M,
y ~ A(M), x ~ J+(y), x ~ M ~
and assume that
which has
already been
ruled out.
COROLLARY
2.12.20
If a closed invariant set (or in particular the space subset in it,
X)
M
i8 asymptotically stable L~d
is locally compact and
~hen the invariant set
A(M) N k M
A(M) \
M
contains a countable dense
i8 parallelizable.
The proof follows from the above theorem, and Theorem 2.11.22.
Remark
2.12.21
The considerations in Section 5.8. show that if
X
is locally compact and
is a compact~invariant, asymptotically stable set, then if
M~X
satisfying conditions of {x ~ S[M,E] : ~(x) = ~}
Le~u-_a 2.8.1 in a neighborhood
for fixed
e > 0
such that
~(x)
N
of
M,
S[M,c] ~
N,
and
sufficiently small, represent sections of the parallelizable flow in (See proof of Theorem 2.7.9.
is a function
the sets
A(M) ~% M
How far the same method of construction can be
extended to non-compact closed sets, depends naturally on whether the flow in A(M) k M
is parallelizable. We shall now prove that uniform asymptotic stability of a closed set
implies that the flow in
M C X
A(M) ~ M 2.12.22
of
is parallelizable, even when the subspace
is assumed to be neither locally compact nor separable.
PROPOSITION Let
with
X
A~) ~ M
A(M)
MC X
be a closed,invariant~ uniformly asymptotically stable set
as its region of attraction.
Then
A(M) \ M
is parallelizable.
211
Proof: on
Since
M
is asymptotically
stable, we can find a function
A(M) and having the properties given in Theorem 2.12.18.
uniformly asymptotically and
such that for any
?(xT)C
S(M,~)
a > 0,
for every
m~
Indeed
stable,
m ~ > 0.
=
there is an there is a
x ~ S[M,~].
=
q
We claim that if
x(A(M) \
{x(S[M,~/:
q < mo'
M,
S[M, ~ t " A ( M ) ,
with the property that
then
r
=
Sn
~(x)
n}.
is a section of the flow in function
is unique, and
A(M) ~ M
~(x), T :A(M) k M § R,
x T(x) ~ S . n A(M) ~
M
with the
such that for
The existence of such a
is parallelizable
(Theorem 2.11.20).
has the properties enunciated above, we consider the set
n
Pn = {x~S[M,a]:~(x) trajectory in
is
Now let
section indeed shows that the flow on S
such that
M
inf{~(x) :p(x,M) = a}
property that there is a continuous
To see that
T > 0
Since
defined
Consider now any set
s
each
e > 0
$(x)
.< n}-
A(M) %~ M
Indeed
P~A(M),
can intersect
S
and
Pn ~
M.
We note now that any
at most at one point.
This is so
n because if any trajectory may assume that
in
x 2 = Xlt
A(M) \
where
M
has two points
t > 0.
But then since
$(x 2) = $(Xlt) < _$(Xl), which contradicts trajectory is a
y
t > 0
in
A(M) ~ M
such that
intersects
xt~P
.
But then
Xl,X 2 S ~ n
the definition of
Sq
on
S O'
M = @, S . n
To see that every
we note first that if x[0,t] ~ B P
~ ~.
then we
x~, Pq,
However,
S
n If
x~P
,
otherwise 6 > 0 implies
n
then we claim that there is a y-(x)~
(otherwise p~,M)
x(-r) = y ~ P n C
< ~
~P . n M
In this case we can set
will be unstable). (such
S[M,a],
t .< 0
but
T > 0
If now
such that xt ( S
n
.
~ = inf{p(xt,M):t T > 0
~).
-= ~P . n
For .< 0},
be such that
and
y ES[M,a]
exists by uniform asymptotic stability),
yT = x ~ S ( M ,
there
then
This contradiction shows that every
212
trajectory in
A(M) \ M
T:A(M) \ M § SQ
intersects
S
n
exactly once.
by the requirement that
is uniquely defined and is continuous. the proof of Theorem 2.7.9.
xT(x) s
n
We now define the function
for
x ~A(M) ~ M.
Then
~(x)
The continuity follows in the same way as in
We have thus proved our proposition.
We shall now prove the following theorem.
2.12.23
THEOREM A closed set
with an open set
N
M
is uniformly asymptotically stable and equi-attracting
containing
for some
S(M, 6)
6 > 0,
~(x)
if and only if there exists a continuous function
as its region of attraction, defined on
N
with the
following properties: (i) ~(x) -- 0
for
> 0
xs
for
x~M;
(ii) there exists strictly increasing continuous functions 8(r), a(0) = 8(0) = 0
~(p(x,M))
such that
.< ~(x) .< S(p(x,M));
(iii) there is a sequence of closed sets En+l~l(En)~
S(M,6 n)
for some
{En}
such that
6n > O ; U { E n : n -- 1,2,3,...} = N,
and this sequence has the property that for any integer
no
~(r),
such that
r
> a
for
a > 0
there is an
x#E n ; 0
(iv) r
Proof:
= e-t~(x)
N
For the proof of sufficiency we remark that (iii) and (iv) imply
is invariant.
attraction and show that that
x s N, t ~ R.
We shall not give complete details as the arguments are similar to those
used in Section 2.8. that
for
N = A(M).
(i), (ii) and (iv) ensure uniform stability, N~A(M).
Since
N
as well as
is invariant neighborhood,
Uniform attraction and equi-attraction
To prove necessity, we consider the region of attraction
it follows
follow from (ii) and (iv). A(M)
and define a
~(x)
213
on
A(M)
as in Theorem 2.12.18.
A(M) \ M
defined by this
continuous map
~(x)
S
n
of the flow in
(Proposition 2.12.22), with the corresponding
T:A(M) \ M + R.
~(x)
We then consider a section
Lastly~we define
= e T(x)
for
x s
for
xEM
\ M,
and
~(x) = 0
This
~(x)
is easily shown to have all the properties
get the sequence
{En }'
we set
E 0 = {x:#(x) ~ i}.
(i) - (iv).
Then define
Note that to En = E0[-n'0]"
These sets are closed and have the required properties.
Setting
#(x)
=
-
i +
~(x)
we obtain the following very important
'
corollary.
2.12.24
COROLLARY A closed invariant set
with an open set
N
containing
M
is asymptotically stable and equi-attracting for some
S(M,6)
if and only if there exists a continuous function
6 > 0, (x)
as its region of attraction, defined on
N
with the
following properties (i) -i < r
< 0
(ii) ~(x) + 0 (iii) for any for
e > 0
x~N ~M,
as
0(x,M) + 0,
there is a
~ > 0
such that
~(x) .< - e
O (x,M) >~,
(iv) ~(x) § -i,
(v) d~ (xt) dt
for
[
as
=-
x § y ~ 3N,
~(x) ( l + ~ ( x ) )
t-O
We shall now give a theorem on the lines of the Theorem 2.12.18 for the case of uniform asymptotic stability.
214
THEOREM
2.12.25
Let the space
be locally compact and separable.
is uniformly asymptotically stable with an open set
M~X
for some on
X
N
6 > O,
Then a closed set
N
containing
if and only if there exists a continuous f~nction
S(M, 6)
defined
~(x)
and having the following properties: (i) #(x) = 0
for
> 0
x~M,~(x)
for
x/M,
(ii) there exist continuous strictly increasing functions ~(0) = ~(0)~
such that
~(o(x,M))
.< ~(x)
.< S ( ~ ( x , M ) ) ,
(iii) there exists a sequence of closed sets
0
n= 1 E
n
= N, such that given any
~(x) > ~
if
~(r), S(r),
x IE n ,
~ > 0
a~d on every
{En} ,
En ~
I En+l '
there i8 an
En,~(x)
no
such that
i8 bounded,
o
(iv) #(xt) .< e-t~(x)
The conditions can easily be shown to be sufficient.
To prove the
necessity we need the followi~g Ipmma.
2.12.26
LEnA
Let
f(r,x)
be a function from
compact separable metric space. (0,1] • X.
X
Let
f(r,x)
Then there exist two functions
(0,i] • X § [0,+~),
H(r)
and
G(x)
6 H(r)
9 G(x)
defined on
(0,1]
(and may even be chosen continuous), such that f(r,x)
X
is locally
be bounded on every compact subset of
respectively, which are bounded on compact subsets of
respectively
where
and
x
(0,1] and
215
Proof:
Since
X
compact sets
Un
is locally compact and separable we can find a sequence of such
that
Un~Un+
1
H(r) = sup{f(r,x) + l : x ~ U n , o and
,f(r,x) G(x) = supt H(r)
required properties.
i ~ r > 0}. Indeed
where
The above defined
H(r)
attraction of the set S(M,a)~
A(M).
M.
Since
We might choose
T(r,x) = inf{T > 0 : x t ~ S ( M , r ) any compact set each compact Kt~S(M,r)
for
and
t >. T.
M
y~S(M,6)
a .< 1. for
For each
t >. T}.
r > 0,
x
A(M)
x.
have the
be the region of
We assert that
T(r,x)
N
such that
is bounded on
To prove this we need to show that for
y+(y)~S(M,
9
~ > 0
define
r.
S(M,~)
= S(xT(x),g)
is a compact neighborhood of
G(x)
r ~(0,~)
there exists a
implies Since
and
is an attractor, there is an
T > 0
such that
r).
For
M,
there is a
x~A(M),
is open, we can choose a
is compact and contained in
N
Let
We note first that by stability of
xT(x) ~ S(M,6).
S(xT(x),g)
H(r)
is defined here as a step function.
and for fixed
K~A(M),
such that
such that
K~A(M)
We now define
i + 1 >. n >. I} r o r
Proof of necessity of Theorem 2.12.25.
2.12.27
> 0
X = n_UI Un .
and
S(M,6).
choose
~ > 0
T(x)
such that
Then its inverse image
(-T(x))
has, moreover,
the property,
that
X
NT(x)~ T(x)
S(M,~). and a
t >. T(x).
Thus we have in fact shown that for each
p(x) > 0,
such that
y ~ S(x, p(x))
Consider now the open cover
implies
{S(x, p ( x ) ) : x ~ K }
x~A(M), y t ~ S(M, r)
there exists a for
of the compact set
K.
By the Borel Theorem, there exist a finite n,,rber of sets, say, S(x I, P(Xl)),...,S(Xn, P(Xn)) Then
x~ K
implies
which cover
xt ~ S(M,r)
For any given integer
for
K.
We can now choose
t >. T.
i n > -- ,
define
T = max(T(Xl),...,T(Xn))
216
~n(X) = sup {p(xT, S(M, i))
We assert that P > 0
~n(X)
such that
. exp(z): T >. 0}.
is continuous on
S(x,p)
A(M).
To see this, note that for
is a compact subset of
A(M),
there exists a
T > 0
s uch
that
p(y~, s(M, ~)) = o for
y~ S(x,p)
and
T >. T.
Therefore,
JCn(X) - #n(y) l = Jsup{p(xT,S(M, i))
- sup(p(yT,S(M, i))
if
y~S(x,p)
we h a v e
. exp (T): 0 .< T .< T}
9 exp (T): 0 .< r .< T} J
This implies that
l~n(X) - ~n(y) J .< exp(T)
sup {p(xT,y~): 0 .< z .< T}
Using the continuity axiom we conclude that the right hand side tends to zero as p(x,y) § 0.
Thus
~n(X)
is continuous on
A(M).
This
important property
~n(Xt) .< exp (-t) ~n(X),
To see this, note that for
t ~ 0
for
t > 0
#n(X)
has further the following
217
~(xt) = sup{p(x(t + T), scM, ~)) = sup{~(x~,
S(M,
(O: 9 >. 0}
exp
i)) exp (T - t): 9 ~ t}
-- exp (-t) 9 sup{p(xT, S(M, i)) exp (T): T >. t}
.< exp (-t) 9 ~n(X)
as
t >~ 0
We now note that
#n(X)
as
=
sup{p (XT, S(M, i)) exp (T): 0 .< T .< T( I, x)}
~CxT, scM, ~-)) = o
for
9 >. TC~, ~ ) .
~us
%n(x) .< exp(T(l,x)) sup{p(xT, S(M, i)):
Since the function f(r,x) of Lemma 2.12.26, ~n(X) / H(I)
exp(T~,x))
9 >.0}
.
has the properties of the function
we can choose a function
H(r),
is uniformly bounded on each compact subset
such that K~A(M).
We now define co
~(~)=
I ~=~ ~n(~) / H
n!,
where
n o >~
0
Then
~(x)
is continuous on
A(M)
and has
~(xt) .< exp (-t) #(x) I
Note that stability,
#(x) = 0 ~(x) § 0
8(r), 8(0) = 0
for if
x~M,
and
D(x,M) -~ O,
such that
~(x) .< 8(p(x,M))
~(x) > 0
for
x~M.
By uniform asymptotic
there is thus a strictly increasing function
218
Further if some
6 .
p(x,M) ~ e > 0, And hence
then for sufficiently large
#(x) > ~ > 0.
n,
#n (x) ~ 6n > 0
for
Thus there is a strictly increasing function
n
e(r), ~(0) = 0 ~ such that
#(x) % u(p(x,M)).
k < inf{~(x):
We now choose
k > 0
such that
p(x,M) = e}
Consider the sets
Pk = {x~A(M):
#(x) < k } ~
S[M,~],
Sk = {x~A(M):
#(x) = k } ~
S[M,e]
and
Then as shown in Theorem 2.12.23, of all those trajectories x ~Pk
we can define
continuous.
in
T(x)
A(M)
Sk
is the section of the flow in
which are not in
by the requirement that
M.
For each
x~(x)~ Sk.
A(M)
consisting
x (A(M),
Then
T(x)
is
Now define
~(x) = #(x)
for
X~Pk
for
x ~ Pk
'
and
(x) = k e x (x)
This
~(x)
has all the properties
required in the theorem as may easily be verified.
Remark
2.12.27
Note that if in the above theorem we assume construct property of the
~(x)
as in Theorem 2.12.23,
(ii) in the above theorem. ~n(X)
would be superfluous
then this
~(x)
M
to be invariant and would need not satisfy the
Indeed if that were the case, then construction and then uniform asymptotic stability will imply
equi-attraction which is indeed not the case.
219
2.12.28
Notes and references The notion of equi-stability seems to be new.
Theorem 2.12.19 seems
to pave the way for the use of the theory of parallelizable flows in studying various problems on asymptotic stability, especially its connection with the existence of the so-called Liapunov Functions.
The exposition in this section is not complete,
but is more general than that of Antosiewicz and Dugundji. Theorem 2.12.23 similar but better than a well known theorem of Zubov ([6], Translation page 52), and is in line with results in Section 2.7. also end of Section 2.7 for further notes.
See
220
Higher prolongations and stability
2.13
The first positive prolongation,
and the first positive prolongational
limit set have been shown to be useful in characterizing various concepts in dynamical systems.
Notable applications being the characterization
of stability of
a compact set in a locally compact metric space, and the characterization dispersive flow.
The first positive prolongation may be thought of as an extension
of the positive semi-trajectory. 2 (E ,R,~) which is geometrically 2.13.1
of a
For example consider a dynamical system described by the following figure
Figure
Y.
The first positive prolongation of the point semi-positive
trajectory
trajectories
yi,Y2,Y3 ,
y+(x), and
Y4"
x
in the figure consists of the
the equilibrium points
0,P, and
Q,
In a way, to get the prolongation
of a point, we might
find ourselves arguing that we move along the positive semi-trajectory the equilibrium point
0.
a trajectory which leaves to
YI'
So we transfer to the point 0,
e.g., we can transfer to
then we approach the equilibrium point
P.
0. YI
From
0
or
Y3"
and the
and approach
we transfer to
So we transfer to
If we transfer P,
and thence
221 to a trajectory leaving
P,
and so on.
If indeed this procedure were laid down to
define the prolongation of a point, then notice that we would have to include the trajectories
Y5
prolongation,
however, excludes the trajectories
of
x.
and
T6
in the prolongation of
x. y5
The definition of a and
~6
from the prolongation
If, however, we wished to include these in a prolongation,
must change the definition of prolongation,
then either we
or in a sense introduce other prolongations
which will do precisely what we did with the intuitive reasoning above.
Just as the
first positive prolongation is in fact a meaningful extension of the positive trajectory,
the 2nd and bigher prolongations which will be presently introduced,
will be shown to be meaningful extensions of the first prolongation. The description of higher prolongations of two operations
The operators
2.13.2
If
S
on the class of maps from
and
we define
~F
SF(x) = u{Fn(x):
Fl(x)
= F(x),
and
into
2x.
by
ur N(x) }
denotes the neighborhood
2.13.4
X
9.
Pr(x) = n { ~ u ) :
N(x)
where
and
F:X § 2x,
2.13.3
where
S
is facilitated by the introduction
filter of
x.
Further,
~F
is defined by
n = 1,2,...}
Fn(x) = F(r(n-l)(x)), n = 2,3,...
In the sequel the following lemma will prove useful.
2.13.5
LEMMA For any
and
r:x § 2 x,
~r(x) is the set of all points
such that there are sequences
{Xn }'{yn}
in
Further,
$F(x)
Xl,X2,... ,xk, Xl=
x, ~ = y .
xs
i8 the set of all points with the property that
x, yn ~ F(x n) ,
y~X
and
y~X
Xn § x, Yn § y"
such that there are points
Xi+l~ F(xi),i = 1,2,...,k-1,
and
222
The proof is immediate and is left to the reader. The following lemma gives some elementary properties of the operators D
and
2.13.6
S.
LE~
(a) 9 2 = 9, (b) If
and
M~X
S 2 = S.
Thus
D
and
S
are idempotent operators,
is compact, then
Dr(M) = U{Dr(x):xs
is c~losed, (c) If
v = ~(x) is a continuous real-valued function on
~(y) .< ~(x),
y ~ Dr(x) U Proof:
(a)
Dr(x)C
D~r(x).
such that {
Let
and
Yn § y"
n r( ~ ), }, {yk }, yk~
then
~(y) .< ~(x)
For each
~F(x)~
= Dr(x).
then indeed
whenever
y ~ ~r(x).
then there are sequences
ynn ~ F( xnn).
Clearly
~r(x)
y ~ Dr(x),
such that
We have thus proved that shows that
If
y~r(x),
If
y ~ r(x),
such that
~r (x).
r:x § 2 x.
Xn § x
{xn}'n {yn }"
whenever
X,
Hence
Further Dr(x).
{Xn}, {yn }, yn ~ Dr(x),
Xk' Yk' Yk % DF(Xk)'
n + Xk' Ykn § Yk" Xk Xnn § x,
Thus
there are sequences
Now consider the sequences Ynn + y"
and
Thus
y ~ DF(x).
This together with the previous observation
9 2 = 9.
Proof of
S 2 --S
is
even simpler and
is left as an exercise. (b) a sequence that
Let {Xn }
xn § x ~ M . (c)
that
Xn § x,
If and
{yn } in
M
Thus
be a sequence in such that y~Or(x)
y~ Dr(x), Yn -> y"
Dr(M)
yn ~ DF(x n). = Dr(x)~
such that Since
Dr(M)
continuous we get by proceeding to the limit
Then there is
is compact, we may assume
This shows that
then there are sequences It is given that
M
Yn § y"
DF(M) is closed.
{xn} , {yn }, yn ~ F(Xn) ,
~(yn ) .< #(Xn). ~(y) .< ~(x).
If
Since y s
~
such
is then there
223
are points
x = xl,x2,...,x n ~ y
such that
xi+ IE r(xi),i = 1,2,...,n-l.
~(y) = ~(xn) ~ ~(Xn_ l) ~ ... ~ ~(x 2) ~ ~(x l) = ~(x).
Hence
This completes the proof of the
lemma.
DEFINITION
2.13.7
A map
(a)
A map
F:X + 2X
(b)
Given
r:x § 2X,
A map
Sr is transitive.
will be called a cluster map if
r:x § 2 x
Dr = F.
DEFINITION
2.13.10
A map
property:
will be called a
r:x + 2X
For any co~pact set
KCX
m~
c-c
such that
if it has the following
x ~K, one has either
F(x)~K,
F(x) ~) ~K # ~.
THEOREM
2.13.11
Let the space
where
X
be locally co~pact.
Let
r
be a
c-c
map.
Then
is compact, then it is connected.
r(x)
Proof:
If ~,
F(x) ~
is compact, but not connected, then we can write
are non-empty compact disjoint sets.
can choose compact neighborhoods UiO
r2_-r.
is transitive if and only if
DEFINITION
2.13.9
if
SF = r.
EXERCISE
2.13.8
or
will be called transitive~ if
r:x § 2X
U 2 - ~.
~UI ~ r(x) = r is connected.
Note that
U1, U 2
r(x)(~ U I # ~,
of but
contradicting the fact that
MI, ~
Since
X
F(x) = M I U
~,
is locally compact, we
respectively such that
UI=~ F(x),
as
r(x)
c-c
is a
~ ~ U I ~ ~. map.
Thus
However, F(x)
224
THEOREM
2.13.12
Let the space is compact, then
MCX
M,
X
be locally compact.
Proof:
Sufficiency.
~(M)~)M
always.
such that
~U
w
r
be a
of
M
such that
Note that for any
c-c
map
F,
x ~ F(x).
n
.
Then there is a sequence of neighborhoods
Then
DF(M)~'O
F(%)
~_~
and sufficiency
Necessity.
Indeed assume, if possible,
that
U
is a
{Xn} , x n + x ~ M,
c-c
assume that as
n
DF(M) = M.
M,
such that
{W }
Zn~ F ( X n ) ~
§ z ~ ~U.
Thus we have proved that
of
that there is a neighborhood M,
and a sequence
F(W)~U.
X
{yn }, yn # r(Xn)
z ~ Pr(x)C
F(x n) 0
M,
such that
~U # ~.
Since
~F(M),
of
Then there is a
x E U, n = 1,2, . . . . n
~U, n = 1,2, . . . .
U
We may assume without loss of
is locally compact).
we must have
But then
but
~U
But then since
Consequently,
there
is compact, we may
z~M.
A contradiction,
This proves necessity and the theorem is proved.
We remark now that help build families of
2.13.13
= M.
n
(because
r(Xn) C~:U ,
{Zn} , z
U
Indeed we may assume that
map and
is a sequence
W
is compact
y n ~ U,n - 1,2, . . . . F
of
is proved.
such that for every neighborhood
sequence
{Un} n
~F(M) = M,
generality
U of
Hence
Consider now a sequence of closed neighborhoods
= M.
If
F(W)~U.
n
F(Wn)~U
map.
c-c
if and only if for each neighborhood
DF(M) = M
there is a neighborhood
Let
c-c
c-c
maps have also the following properties, which
maps.
LEMMA (i) Let {F } c-c
(ii) If (iii) If
~{ A,
be a family of
c-c
maps
Then
F =UF
map. rl, F 2
r
is a
are c-c
c-c
maps,
map,
then so i8 the map
then 8o are
SF
and
F = r I o r2. Dr.
is a
225
Proof.
(i)
that F ,
Let
F(x) ~
be a compact set, and
~K # @.
such that
F (x) ~
K
Indeed if
Fe(x) ~ K .
~K # ~.
Thus
need consider the case that
F2(x)~
I(K),
Fl(Y) ~
~K # @,
F1 o F2(x ) ~ SF
F2(x)C
then there is a and since
~K # ~.
and and so
I(K),
Thus
y ~F2(x)
Fl(Y)~
F~ F
Let
x s
where
without loss of generality that {xn } in
K,
F(xn) ~ K ,
Xn § x, and
K
~F(x) ~ ~K # @.
This proves that
z
~F
K,
compact.
c-c
But then
F
is a c-c map,
To show that
DF(x) ~ K,
But then
This is
and
(iii) If
yn §
We
we have
DF
we may assume
then there is a sequence y ~K,
{Zn }' ZnC F(Xn),
We now prove the following interesting
Yn ~ F(Xn)" and
Since
Zn s ~K.
z s DF(x),
Since
so that
map. theorem.
THEOREM
2.13.14
Let be a
F2(x)
~F(x) C K ,
§ z ~ 3K. is a
(ii)
K
FI(Y)~K.
If
{Yn }' Yn ~
n
and
(i) and (ii).
is compact. If
map.
F 1 ~ F 2 (x) ~ K
If
there is a sequence
is compact, we may assume that
we must prove
map, we must have
c-c
is a c-c map.
x ~ ~(K).
c-c
x~K,
FI(F2(x) ) = F
and a sequence
Xn~K ,
is a
such that
F = F1 o F2
F(x)~K,
is a
where
is indeed a c-c map by the assertions
is a c-c map.
~K
x~K,
~K # ~,
If
then there is at least one map, say
[i o F2(x) = F I ( F 2 ( x ) ) ~ F 2 ( x ) '
so, because
then
F(x) ~ K ,
Since
F(x) ~
xs
X
be locally compact, and
M
a compact subset of
X.
Let
F
map which is moreover a transitive map as well as a cluster m~p.
c-c
Then
F(M) = M
if and only if there exists a fundamental system of compact neighborhoods
{Un}
M
of
Proof:
Let
such that {W }
F(U n) = U .
n
be a fundamental system of neighborhoods
of
M.
Since
n
~F(M) = F(M) = M,
by Theorem 2.13.12,
such
Wn .
that
compact.
r (K)~ Setting
Now n o t i c e
F ( K n) = Un ,
that
we g e t
we h a v e r (K)
compact neighborhoods are
a fundamental
closed system
Kn
of
M
a n d may b e c o n s i d e r e d of
as
compact neighborhoods
226
of
M
such that
F(U n) = U n.
as
F
is transitive.
This is so because
F(U)
= F(F(Kn)) = F(K n) = U n,
We shall now apply the theory constructed above to a dynamical system (X,R,~).
2.13.15
The higher prolongations (Definition). Consider the map
through each point
x ~ X.
Further, notice that S y+ = y+ .
x.
it is denoted by
~ 2x
Then since
which defines the positive semi-trajectory y+(x)
so that
+ ~$7 + E ~y+ = DI,
and call
y+
of
x.
D:,
D +.
Indeed
D +1
is a cluster map
D2
we define
and call
D:+I = ~ D : ,
This defines a prolongation of
(x)
and call x
as follows:
D+~ = DS D: -i" < =,
D:(x)
we set
If If
~ a
as
D
is idempotent, but
We, therefore,
D+(X)n
D:(x)
of
x
then having defined
for every ordinal
This defines for each
x ~X,
D:(x)
for any ordinal
~
are closed.
THEOREM Let
~
+ (i) D~
(ii)
D +2
be the first uncountable ordinal number. Then =
{D+: a
Using
D:_I, we set D+e
for every
e.
We give below some properties of these prolongations.
2.13.16
n.
a prolongation
Notice ~hat each of the map considered above is a c-c map. prolongations
Having
for any ordinal number
is not a successor ordinal, then having defined SD +
x.
to be the nth-prolongation
for any positive integer
is a successor ordinal,
D
consider the map
as the 2nd prolongation of
transfinite induction, we define a prolongation a
as the first positive
This is clearly the same as defined in Section 2.3, where
and denote it by
defined
is a c-c map.
is a transitive map, i.e.,
D~(x)
D I+ is not transitive as simple examples will show. DS
+ 7
is connected,
y+ (y+ (x)) = y+ (x),
We now set
prolongation of
r
P2 > P3 > "'" > 0.
To the right
of
{Pi },
and
PI'
Between
we introduce a sequence PI
and
P2'
{P2k }
n + P2k" P2k
between
PI
§ P2"
and
between
P2k
Between
P2
P2'
and and
P2(k-l)' P3
Then for each such that,
PIn § PI"
P2k'
we introduce a
i 2 P2k > P2k > "'" > P2k'
and then between any two successive points we introduce a monotone
Having introduced a suitable sequence between say Pn
PI'
we first introduce a sequence of points as
decreasing sequence converging to the point on the left.
sequence between
>"'>
{P2n },
2n
n
and
PII > PI2
we first introduce a sequence
PI > P21 > P22 > P23 > "'" > P2' P sequence
such that
and
Pn+l
Pn-i
We now proceed inductively. and
Pn"
we introduce a
similar to the one introduced between
Pn-i
and
Pn,
then between each pair of successive points of this sequence, we introduce a monotonic decreasing sequence converging to the point on the left. introduce the dynamical system on the real line.
Now we are ready to
Each point of the countable set of
points introduced on the line is an equilibrium point.
There are no other
230
equilibrium point, and the motion between any two successive equilibrium points is from left to right.
It is easy to see, that each point
{P }
of the first sequence
n
introduced above has the following property. order
i,
but not stable of order 2,
order i), but is not stable of order 3, of order
n+l.
The point
stable of order
0
P3
P1
is not stable,
P2
is stable of
is stable of order 2 (and hence also of Pn+l
is stable of order
is stable of every integral order
n n,
but not stable but is not
~.
If we consider example 2.13.21, then it is an easy matter to show that no continuous scalar function satisfying conditions of Theorem 2.12.10 exists for the uniformly stable equilibrium point
-i.
An example in the plane, e.g., example
1.5.32(v) and figure 1.5.35 can be used to establish the same thing. the point
0
In fact, even
in example 2.13.24 which is stable of every integral order
n
is such
that no continuous function satisfying conditions of Theorem 2.12.10 can exist for this point.
The question obviously arises, as to what are the implications of the
existence of a continuous function satisfying given closed set
M.
the conditions of Theorem 2.12.10 for a
The answer for a compact set
M
in locally compact spaces
X
is given by the following theorem.
2.13.25
THEOREM Let
X
be locally compact, and let
MC.X
be compact.
Then the following
are equivalent: (i) There i8 a real-valued function 8atis~ing conditions of Theorem 2.12.10 which is continuous in some neighborhood of
(ii) M
M,
possesses a fundamental system of absolutely stable compact ne~hbor-
hoods j
(iii) M
i8 absolutely stable.
We shall need the following ipmm,, whose proof is ~-,,ediate from the definitions.
231
LEP2~IA
2.13.26
Let
v = r
Theorem 2.12.10. the set
If
{Ua:a > O}
us = {x~X:r
be a real valued function satisfying conditions of M
is compact, and the space
Let
m
M,
where
0,
U
be a compact neighborhood
and
{U :0 < s < m },
0
U s = {x~ X:r
~ ~},
neighborhoods
of
X,
S
M.
We will now show that each
by means of
where
0
To do t h i s ,
~(x) =r
for
U
is absolutely stable.
we c o n s i d e r
X~Um
,
and
the function ~(x) = mo
We shall
~(x)
for
x~U m .
o
o
This is a continuous
function which is decreasing along the trajectories.
0 < a < 8 < mo,
is indeed a compact neighborhood of
UB
decreasing along the trajectories, D (U s) # Us,
then there is a
there is an
x ~Ua,
~(y) ~ ~(x) ~ a,
of
is a fundamental system of compact, positively invariant
show this by u s i n g lemma 2 . 1 3 . 6 ( c ) . defined on
is locally compact, then
is a fundamental system of ne~hborhoods of
Proof of Theorem 2.13.25: M.
X
and a
we get
B > 0
such that
y ~D~(x) O SU B.
and, on the other hand,
D~(U s) = Us, i.e., each (ii) implies (iii) implies
y ~ D~(x),
then
+ D~ U s ~ U
Since
~(y) B.
r
~(x). + D~
Since
is If is a c-c map,
On one hand, therefore, ~(y) = B > s.
Us
is absolutely stable.
(iii).
This is ~mmediate.
(i).
Us.
For
Using Theorem 2.13.14 (since
This contradiction shows that
+ D~
is a c-c map which is
moreover a transitive as well as a cluster map) we first construct a fundamental system of absolutely stable neighborhoods
UI
, n = 0,1,2,...,
such that
2n
Ul ~(Ul 2n
).
We now extend this system I of absolutely stable compact neighborhoods
2n-I
to one defined over the diadic rationals, = J/2 n, n = 0,1,2,...; j = 1,2,...,2 n, corresponding
i.e., numbers of the type in such a way that (a) the compact neighborhood
to any diadic rational is absolutely stable,
(b) if
s < B
are diadic
232
rationals,
then
U~a
I(Us),
(c)
M = O{Ue:~
possible by using Theorem 2.13.14. diadic rational}. ~(xt) ~ ~(x) .
Clearly
Now if
~(x) = 0
This is so, because if
x6 U ,
xt6 U ,
on
we assume that this is not true.
{x n}
in
UI
such that
diadic rationals Xn ~ U~ 1 ,
But
U i~
such that
U 2C
2.13.27
x ~ U 2-
such that Since
[(U 2 ).
U i
l(U i) ,
which contradicts
x ~ M.
and
is closed,
Then
Xn + x.
~(x) x~Ue,
and a sequence
then we can choose
Then for large x
§ x,
then choose 1
then
is continuous
x~U
n
~ > ~x' Xn~U
t > O,
,
is positively invariant,
~ < ax'
e < ~i < ~2 < ~ X "
If
U
Then there is an If
Indeed this is
v = ~(x) = i n f { a : x 6 U
Finally, to see that
If again
a > ~i > ~2 > ex"
define
then since
~(Xn) § a # ~(x) = ax"
el' ~2'
whereas
contradiction as al,a2,
~(xt) $ ~(x).
X~Ul,
if and only if
we have U I,
hence
diadic rational}.
for large
x n' 9
This is a
el diadic rationals n,
whereas
x6Ue2.
This completes the proof of the theorem.
Prolongations and stability of closed sets. Although Theorem 2.6.6 gives an excellent characterization
of Liapunov
stability of compact sets in locally compact spaces, a similar characterization not available for closed (noncompact)
sets, or in general metric spaces.
is
Indeed we
defined several concepts of stability of closed sets in Section 2.12, and it appears that if we are to reach at a characterization we must first change the definition of prolongation for noncompact sets. The following lemma gives an insight into what may be done.
2 .13.28
LEMMA If the set
MCX
is compact, then
D~(M) =O{y+(S(M,6)):
6 > 0 }
The proof, is elementary and is left as an exercise.
We only recall that
D~(M)
is
by definition the set U {D~(x):x ~ M}. It is now to be noted that M,
which are not compact.
DiS)
need not even be closed for closed sets
And further, in general, if for any closed set
M,
we
233
have
DiS)
= M,
then the set
M
need neither be stable or equi-stable.
W e D.OW
introduce the following definition.
DEFINITION
2.13.29
M in
Given any non-empty set
X,
we shall call the set fl (y+(g(M, 6 ) ) : 6 > O}
as the uniform (first) (positive) ~rolongation of 9 Lemma 2.13.28 says that if
M
M
and denote it by
D+(M). U
is compact, then
D+(M)u = D~(M).
The uniform prolongation has further the following properties
LEM94A
2.13.30
(i) For any non-empty set
Me
X, D+(M)
is closed and positively invariant,
U
(ii) D+(M) = {y ~X:
there are sequences
U
such that
P(Xn, M) § 0,
ciii
and
x t n
{xn}
in
X
and
{tn}
in
R+
§ y}, n
U
The proofs are !mmediate consequences of the definition. The uniform prolongation is useful in characterizing
the equi-stability
of
a closed set.
2.13.31
THEOREM A closed set
M CX
i8 equi-stable if and only if
This is an ~mmediate consequence of the definitions details to the reader.
D+(M) = M. U
and we leave the
We note that Theorem 2.6.6 of Ura falls as a corollary of
this theorem, when we note Proposition 2.12.2.
2.13.32
Notes and References T~otion
of higher prolongations
is due to Ura [4] who also showed their close
connection with stability and introduced the notion of stability of order exposition here is based on Auslander and Seibert
[2].
enumeration of Auslander and Seibert for prolongations.
a.
The
We have followed the Ura's enumeration is different.
234
For example the 2nd prolongation of Ura is is what Ura labels as
D
where
~
D +I o D +I.
The prolongation
is the first countable ordinal.
D +2
here
Ura [4]
(page 195) also showed that the prolongations introduced here are the only ones which lead to different concepts of stability.
The notion of a c-c map is one of
the axioms of Auslander and Seibert for an abstract prolongation.
We show that this
is the concept which leads to various properties which are needed for results on stability.
Thus sections 2.13.2 ~o 2.13.14 are independent of the notion of a
dynamical system.
For example Theorem 2.13.12 contains as a particular case Ura's
characterization of stability : Theorem 2.6.6. and Seibert.
Theorem 2.13.25 is due to Auslander
235
H ~ h e r prolongational limit sets and generalized recurrence.
2.14
In Section 2.3 we introduced the first positive prolongation, and the first positive prolongational limit set, and we studied some of their properties. We introduced the higher prolongations in Section 2.13. J
We shall now introduce
also the higher prolongational limit sets and study some of the properties. We shall then use these to characterize the notion of generalized recurrence introduced by Joseph Auslander.
2.14.1
DEFINITION The first positive prolongational limit set
is defined by
x n § x, tn § + %
such that denoted by 2.13.2,
there are sequences
J+(x) = {ys
J+(x).
and
we define for any
{xn}
in
of any point X,
and
x~X
{tn}
in
R
In Section 2.3 this set was simply
x n tn + y}.
Using now the operators
Jl(X)
S
and
~
introduced in Section
x~X
=
and if
a
is any ordinal number, and
J~ +
J+(x)a = D(U{SJ~:8 < a})
(x)
has been defined for all
i, then y ~ J (x) if and only if there are sequences {x }, k a n {yn},Yn~ j Bn n (Xn),X n § x , Yn § y" where Bn are ordinal n ~ b e r s less than % kn
are positive integers.
rl=r.
Recall that for any map
r:x § 2X, rn = r o rn-l,
and
where
236 We leave the proof to the reader.
2.14.3
It is also to be noted that
LEP2~4 For any ordinal
{Xn},{y n}
in
xn + x ,
Yn §
if and only if there are sequences k and Yn~DSn(xn), where for each
e
k
i8 a positive integer. (In this lemma
e > I, y E D + (x)
X such that
n n,
i8 an ordinal less than
8n
and
n
and hereafter
is simply written as
D+ e
D
to facilitate the use of upper indices.)
e
The following lemma now expresses some elementary properties of prolongatioDs and prolongational limit sets.
LEM~A
2.14.4
For any
x ~ X,
(i) J e (x)
and any ordinal
e
is closed and invariant,
(ii) J (xt) = J (x)t = J (x), for all e
e
e
(iii) D (x) = y+(x) U J e
t~R,
(x), e
(iv) De(x)
is closed and positively invariant,
(v) If the space
X
is locally compact, then
D e (x), Je (x) are connected,
whenever they are compact (if one is compact, then so is the other), e~d if D (x)
(J (x)) is not compact it does not possess any compact components.
e
e
Proof:
(i)
Jl(X)
has been proved to be closed and invariant
(Section 2.3).
J (x) e
is closed by construction. 8 < e.
Let
Y~J e ( x ) '
8n < e
and
kn
and
To prove invariance, let t~R.
be invariant for all k Xn + x , Yn § y' Y n ~ J s n ( x n )' where n k Then by the induction hypothesis yn t ~J~n(xn).
Let
is a positive integer.
Js(x)
n
Since
yn t § yt,
we have
yt E Je(x),
and the result follows.
is a trivial consequence of invariance of that
Jl(Xt) = Jl(X)t
that
Js(xt) = Js(x)t
Je(x).
To see
(ii) Je(x)t = Je(x)
Je(xt) = Je(x)t,
note
(this is an easy consequence of the definition). Now assume k for all 8 < e. Let y eJe(xt). Let yn 6 JSn(xn t) (where n
237
< ~,
t positive integers) such that x t § xt and Yn § y" Now n k k n Xn § x, and yn(-t) ~ Jsn(xnt)(-t) = Jsn(xn ), by the induction hypothesis. Since n n yn(-t) + y(-t), so y ( - t ) ~ J (x) and y ~ J (x)t. Hence J ( x t ) C J (x)t. Now
and
Ja(x)t = J a ( x t ( - t ) ) t C J
Section 2.3 we proved t h a t
true for all
8 < ~.
(xt)(-t) = J (xt).
This proves (ii). (iii)
Dl(X) = y+(x) U J l ( x ) .
Notice that if
In
Now assume t h a t t h e r e s u l t i s
Y'6D~(x'),
t h e n by ( i i )
Y'Eu
or k
y'~ J~(x')
where
m ~ k.
Now if
y~D
(x)
'
(where many
~
8n < ~, k n n,
then
positive integers).
(x).
y ~J
If
y ~Dl(X ) = y+(x) U J l ( x ) .
If
Yn ~ y + ( x )
~
let '
x ~
~ x, Yn § y' Yn 6 D~n(x ) n
~n
Yn ~ J8 n(xn ) n
(s ~
k n)
for infinitely
many
n,
for infinitely
then
n
I n e i t h e r case
y 6u
U J a (x).
D~(x) C y+(x) U J a ( x ) .
Since
D (x) = u
This completes t h e p r o o f of ( i i i ) .
~ J (x).
n
y+(x) U J a ( x ) l ~ D (x)
i s an immediate consequence of ( i i i ) ,
and
D (x)
Thus
i s o b v i o u s , we have (iv) P o s i t i v e i n v a r i a n c e
i s c l o s e d by d e f i n i t i o n .
(v) The
p r o o f of t h i s s t a t e m e n t may e a s i l y be c o n s t r u c t e d by the method adopted f o r t h e p r o o f of a s i m l a r
s t a t e m e n t about
in Section 2.3.
2.14.5
A+(x)
i n S e c t i o n 2 . 2 , and about
Dl(X)
and
Jl(x)
This we leave to the reader.
Exc~8~ Show that for any ordinal
J~(x) = ~{D (xt):t~ R}
We now recall some of the notions of recurrence that have occurred earlier, namely, a rest point, a periodic trajectory (or periodic point), a positively or negatively Poisson stable motion (or point), a non-wandering point. these concepts are respectively equivalent to
x ~A-(x)~
and
Now let such that
x~J~(x) V
x = xt
which is equivalent to
for all
We recall that
t ~ R,
for all
x~X
and all
or
X~Jl(X).
denote the class of real-valued continuous functions
f(xt) ~ f(x),
x~ A+(x)
t > 0.
f
on
X
238
DEFINITION
2.14.6
Let all
R
f ~ V,
denote the set of all points
and all
t ~ 0. R
x ~x
such that
for
f(xt) = f(x),
will be called the generalized recurrent set.
We have immediately
2.14.7
LE2~Z4 R
Proof-
includes the non-wandering points in
Let
x ~ Jl(x).
there are sequences and since
f
Let
x § n
t > 0,
xt, t § n
and +~,
f ~ V. and
X.
Then indeed
x t § x. n n
x ~ Jl(Xt),
Then indeed
and
f(Xntn)
~< f(Xn),
is continuous, we have
f(x) ~ f(xt)
As
f(xt) .< f(x)
holds by hypothesis, we get
f(xt) = f(x).
Thus
x~R.
Now we have
THEOREM
2.14.8
R
is closed and invariant.
Proof,
That
T > 0.
Then for any
Secondly, let
R
is closed is clear. f~ V
T < 0,
f((xT)t o) < f(xT).
and
To see invariance, let first
x ~ R,
f((xT)t) - f(x(T + t)) = f(x) = f(xT). xT
Define now
R.
Then there is an
g@V
by
g(x) = f(x~)
g(xt o) = f((Xto)T) = f((x~)t o) < f(xT) = g(x).
f~ V
Thus
This contradicts
xT~R.
to > 0
and a
for any
and
x~X.
such that
Then
x~R,
and the
theorem is proved. It is clear that if d
are real numbers with
f ~V, c ~ 0.
then so are
tan f
and
cf + d,
where
This remark and the above theorem yield
c
and
239
LE~Z4
2.14.9
Let
all
xs
real
be real numbers.
a, bj a < b Then
if and only if
xER
Set
Va, b = { f 6 V : a
,< f(x) ,< b e
for
f6Va, b
andall
forall
f(xt)=f(x)
t.
From now on we shall assume that the space
X
is locally compact and separable.
The following theorem shows that in the class
V
of functions
there is a
function which is constant along any trajectory in the recurrent set, but is strictly decreasing along any trajectory which is not in the recurrent set.
THEOREM
2.14.10
There i8 an
fs
such that
(i) If
xCR,
then
(ii) If
x ~R,
and
Proof:
Let
C(X)
for all real
f(x) = f(xt)
t > O,
then
real-valued
functions on
the topology of uniform convergence on compact sets.
set in
V'.
Then
x~R
V' = V_I,I.
and
f(xt) < f(x). ~
denote the continuous
dense subset and so does
t,
Let
if and only if
Then
C(X)
{fk } , k = 1,2,...,
fk(xt) = fk(x)
for
X,
provided with
contains a countable be a countable dense
k = 1,2,...,
and
oo
real and
t.
Set
g =
Ig( x) l -< i.
fk(xt) = fk(x) {tn}
in
R+
[ k=l Thus
for with
.~
1 --{ fk" 2~ g~V'.
Since If
k = 1,2,..., t
§ + ~
Ifk(x) I .< I,
it follows that
g(xt) = g(x) , and so
such that
xER.
for all If
x~R,
g(x) > g(xt I)__
t > 0 ,
g
is continuous
then
there is a sequence
> g(xt 2)_
....
Define
n
!
f(x) = I e-tg(xt)dt"
Then indeed
f s V',
and
f
has the properties
required in the
J O
theorem.
We shall now obtain a characterization limit sets.
First,
the following lemma.
of
R
by means of the prolongational
240
LE2~4A
2.14.11
If
f ~ V,
and
y ~ n a ~),
then
f(y) ~ f(x).
This is an immediate consequence of Lemma 2.13.6 and the definition of
D (x).
DEFINITION
2.14.12
The set of all points R .
And we set
R' = U { R
x ~X
:~
such that
xtJ
(x)
will be denoted by
an ordinal number} .
The following theorem characterizes
R.
THEOREM
2.14.13
R = R'.
That is,
x~ R if and only if
for some ordinal
xs
~.
For the proof we need the following topological theorem.
2.14.14
THEOREM Let
X
be a locally compact, separable metric space and let -~
quasi order on
x.
Let
there i8 an
in
C(X)
(ii)
f(y)
f
x
and
y
in
such that
X
such that
(i) if
z ~ z',
x~
y
then
be a closed
does not hold. f(z)
Then
-< f(z'),
< f(x).
Proof of Theorem 2.14.13. We first show that any real f ~ V,
R.
t, x ~ J~(xt) = J (x),
we have
Thus for each proves
R'~
R'~
and only if
f(x) ~ f(xt). f~ V,
R.
x ~J
(x)
for some
and in particular this holds for However, we have
f(xt) = f(x)
To prove
Indeed let
R~R',
y6 D' (x) = ~ D ~ (x).
for all
f(xt) .< f(x)
t > O,
~
~
o n
Then for
t > O.
Then if
by definition of
and, therefore,
we define a relation Then
~.
X
is a closed quasi order on
x(R. by
y~ X.
f.
This x
if
Observe
241
that
xt~
x ~
y.
xt~ e.
x.
x,
x~X,
Note now that if If
Then
x ~ xt,
then
y(x)
"%us 9
x~R'
and
t > 0.
xtR',
and
xs
x ~ X + ( x t ) U J (xt).
case
2.14.15
whenever
x~
t > 0, Thus
Thus either
is periodic and so
If
X~Jl(Xt) C
and this is a contradiction.
then
x~D xEJ
y
but not xt~
(xt) (xt)
J (xt).
x.
for or
y'x,
we write
To see this note that
t > 0
and some ordinal
xEX+(xt) 9
In any case then
In the second x~J
(~t) = J (x).
The rest follows from Theorem 2.14.14.
Remark By a quasi order on
X
one means a reflexive, transitive, but not necessarily
antisymmetric relation.
2.14.16
Notes and References This section is almost exclusively a reproduction of results of Joseph Auslander
[3].
The only exception is the statement (v) in Lemma 2.14.4.
of the statement about
D (x)
follows from Theorem 2.13.11 as
However, the remaining parts do need a separate proof. to construct examples of c-c maps F(x)
has a compact component.
F
such that
F(x)
Notice that first part D
is a c-c map.
Indeed it is not too difficult is closed but not compact, and
242
2.15
Re lative Stability and Relative Prolongations.
We shall assume in this section that the phase space
2.15.1
X
is locally compact.
DEFINITION Given a point
prolongation of + D (x,U) = {y~X:
x
x~X,
and a set
with respect
.for each
nj
2.15.2
DEFINITION
such that
Given a compact set
X,
to the set
there i8 a sequence
t >~ 0 n
UC
x
n
MC-X,
U
is the set
and
~ > 0,
such that
and a set
y+(S(M,~) ~
M
with respect to
U
and
U,
M
is said to be
if given an
c > O,
U)CS(M,E).
Further, in definition 2.15.2, if
relative stability of
{tn} , Xn~ U,
the set
U~X,
It is clear that if in the definition 2.15.1, D+(x,U) = D~(x).
g i v e n by
x t § y}. n n
(positively) relativel~ stable with respect to the set there exists a
D+(x,U)
and a sequence
{Xn} ,
+ x,
the (first) (positive) relative
U U
is a neighborhood of is a neighborhood of
is the stability of
M
x, M,
then then
as defined in
Section 2.6. We have now the following theorem.
2.15.3
THEOREM A compact set
if and only if
2.15.4
M~X
M~D+(M,U).
is relatively stable with respect to the set Here
U~X
D+(M,U) = U{D+(x,U) :x ~M}.
Remark In the above theorem or definitions~
or the set condition
M.
If, however,
M~D+(M,U)
by
one obtains Theorem 2.6.6.
U~M,
the set
U
need not contain the point
x
then in the above theorem one may replace the
M = D+(M,U).
Further,
in case
U
is a neighborhood
of
M,
243 Proof of Theorem 2.15.3:
Sufficiency:
Let
M~D+(M,U),
be not relatively stable with respect to
U.
{xn}
{tn} , tn ~ 0,
in
U,
xn § x ~ M ,
may assume that
and a sequence
H(M,e)
is compact.
and let, if possible,
Then there is an
Thus the sequence
such that {x Z } n
to a point
ys
Then
y
~: D+(x,U) ~--- D+(M,U),
e > 0,
but
M
a sequence
P(Xntn, M) = e.
We
may be assumed to converg
n
y~M.
This contradiction
proves sufficiency.
Necessity:
Let
M
D+(M,U)~- S[M,e]
be relatively stable with respect to for arbitrary
~ > 0.
Hence
U.
Then clearly
D+(M,U)CO{S[M,e]:e
This
> 0} = M.
proves the theorem.
The concept of relative stability may be motivated by considering the example of a limit cycle
C
in the plane, with the property that all trajectories outside
the disc bounded by the limit cycle
C,
have C
as their sole positive limit set, and
all trajectories in the interior of the disc bounded by Notice that if
U
C
tend to an equilibrium point.
is the complement of the disc bounded by
stable with respect to cycle, then
C
U.
Notice also that if
C
C,
then
C
is relatively
is an asymptotically stable limit
is stable with respect to every component of
R 2 \ C.
These considera-
tions lead to the following definition and theorem.
2.15.5
DEFINITION Let
M~X
be compact.
We say that
M
is component-wise stable if
relatively stable with respect to every component of
M
is
x \ M.
We have then
2.15.6
THEOREM Let a compact set
MC-X
be positively stable.
stab le.
The proof is obvious and is ommitted.
Then
M
is component-wise
244
The comverse of Theorem 2.15.6 is in general not true.
To see this, we consider
a simple example.
2.15.7
Example Let
integer,
X~E
2
,(the euclidean plane) be given by The space
or y = 0}.
X
i X = {(x,y)~ E2:y = n'
X
any
is a metric space with the distance between any two
points being the euclidean distance between the points in system on
n
E 2.
We define a dynamical
by the differential equations
~ = O,
@ = 0
if
y = 0
~=
~=0
if
y#0
and
Then the set
I,
{(0,0)}~X
is component-wise stable, but is not stable.
The question now arises, as to when the converse of Theorem 2.15.6 is true. For this purpose the following definition is convenient.
2.15.8
DEFINITION Let
M~X
be compact.
We shall say that the pair
(H,X)
i8 stability-additive
if the converse of Theorem 2.15.6 holds for every dynamical system defined on ad~ts
x which
M Qsaninvariant set. In this connection the following theorems are important.
2.15.9
THEOREM The pair
(M, X)
is stability-additive
if
x\ M
(M, X)
is stability-additive
if
x ~ M
has a finite number of
components. 2.15.10 THEOREM
The pair
is locally connected.
245
The proof of Theorem 2.15.9 is immediate and is left as an exercise.
We
prove Theorem 2.15.10.
Proof of Theorem 2.25.10: Let system on
X
there is an that
and let e > 0
M
M
be a compact invariant set for a given dynamical
be component-wise stable.
such that
S[M,e]
Since
and hence also
only a finite number of components of
X~
M
M
H(M,e)
is locally compact, is compact.
can intersect
We claim
H(M,E).
otherwise, if an infinite number of components of
X\M
may choose a sequence of points
such that no two points of the
{xn}
sequence are in the same component. x
n
§ x~H(M,e).
Since
N,
such that
N
no
such that
Xn~ N
component of X \ M
X
61,62,...,6p
for
n ~ no
If
such that
2.15.11
CI, C2,...,C p
y+(S(M,6.)I ~ C i) c we get
Y+(x) C S ( M , ~ ) ,
last two cases
and hence all
then we
is compact we may assume that
X~
M.
xn
x,
say
Now there is an integer for
n ~ no
belong to the same
Now notice that every component of are the components of
X \ M
which
then by component stability we have positive numbers
last assertion, note that if
not intersect
H(M,c)
which is a contradiction.
6 = min (51,62,...,5p)
and hence
Since
is a subset of a component of
X\M,
H(M,~),
H(M,e)
H(M,~),
is locally connected, there is a neighborhood of
is an invariant set.
intersect
in
intersect
For
H(M,~)
y + ( S ( M , 6 ) ) C S(M,e), i.e., x~S(M,6),
or
S(M,~), i = 1,2,..,p.
x
then either
x~C.
l
is stable. for some
is an element of a component of
and hence is contained in
~+(x)CS(M,~).
M
If now
S(M,e),
or
x~M.
To see this
i = 1,2,...,p
X \ M
which does
In either of the
The theorem is proved.
Notes and References The concepts introduced here are from Ura [7].
We refer the reader to this
paper for a detailed discussion of these concepts and their relation to saddle sets. remark that one can in a similar fashion define the concept of relative asymptotic stability and discuss many similar problems.
We
246
CHAPTER 3 THE SECOND METHOD OF LIAPUNOV FOR ORDINARY DIFFERENTIAL EQUATIONS.
Dyne~ical systems defined by ordinary differential equations.
3.1
In this section we shall prove theorems for existence~uniqueness~ and extendability of solutions of ordinary differential equations. Consider the autonomous differential equation 3.1.1
~ ffi f(x)
where
x
point
x~
and
f
are
in the
n-vectors.
Under certain conditions, given any
n-dimensional Euclidean space
components of the vector
x ,
E ,
spanned by the
the differential equation 3.1.1 defines
a differentiable function (solution) 3.1.2 such
x = x(t, x ~
that ~(t, x~
3.1.3
= f(x(t,
of a certain interval
on all points
x~ (a, b)
which is such that
t ~ (a,b) and o
x~
3.1.4 If for any point
ffi
X(to, x ~
x~ ( E
.
there exists a unique solution 3.1.2
of 3.1.1 which satisfies 3.1.4, which is a continuous function of t
and which is defined for all
3.1.2 induce on
E
a dynamical system.
t ~ R ,
a flow satisfying the
x
o
and
then, clearly such solutions axioms i.i. 2
a n d thus define
247
In this section various sufficient conditions for an ordinary differential equation to define a dynamical system will be given.
Some theorems
are standard and may be found in any modern work on differential equations, others have a more specialized purpose. For the sake of convenience we shall derive these conditions for existenc%uniqueness~and
continuity of solutions in the formally more
general case of the differential equation 3.1.5
i-- f(x, t)
with initial condition 3.1.6
X(to, x ~
From now on we shall denote with
t ) = x~ o
o
x = x(t) = x(t, x , t )
a solution of
o
the equation
3.1.5 which satisf~e~the initial condition 3.1.6.
We shall proceed next with the p r o o ~ o f
the basic existence
theorems. The first existence theorem that we shall present is the classical result due to Peano and its proof is based upon the following basic lemma on uniformly bounded and equicontinuous families of functions. Note that a family on a bounded interval > 0 , there is a implies
[a,b]
F = {f(t)}
such that
for all
tl, t 2 ~ [a,b],
f(t) ~ F .
sequence of continuous functions on a compact set convergent on
~ ,
f(t) defined
is called equi.Fcontinuous if for each
~ = 6(e) > 0
Jf(tl) - f(t2) j < c
of functions
JtI - t2J < 6
In particular if a ~
is uniformly
then it is uniformly bounded and equicontinuous.
248 LE~N~A (Arzela', Ascoli).
3.1.7
Let of functions
be an equicontinuous, uniformly bounded family
F = {f(t)}
defined on a bounded interval
f(t) ,
exists a uniformly convergent sequence
{fn(t)}
[a, b] 9
of functions
Then there fn(t) ~ F .
We are now in the position of proving the basic existence theorem.
3. i. 8
THEOREM (Peano 's existence theorem). Let
be continuous on a parallelepiped
f(x,t)
fl C E x R
defined by the relations: t
0
~< t ~< t
+
a
]x - x ~
,
o
~< b
Let M =
max (t,x) ~
[f(x,t)[
a = min (a, b/M)
Then the ordinary differential equation on
[ t o , t o + a]
Proof. an
Let
. 3.1.5
has at least one solution
.
Tab
denote the closed interval
[t o - a, to + b] .
n-dimensional continuously differentiable vector
the interval
T
x~
Consider
, defined on
, (E > 0 , sufficiently small) which satisfies the Eo
~.~rt, o~s
]x~
~(Co)=f
- x~
~ b ,and
we shall construct a vector
x~(t)
while
x~(t)
, x o (to) = x o ,
(~%)
]f(x~ x~(t)
= x~
,
t)] ~ M for all
on the interval
t ~ T
s
is a solution of the integral equation
T Ee
t ~ Tao
.
Next
as follows:
249
3.1.9
(t) = x ~
x
It
+
f (x6(~-6), T) dT
t o
with
0 < 6 ~ e
on the interval
T
It must now be shown that such a
oe
solution
x~(t)
of the integral equation above indeed exists on the whole
interval
T
Clearly,
oe
Then on the interval T Ix6(t) - x~
~ b .
ee 1
such
x6(t)
,x6(t)
~ CI
exist, on
Toe I
and satisfies
where
e I = min(e,6)
the conditions
Then clearly this solution can be extended in the same
fashion on the interval
Tee 2
where
~2 = min(e,
26)
etc.
By repeated
application of this procedure it is possible to construct on x 6 (t) , with
continuously differentiable function
If(x~(t), t) J = l~6(t) l ~ M
family of continuously differentiable equicontinuous. that
functions
= x(t)
exists uniformly on
f(x~(n)(t - ~(n)),t) 3.1.9
it follows that the 6 x (t),
0 < 6 ~ e
T
is
{6(n)}
such
9 ee
From the uniform continuity of
equation
and
3.1.9,
Then from Lemma 3.1.7 there exists a sequence
lim x6(n)(t) n § ~
a
ee
x 6 (t o ) = x o
Jx6(t)-x~ I ~ b , which satisfies the integral equation Now since
T
tends uniformly to
with solution
x 6(n)
f(x,t)
it follows that
f(x(t),t)
as
n § ~ ,
thus
tends to integral equation
t
x(t) = x ~ +
3.1.10
f
t
which in the domain
~
f(x(T),
Y) dT
o
is equivalent
to the differential
together with the initial condition 3.1.6.
Thus
x(t)
equation 3.1.5
so constructed
is a solution of 3.1.5, which proves the theorem.
The integral at the right hand side of expression 3.1.10 is defined for a much larger class than the one of continuously ~iifer~,~a&l, {vnct(o,s x(t).
This fact allows us to define solutions "in the Carath~odory
250
sense" of the differential restrictive
conditions
equation
3.1.5 which exist under less
than the one required by the previous
the next theorem we shall state the classical of such solutions
3.1.11
In
for the existence
sense".
THEOREM Let
continuous in x.
in the "Carath~odory
conditions
theorem.
be defined on
f(x,t)
x
for each fixed
If on the interval
t
~
defined as in t
3.1.8 ,
for each fixed
there exists a function
such that for
If(t,x) l .< re(t)
then there exists on some interval x(t)
differential equation
with
and measurable in
[t o , t o + a]
m(t) ~ L1[to, to + a],
continuous function
n ,
an absolutely
[t o , t o + 8] (8 > 0)
such that
3.1.5
(x,t) ~
x(t o) = x
for all
0
and which satisfies the
t ~ [to, to + 8], but a set of
Lebesgue-measure zero.
3.1.12
COROLLARY Let
continuous in
f(x,t) x,t
be continuous in
in
E x R .
there exists a solution
x(t)
open interval and is such that
3.1.5 relative
the integral equation
(for fixed
Then for all
(x ~
x(t o) = x
o
o
3.1.12
t)
and piecewise
in
E x R ,
3.1.5
on an
.
the solution of the differential
to the initial condition
3.1.10
t )
which satisfies equation
In the case of corollary equation
x
3.1.6 is equivalent
at all points of continuity
of
f(x,t)
to .
251
We shall now investigate the relations between the rectangle in which the system is defined and the number
u
defined in Theorem 3.1.8
which defines the interval of definition of the solution. particular, concerned about the properties of • in the whole space
E x R .
when
We are, in
f(x,t)
is defined
This problem is called in the literature:
"Problem of the extension of solutions of an ordinary differential equation." Suppose that
f(x,t)
is defined on
E x R
of the ordinary differential equation on an interval
[a,8] 9
is defined on some interval x2(t) ,
3.1.5.
Then the point
is possible to find a solution
xl(t)
[~,6]
defined on the interval
and let
[~,6]
x2(t) = xO(t)
for
t ~ [a,B]
xZ(t) = xl(t)
for
t ~ [ ,~]
interval which is larger than either
is in
[a,~]
is defined
E • R
and it
Clearly the function
which is defined on an [~,~] .
Such a solution
is called an extension of either one of the solutions
xl(t) .
and which
by the relations
3.1.5 or
x~
xl(8) = x~
8 < 6 .
is a solution of the differential equation
x2(t)
~)
such that
[~,8] U
be a solution
Assume that
(x~
with
x~
x~
and
This process of extension may be applied at either end of a closed
interval and a given solution extended to a larger interval.
By repeated
application of the above process a maximal interval of e ~ s t e n c e given solution can be constructed.
of any
Obviously such a maximal interval of
existence is open. For the case of solutions defined on the maximal interval of existence the following theorem holds.
3.1.13
THEOREM Let
~C__ E • R.
Let
F ~ ~
be compact.
Fix
(x~
0
~
r.
Let
252
I
be the maximal interval of existence of a solution
= (t-,t +)
x
differential equation for
(x~
for
~ ~ t < t+ (t- < t ~ r)
Proof.
where
3.1.5
Then there exists
We need show only, that if
lies in a compact subset t- < t
< t+ ,
N
of
~ > to, such that
Ix ,
t+
~
for
then the interval
is defined and continous in
(x,t)
rE
of the
x(t)
is finite and if t ~ [to, t+)
(t - , t+)
~,
x (t,x~
(F)
(t, x(t))
where
cannot be a maximal interval
o
of existence.
We will show first that in such a case
For this purpose, x(t n) § z I
and
let
{t n} , {r n}
X(Tn) + z 2
and
lim x(t) t+t+-0
exists.
be any two sequences, such that zI # z2 .
Clearly
zI , z 2 ~ N C ~
.
We have of course
x(t n) = x
fln f(T,
+
o
x(r)) dr
o
fln
x(T n) = Xo +
f(T, x(r)) dr
o
so that
llx(tn) -X 1
and
be such that
D = {(x,t):
Ix - x~
0 < b - tn ~ e/Tme
and
that this is false let then
~ e,
for
T:t
t
n
< T < b
for
0 ~ b - t ~ e}
Ix(t n) - x~
Ix(t) - X(tn) [ < me(b - tn) .< el2
Assume
If(x,t) l ~ m e
~ e/T
.
Let
Then
.< t < b
be the smallest
T
for
n
which
Ix(T) - X(tn) I = m(b - t n)
Ix(t) -x~ and then
e/2
9
.< el2 + Jx(t n) - x ~
.< e
Thus
for
t
.
~ t o
x(t, to, x ~
x E
,
which satisfies the differential
on an open i n t e r v a l and 8uch t h a t
Furthermore these solutions and
for fixed
Let
there exists a unique function
t
x
x ( t o) = x
o
.
are continuous functions of
.
o
Proof.
From the previous theorem we have the existence of at least one
solution corresponding to each pair
(to, Xo)
Hence we need only to
concern ourselves with uniqueness and continuous dependence on initial data.
261
Suppose we have two solutions 3.1.5
having the values
x
and o
respectively. d
d~
IIx(t)
Yo
x(t)
and
y(t)
of equation
respectively at
t
and
o
T
o
We observe that at every point where
- y(t)II
2
exists, that
d-td llx(t) - y(t) ll 2 = 2 ~ x ( t )
- y(t), f(x~t~, t ) - J(y(t),t )>
.
Hence it follows that
IIx(t)
- y(t) l]2 =
ilXo _ y ( t o )
l] 2
t
+ 2
s
<x(T)
t
and by
- y(T),
((•
T ) - ~r
) > d~
o
3.1.28 ft
[Ix(t)
-
y(t) I I 2 ~
IIx ~ - Y ( t o) II 2 + 2
~(~) t
From the Corollary
IIxCt)
- y(t)II
3.1.24
2
~
N
x
k(i)0
If in addition the solutions of x(t,x~
o) =
sets in
lim xk(t,xk~
)
are unique, then
~ = f~
and the convergence i8 uniform on compact
I o . X
A version of this theorem on a given closed interval
[a,b]_~ C I o X
is due to Kamke
[ ~ ] , while the proof of the complete
theorem on
I o X
can be, for instance,
found in Hartman
[4, Ch. II, Theor.
3.2].
The local version of this theorem is on the other hand an immediate
consequence
of Theorem
We shall now illustrate tions of the differential Peano's existence
4, Ch. I . Theor.
some qualitative 3.2.1
theorem are satisfied,
property of uniqueness. following
equation
3.1.8 [Hartman
w~e~ but
properties
2.8].
of the solu-
the hypotheses
o~
the flow does not have
We shall begin our presentation
with the
284 3.2.6
DEFINITION Consider the differential equation
the Peano
existence condition.
Through all points
x(t,x~
there exist at least one solution equation
3.2.1 which satisfies
o)
(x~
~ E • R
of the differential
3.2.1 . Consider now the set of all such solutions
through a given point
(x~
" For a given
x(t,x~
)
(x~ o) 6 E x R we shall
define:
t+Mo = sup t+(for all
x(t,x~
)
with
X(to,X~
o) = x ~
t-M= inf t-(for all o
x(t,x~
)
with X(to,X~
o) = x ~
t+mo = inf t+(for all
x(t,x~
o)
with X(to,X~
o) = x ~
t-mo = sup t-(for all
x(t,x~
o)
with X(to,X~
o) = x ~
o
~.2.7
3.2.8
IM--o [toM'to +M]
and 3.2.9
im
o = [to- m 'to+ m ]
3.2.10
DEFINITION ~ The set
x(~'x~
T(~,x~
o) C E which is the union of all points
) reached by some solution
equation
3.2.1 at the time
t = 9
set of the differential equation
x(t,x~
) of the differential
i8 called the reachable (or attainable)
3.2.1 from the point
(x~
at the o
time
t = T .
The set
T(xO,to, i~ ) = tEIMU T(t,xO to ) is called solution o
funnel ~ of the differential equation
3.2.1
through the point
(x~ o
* Our definition of solution funnel is different from the original definition due to Kamke [ 2 ] in the sense that Kamke defines the solution funnel only on the interval Im . O
285 The set
T(z,x~
o)
can also be called a cross-section
of the solution funnel of Equation instant
3.2. i
(x~
through
at the
o
t = 9 . The set
T(x~
=
U t~[a,b] ~ I M
is called
T(t,x~
segment o[ the solution funnel.
Next we shall investigate some qualitative properties of solution funnels, their cross sections and their boundaries.
The first
theorems are due to Kamke [2].
3.2.11
THEOREM Let the vector-valued fUnction
region
~ C E • R . Let (x~ the segment ofAsolution funnel T(x~
Proof.
If
T(x~
E ~,
f(x,t)
least one
is a compact set.
is not bounded, then through the point
instant
xl(x~
, x2(x~
x~ t) , ...
tk E [a'b] C Imo such that
3.2.2 ,
t~l > k .
however, there must exist a subsequence of the
{xn(x~
above which converges uniformly to a solution curve of the Equation
3.2.1
Ix(x~ Thus, because of Theorem
T(Xo, to,[a,b])
and such that for some
t)[ § ~
as
3.1.13, the solution
exist on the whole interval Hence
~
3.2.1 such that for each of these curves there exists at
From Theorem
x = x(x~
the
0
Ixk(x~
sequence
[a,b] C_ Im
Then if
there exists a sequence of solution curves of Equation
be continuous in the
T E [a,b]
t § Tx(x~
t)
does not
[a,b] which contradicts the hypothesis.
is bounded.
286
Consider now Through each point
x
x n = xn(x~ 3.2.2
a
n
sequence
{xn}
C T(x~
, [a,b]), x n + x .
of such sequence there exist at least one solution
t)
of the equation
3.2.1.
Because of Theorem
it follows that there existsa subsequence of
to a solution curve of x ~ T(x~
3.2.12
3.2.1
{x n}
which joins the points
x
which converges
O
and
x .
Thus
and the theorem is proved.
THEOREM
Let f(x,t,u)
~
be a real number.
continuous in the set
Consider the vector-valued functions
~ C E • R for each
Assume
~ ~ [Uo, Ul] .
that on each compact subset of
f(x,t,~) -> f(x,t,~o)
holds uniformly ~n(x,t) ~ - / o . D e n o t e solution funnel through the point equation
~ = f(x,t,~) .
for all points segment
(x,t,~)
T~(x,t,[c,d])
such that
e > 0
sufficiently near to and
Then
.
there exists a
(x~
as
~ > 0
there exists a
implies T (x,t,[c,d])
.
If the theorem were not true there would exist a real number and a sequence
exist in (x~
T~o(X~
T~(x,t,[c,d]) § T~o(X~
(xk, tk,~k ) § (x~
least one of the integral curves
T
for the ordinary differential
Consider the segment
(x,t,~) C S((x,t,~),~)
S(T o(X~
Proof.
(x,t)
the segment of the
Tu(x,t,[a,b])
in the sense that for each
(x,t,~) § (x~
> 0
with
[c,d]
x = x(tjxk, tk,~k)
or would not belong to an .
so that for each
This contradicts Theorem
either
would
e-neighborhood of 3.2.2.
~o From the Theorem
3.2.12
it immediately follows that:
k not
at
287
COROLLARY
3.2.13
Consider points M ( x , t ) C T(x,t, [c,d])
Consider the set
x s T
(x~ ~o such that M~(x,t)~
%T
(x~
=
i
~o
then
as
M (x,t) § ~T (x~ u ~o
~ §
~o
D
We can now proceed with the proof of the renow"~d Kneser on the structure of the cross-sections of solution funnels.
Theorem
See Theorem
3.4.37 for a full proof using the concepts of weak invariance.
3.2.14
THEOREM Let the vector-valued function
region
~ C q E • R,
through
(x~
f(x,t)
then e~h cross-section
: T(x~
~I
TM 0
be continuous in a
of the solution funnel
is a compact, connected set. '
The proof of this theorem is u ~ l l y
based upon the following
local theorem, which we state without proof since a more complete proof is in 3.4.
3.2.15
LEMMA
Let the vector-valued function set
D: It - tol $ a, lx - x~
y = min (a, ~)
and
Let
~ 8 9
Ic - t I ~ Y.
Then
f(x,t)
be continuous in the
If(x,t) l ~ M
in
D,
the cross-section of the solu-
0
tion funnel
3.2.16
T(x~
)
is a compact, connected set.
Proof of Theorem 3.2.14 From Lemmm 3.2.15
and all
T ~ (to,~)
it follows that for a certain
the theorem is true.
that the theorem is also true for all since
T(x~
~ ~ (to,to+m )
Because of continuity it follows
T ~ (to,~+e) , (e > 0) .
is a compact subset of
~
In fact,
there exist real numbers
=. i
and
8i
such that each set
288
with
(p,x') ~ T(x~
H = U DiC for all
P) , is such that
T(x~
P) .
By continuity
(x,t) ~ T(x~
P)
and that
D i C T(x~
(x,t) E H
]f(x,t) I < M ,
there exist
and
T(x, t, [la, p+y) )
with
T(x,t~T)
T [ [~,p +y)
y
as
in Lemma 3.2.15. Consider now the cross-section
with
Clearly
T(x~
Z) =
Suppose now that
U (T(x,t,z):
T(x~
(x,t)
s r(x~
T) = TI U T 2
where '
~i N ~2 = 0 9 have
If
T(x,t,T) A T i # # ,
T(x,t,T) ~T.I
Since
T. ~ ~ (i = 1,2)
Thus one can divide the points of
T(x~
n)
T(x,t,T)
is a continuuum
a common boundary, point b.
Assume that
T(ci,[p,T])
From Corollary
converge to
T(b,~) C TI,
3.2.17
Remark.
and (xl,tl) { ~T(x ~
b
belongs to the first class.
C~_ ~T(x~
b
which belong
as
ci + b .
TI
and
There exist then points
T(ci, T) C T 2 , T2
which
and proves the theorem
[to, to +m ) .
Theorem 3.2.14 does not
imply that, given
to, I~), there exist a solution
the differential equation x(t,x~
or to
then the two classes must have
which are arbitrarily close to
on the whole interval
TI
3.2.13, then the segments
T(b,[~,T])
contradicts the assumptions made on
3.2.15, we
T(x~
belongs to
Clearly there exist then points ci, arbitrarily close to to the second class.
and
l
then, because of Lemma
into two classes according to whether T2 .
, z ~ [~,~+V) 9
lo +M)
3.2.1
with
for all
(x~
x = x(t, x~
x i = x(t I, xO,to ) ,
o) )
of
such that
t E [to, t I + e] , e > 0 .
pointed out by Nagumo and Fukuhara with the aid of an example.
This fact was
289
An additional
example was p r o p o s e d by
Digel.
!
The following example is to show how for
t ~ I ~ the funnel
section need not be closed nor connected.
3.2.19
Example ~(t) = f(t,x,y) 9(t) = g(t,x,y)
where and for
g ~ 0
for
t ~ [0,5]
t • [0,4] (2_x)i/2 f (t,x,y) =
1 .< x .< 2
x
O.<x. 4 +
2
The (x, y)-cross-section is:
[O, t2/4] x {0}
for
t 610,2] ,
[0,-t2/4 + 2t -2] x {0}
t
[o,2] x {0}
t ~ [4,5],
( [2,4],
{(x,(1-sin(2 x) sin2(t+5))-l-1) Ix 6 [0,2]}
t ( [5,5 + ~ ) ,
{(x,(l-sln(2 x)sin2(t+5))-l-l)Ix ~[0, i) U (1,2]} Note in particular for
t = 5 + ~,
t~5+~.
the cross-section is
[0, i) x {0} U (1,2] x {0} which is a bounded but not closed set. 3.2.20
Remark.
The following remarks on cross-section of solution funnels
are due to Pugh [I] and Nagumo and Fukuhara. i)
There exists a
T(T,x~
)
Let
Y ~ ITM. o
which is not arcwise connected.
ii) There exists a non-simply-connected continuum which is a ill)There exists a continuum which is not a equation.
T(T,x~
o)
T(l,x~
for any differential
291
iv)
Any clpolyhedron is a funnel-section.
We shall now discuss a theorem due to Fukuhara on the qualitative properties of the boundary of the solution-funnel.
See 3.4.33 for a simple
proof using results on weak invariance.
THEOREM
3.2.21
Let the vector-valued function region
~ C E x R . Let
(xl, tl ) # (x~
equation
6 ~ .
Then there exist;a
o
x = x(t, xO,to)
3.2.1,
that for all
(x~
t s [to, tl],
f(x,t)
Let
be continuous
~n the
(xl,tl) C ~T(x~
l~) ,
solution of the differential
such that x(t,x~
x i = x(tl,xo, to ) ) ~ ~T(X~
and such
.
This theorem is true for even more general flows than the ones defined by the solution~of an ordinary differential equation without uniqueness.
This will be proved in Section 3.4.
The proof of Theorem 3.2.21 is
therefore omitted. Another property of differential equations which we must mention is that of the differenti
ity of the solutions with respect to their
initial conditions.
property
and 3.2.29. 3.2.22
I~5
Th~s
is
proof is based upon the following lemma.
Ler~na
Let the conditions of Theorem be
illustrated in Theorems 3.2.24
El-approximate* and
on an interval
[a,b]
3.1.8
hold.
Let
~2-approximate solutions of
containing
t
. And let
N
Xl(t) , x2(t)
3.2.1
defined
be a compact subset
o
of
~
such that
(t,xl(t)) , (t,x2(t))
remain in
N
for
t s [a,b] .
Then * Def. A function ~(t) , defined and continuous on an interval I T is called an e-approximate solution of the differential equation 3.2.1 if the following conditions hold: i) ($(t),t) E R for t G 17 . ii) ~(t) is continuously differentiable on I ~ \ S , where S is a finlte set of numbers, ill) Ilk(t) - f(~(t),t)]] ~ E for t 9 I~\ S .
292
IlXl(t)
3.2.23
-x(t)ll is
"
h
h .
Using this fact the relation
x(t,h)-x(t) h
,
O
3.2.28
yields
- z(t) ll= < E(h) + I t llJf(T,x( z))ll 11x(~'h)-x(T)h t
- z(~)H at
O
where
e(h) § 0
as
h § 0 .
Using now Corollary
3.1.24
we get
t
]I~
-
z(t)ll
O h
=
z(t)
Differentiability with respect to
follows
t O
by considering the system = f (t,x)
s
The details are left to the reader.
296
The following more comprehensive theorem can be proved by repeated application of the processes explained in the above proof.
THEOREM
3.2.29
Let
possess continuous partial derivatives of order
f(t,x)
in all its arguments in a region x(t,x~
r
of
o)
~ C E x R.
r
Then the solution
possesses continuous partial derivatives of order
3.2.1
with respect to all its arguments and 3r+ix(t,x~
)
3t r+l
also exists
and i8 continuous.
The last property that we want to mention about the solutions of differential equations is the continuity properties of the maximal interval of existence
I~ .
This result is proved for the special case of differen-
tial equations with uniqueness.
THEOREM
3.2.30
Let that for all
f (x, t,u) (x~
U)
has a unique solution
be continuous in a region ~
(fixed
x = x(t,x~
u ) .
~ C E x R x ~ such
the differential equation Let
I o
3.2.1
be the maximal interval
of existence of such a solution, then, in addition to the fact that is continuous in
x(t,x~
i8 a lower 3.2.31
[upper]
Io, t+ = t+(x~
U)[t - -- t-(x~
8emicontinuou8 function of its arguments.
Notes and References
Most of the results presented in this section will be discussed again in Section 3.4. Theorem 3.2.2, 3.2.11, 3.2.12 and the proof 3.2.16 are due to Kamke [2]. Theorem 3.2.14 is due to Kneser [i]. Theorem 3.2.21 is due to Fukuhara [3].
297
3.3
Continuous flows without uniqueness. In many situations, for instance in control problem~, one has to cope with
differential equations which do not have the property of uniqueness of solutions. In this section we shall extend some of the results these more general cases.
presented in Chapter i to cover
In the literature not much work has been done along these
lines and the results that will be presented are not complete. of the references have been impossible to consult.
In addition, some
We present these preliminary notes
since we believe that this will be a future fruitf~l research area. l
In this presentation we will not define abstractly the properties of flows without uniqueness, but simply introduce a suitable notation, derive from the theorems presented in the previous section the suitable properties that the flow must have and discuss and extend
thole
properties presented in Chapter i and which have
particular interest for stability theory, We shall present some system~of axioms defining dynamical systems which are more general then the one presented in Chapter i. We shall first define the concept of local dyn~mlcal systems. This is essentially a generalization of the flow defined by an autonomous differential equation whose solutions have the uniqueness but not the necessarily global existence property.
Local dynamical systems were introduced first by
r. Ura [7]. We shall give next the following definition due to G. Sell [Spl].
3.3.1
DEFINITION A transformation
I:X x I
+ Xj
where
Ix = (t~t +)
X
is said to define a local dynamical system on i) H
O~Ix~__R,
if it has the following properties:
is continuous
ii) ~(x,O) = x iii) if
X
is such that
for all
t ~ Ix,S E I
and X
x~ x then
t + s~ I X
~
n(~(x,t),s)
= n(x,t + s)~
298 iv) either 9 ~I
t~ = + ~ (t~ = - ~) or for all co,pact sets FcX, x ~ t h e r e such that
x
v) fhe interval and
Xn § x.
I
x
Then
for
g(x,t)~ CF is
lower semi-continuous in IxClim
inf
emist4
9 ~ t < t+ (t- < t ~ T), x, i.e.~ if
xn ~ x
Ixn .
The relationship between the flow defined by the solutions of an ordinary differential equation with uniqueness property, but not necessarily with global existence,
can be clarified as follows:
The property i) follows from Theorem 2.1.26,
the property ili) from Theorem 3.1.18 or any theorem on uniqueness
(3.1.46),
property ii), for example, from Theorem 3.1.8 with the usual conversion
t
o
and the
= 0.
The property iv) follows from Theorem 3.1.13, while v) is expressed by Theorem 3.2.30.
3.3.2
Remark Notice that the local dynamical system
even its particular form (E, R,H). system
(E,
(E, Ix,N )
(X, R, P), C
X'
9)
defined in 3.3.1 and
is more general than the dynamical system
In fact, Theorem 3.1.67 does not necessarily hold for a general dynamical
Ix,q).
We shall now discuss a few axiomatic M.
(X, I
I. Minkevich, where
X
for instance,
systems for flows without uniqueness.
considers the flow without uniqueness
is a compact metric space and
is a set of nonempty closed subsets of
X
and
P: X • R + C
is a multivalued map.
it is metrized by the
Hausdorff metric.
3.3.3
DEFINITION A multivalued m~o
P:X • R § C
i8 said to define a flow without uniqueness
if the following conditions are satisfied:
299
i) P(x,O) ={x}
xEX~
for all
ii) P(P(x,t),s) = P(x,t + s) iii) ys
x~e(y,-t)
implies
x~X,
for all
for all
and
t,ss
x,ys
and
P(x,t),(x~X,t ER) is continuous in
iv) the map
t
with
ts >. O,
t~R~
for each fixed
x.
These axioms are similar to the ones used by E. A. Barbashin [5,7,8].
Other
axioms for a flow without uniqueness (dispersive flow) have been proposed by B. M. Budak.
3.3.4
They are the following:
DEFINITION Let
X
A, B C X
be a metric space,
and
N(B,E) the
E-neighborhood of
B.
Let
~(A,B) = inf{e:A~N(B,e)}
and
~(A,B) = max(~(A,B),~(B,A)}
A mapping
P:X
i8 said to define a dispersive ~ n e ~ i c a l
x R -+ Z
system if the following
conditions are satisfied: i) P(x,O) ={x} ii) P(x,t)
for all
x~X,
i8 a nonempty compactum for all
iii) yEP(x,t)
x~P(y,-t)
implies
x ~ x, t ~ R~
x,y~X,
for all
t~R~
iv) P(P(x,t),s) = P(x,t + s), q,
V# x + y
and
~d
t § s
a(P(x,t),P(y,s)) § 0
implies
for all
x,yEX
t,s~ R)
ui) t + s
implies
~(P(x,t),P(x,s)) -~ 0
vii) A motion through
x ~X
is a mapping
for all P
X
x~X
:R § X
and
t,sER.
such that
a) Px(O) = {x } b) t < s
implies
Px(S)C'P(Px(t) ,s - t).
trajectorH, through
x.
The set
Px(T)
i8 the
300
Quite recently I. v. Bronshtein and I. V. Bronshtein
[1,2,3,4,6], K. S. Sibirskli and
[i], K. S. Sibirskii, V. I. Krecu and I. V. Bronshtein
[i] and
K. S. Sibirskii and A. M. Stakhi have presented a series of works in which a class of generalized dyD~mical systems defined as semigroups of multivalued mappings is investigated. We shall present next the definition given by Bronshtein of semigroupSof multivalued mappings.
3.3.5
DEFINITION Let
T
be a topological space,
identity element
e,
P
S
a topological 8emigroup
a mapping such that for each point
the image set
P(x,s)C T
x ~ T,
with an and each
element
s~ S,
is a nonempty compactum.
(T,S,P)
will be called 8emigroup of multivalued mappings if the following
The trip let
conditions are satisfied
i) P(x,e) = {x}
for all
ii) P(P(X,Sl) ,s2)) = P(x,s I + s 2) iii) for all set x
x ~T
P(x,s) in
in
T
and
T,
s{:S
x ~T, for all
and for any neighborhood
there exists a neighborhood
and a neighborhood
particular,
xs
N(P(x,s))
Q(x)
of the element
U(s)
P(Q(x), U(s))C N(P(x,s)) From these axioms Bronshtein
and all
Sl,S2~S
s
of the
of the point in
S,
such that
.
[2] derives various interesting theorems.
In
he shows that the axiom (vi) of the Definition 3.3.4 is a consequence of
the first five axioms.
i.e., a topological space with binary associative multiplication continuous in the f~m~ly of components.
operation which is
301
Quite recently E. Roxin [5,7,8] has introduced a set of axioms defining a "general control system."
These systems may have rather important application in
the study of the qualitative properties of differential equations without uniqueness. In what follows, we shall briefly present some of the results obtained by Roxin.
3.3.6
DEFINITION
Let
X
be a locally co,pact metric space.
p(x,y) =
Let for
x,y EX
d(x,y) 1 + d(x,y)
where
is the ~ v e n metric on
d(x,y)
Let f o r
and
x,ys
X.
A,BCX
p(x,B) = p(B,x) = inf{o(x,y);ys O (A,B) = sup{p(x,B);x~A} p(A,B) = p(B,A) = max{p (A,B),p (B,A)}. S(A,s)
The triplet
(X,R,F)
= {x~X;o(x,A)
< s}
i8 called general control sHstem if the following axioms are
satisfied. i) F ( X , t o , t ) and
is a closed none~pty subset of
defined for all
to,t &R~
for all
ii) F(X,to,to) = {x}
iii) for
x,
t o .< t I .< t 2
and
x &X
and
to ~ R~
xs
F(X,to,t 2) =~{F(Y,tl,t2) :y~F(X,to,t I)},
iv) for each
y~X,
t o .< t I
there existssome
Y ~F(X,to,t I),
x~X
such that
xEX
302
v)
x~X,
for each
It-ill < ~
t o ~ tl, ~ > 0
there exists a
6 > 0 such that
i.plies
p(F(X,to,t ), F(x,t ,tl)) < e, O
~)
for each
~(x,y)
xs
t .< s, ~ > O,
there exists a
< 6, It - ill < 6, Is - sXl < 6,
~ > 0
such that
t x .< s I
i.p lie8
p*(F(y,tl,sl),
F(x,t,s)) < E
.
The principal results proved by Roxln [5] for the general control system (X,R,F)
3.3.7
are the following:
THEOREM If
3.3.8
AC
is co.pact and
X
t >. to,
then
F(A,to,t)
is co.pact.
THEOREM If
A ~X
is a continuum and
t o .< tl,
F(A,t o, [to,tl]) = ~ {F(A, to,t) :t ~ [to,tl] }
Notice that
F(X,to,t )
then
is a continuum.
has been so far defined only for
t ~ to .
Both
for the theory of control systems as well as for the study of the qualitative properties of differential equations it is important to define the multivalued map for
t < t . o
3.3.9
DEFINITION Let
G(X,to,t)
be defined by
y6.G(X,to,t)
0
i8 a family of curves such that for each
x~A
a
y(.,x)~ F
that i8, there exists an element
of
F
P(F)
We define
[o,T)
and
[0,6(x))~Dy(.,x);
beginning at each point of
t # O.
P(F)
such that
for some
A .
there exists a
and
F
which are p iecewise elements of
is the smallest family of curves such that if
and
yI(T) = Y2(0) ,
a(t) -- Yl(t)
where we define
and
~
or
F; =that is,
and
then the curve
t s Dyl
for
Y1
for
[0,T]
to be the family of curves with domain
0 < T ,< %
T I = sup Dyl
y(t,x) ~ A
ge do not aasume
Y2
are in
i8 in
~(t + ~) = Y2(t)
F
P(F) for
t s Y2 3.4.3
LE~I4
For any set
A
let
F
be a A-family of curves.
y*~ P(F)
there exists an extension
of
y
Then for any
y~F
P(F).
which is maximal for
The proof by Zorn's lemma is similar to the proof of L~mma 4.3 in Strauss and Yorke [2].
3.4.4
DEFINITION We say
(A)
if
y
y
is an e-solution or (a right E-approximate solution) for
i8 a curve with
0 = inf D
Y t ~ Dy,
continuous and, for almost all
dy(t) dt
I
Note that a e-solution for
-
< sup Dy, 0 s dy(t) dt
y
Y
is absolutely
exists and
g(y(t)) I g c: is actually a solution (on some
c = 0
[O,T)
or
[o,T]). 3.4..5 LE~MA Let set
FE
A
be any subset of
of all e-solutions of
and any maximal
y ~ F
e
,
Rn (A)
and
g
be continuous on
is a A-fcynily.
there either exists a
A.
Assume the
For any co,~act set
t E D
y
,
C~A
t > 0, such that
308
Proof.
Suppose
7(t)s C
bounded and so,is uniformly
for
t fD . y
continuous.
Since
Therefore
g
is bounded on
if
D
'
T # ~, 7
can be extended
to
be extended by "piecing" D
7
# [0,=),
3.4.6
[0,T]
7
continuously.
together with some
= [0,T)
where
7
If
D
7
= [0,T],
=(',7(T)) ( F . E
then
7
Therefore
can
if
DEFINITION ^
is locally compact (locally closed) if for each
such that
b = b(x) > 0
B(b,x) O
A
is compact. A,
N
G(K) = sup x ~ K
C
such that
An
N
is compact.
Let
x EA
there exists
It i8 easy to see that if
i8 a compact subset of a locally compac$ set
3.4.7
is also
7 is not maximal.
We say
of
C, d7 dt
C
then there is a closed neighborhood Ig(x) l"
LEMMA Let
family
FE
C of
be any compact subset of the locally compact set e-solutions of
a compact neighborhood and each maximal Proof. is compact.
N
of
b > 0
By 3.4.5 if
T = inf {t > 0:0(t)
C
~ }
is a
and a
we have
~(',x),
Choose
(A)
A-f~ily of curves. K = K(C) > 0
[0,K]~D~(.
such that if
sup D~ < = < ~.
for
In fact
N xEC
x)
is and
A.
Assume that the
Then there exists
such that for each and
for
~(t,x)6 N
{y:d(y,c)
~ b}
then
~ = ~(',x)~ FE,
b .< 10(T) - 0(0) I .
. ~.
LE~14 Let
g
E-solution8 on for all
+ E]-I
t~ A.
be continuous on the set [0,~].
Then
Let ~n + ~
~
A
and let
be a function,
uniformly on
{~n } be a sequence of
~ = [0, O, and we would
311
be able to choose
t
arbitrarily
o(t) = d ( x +
(iv) for each
If
Xo ~ A
-g(x)
is subtangential
[0,6)
(A)
for
on
#(',Xo)
(v) implies and
Write
x I = 0.
-g(x)
with
(-6,6)
x = (Xl,X 2) ~ E 2
Xo ~ A
to
and A
given by
g(x)
x s A ~ ~A, for some
and
with no solution
T-~(',xo)
Hartman
x
o
~A
there is
6(x o) > 0. 2
such that
but if in
A
(Xl,X2)~ S
for all A
x~A.
Then
g(x)
is not locally compact, defined on any interval
6 > 0.
We first state Lemma 3.4.13 which is essentially in
is
are both subtangential
then for each
S~E
x
~A, the~
~ = -g(x)
~ (t) = ~(-t)
-g(x)
g(x) = (0,i)~ E 2
for all
for all x in A
for the equation
~
and choose
A = E2 - S
t g(x), @.
A , locally compact,
Therefore if
defined on
Let
to
But then
for all
are subtangential
there exists an (-6,6)
A
0.
(-6,0].
to the locally compact set a solution
t g(x), ~(t,x))
there is a solution
on some interval a solution of
small such that
[4], p. 14 .
where differentiable, converge to
y ,
to Theorem 3.2
We need to consider e-solutions which need not be every-
but Hartman's proof adapts easily.
yi § y
equivalent
if for each compact subset
K
We will say the curves of
D
,
K
Yi
is a subset
Y of
D
for all but finitely many i and Yi(t) § y(t) uniformly for t ~ K . Yi 3.4.13 LE~94A. Let g be continuous on the locally compact set A and assume
the family and
F
xn § x
of solutions of
with
Xn ~ A
and
(A)
is a right
x~A.
Let
A-family of curves.
Choose
be a maximal
~n = ~e ( " X n )
~
n
+ 0+
~n-Solution.
n
Then there exists a (right) maximal solution {~nl}C{~n}
such that
Although each
x,
~n. + ~ " 1
d(x + sg(x), A )/s
Furthermore
and a subsequence
sup D~ > 0.
is assumed to tend to zero in theorem 3.4.11 for
it need not do so uniformly for
use polygonal approximations
~ = ~(.,x)
x
in a compact set.
We, therefore,
cannot
with a "finite number of corners" as is done, for example, by
312
Coddington
and Levinson
Since K
A
Then
A CU
and
(Lemm8 3.4.9), changing of
g
g
is
3.4.14
U
is compact.
A g
on
].
is locally compact,
B(2b(x),x) ~ A
~
[2,p.6
for each
Let
is relatively
closed in
A.
there exists for some
U.
b = b(x)
x~A,
d(x,y)
> 0
such that
< 2b(x)}.
By Tietze's
extension
theorem,
to a function
g:U § R n
without
We shall assume for the duration of the proof that the domain
for the equation
(A).
Proof of Theorem 3.4.11. x (A
and
~e(.,x) (with range in U). We willlet Define sufficiently
X(u,x) = x + ug(x)
6 = p(s) + sG(B(b,x)),
s
Fe = {r and
a particular Fix
,x) :x s A}.
p(u) = d(X(u,x), A).
x ~ A and
Choose
is sufficiently
.< e/2
[0,s] = D#e(.,x )
for all
y s B(6,x).
small that there exists
x ~ K
such that
= d(X(s),x ).
and define
Ce(t,x) = (s - t)s-lx + ts-lx *
Ce
an
is an E-solution.
[,e(t) - r
.< Ix* = o(s)
For
m
for
t ~: [O,s].
t ~[O,s],
X I ~ IX
+ slg(x)[.
0
small that
p(s)/s
Let
U = {y:
may be extended continuously
We now define for each
and if
x
lX (S) I * ~ (S) I
313
Hence for
t E (0, s)
[g(+e(t)) - ~
d
0e(t) l -< Ig(0e(t)) - g(x) l + ]g(x) - (x* - x)s-ll E .< ~-Il(x + sg(x)) - x* Is-i
E
Fe
is, therefore,
exists an
P(F)
lim o(t) t+T
C FU . s
A- family of
Suppose
P(-)
,
such that for some
o
y s U .
for each
t
n
[0, t
~T
is
,
Ln
~A
U ~ A .
0 s t ~ T
for o
s C D in
F
e
and since
e -solutions n and
x
~ x n
a maximal
DO .
o
A
((t,T)
n is
{en}C
in
Choose
{oi(ti,)}
(0, ~)
t~
Fe,
such
closed
0E(',Y) s Fe
FU e
Then if y
t, ~ t
g
and
is now
U).
t
A .
' By
E (t,T)
o(t*)
that
relative
[t,, t ]
o(t to
We may extend
) E A.
U,
each
o
the maximality
is some
x
y
o
in
U N A
converges
with
en § 0+
proof of the theorem.
(with
For each
to
is
y
also
in
)
Therefore
each
e
O( " , x o) = O
x i = Oi(t i)
so
o(T + s) = 0e(s,y)
P(F
and
{ % ( ",xo)},
a sequence of maximal
o
1
E Do .
By choosing §
in
by setting of
6 A
But
n.
and
U Fe
Let
is in
Applying Lemm~ 3.4.13 with the domain of g e n we find some subsequence {ci} = { 0 } of { % ( ",x )
t
there
T = sup D o
F
solution
Yi = ~ and
t
contradicting Oe(',y)' is maximal in U .
Choose
(domain of
there is
for
By choice of
exists
Choose
P(F e)
U
We now show that
3.4.3
0e s F e
- t,] = DOe there
By Lemma
which is maximal in
is not maximal in
O(t) = 0e(t - t, , o(t,))
and
e
e-solutions with range in
exists and equals some
definition of
~
E-solutions.
c = c(',x) E P(Fe)
g-solution
be the set of all right Note
a
(s) s-i
i, are
O(t)
range there
in
U)
are
ti,
uniformly and
ti, of
in
A .
But by choice
so
O(t)
6 A
for
all
equal to
U
converges
to
on c o m p a c t ti,
subsets
~ t < ti
,
of where
S(en. , x i) ~ en., ti, 1 1 t ( Do , completing the
§ t
314
The usual global theorem for right-maximal solutions says if ~(t) A
T = sup D~ < ~,
then
~
A
as
t § T.
as
Let
Let
r be a right-maximal solution and let
A,
that is,
A
must remain away a locally compact set.
(letting all
ha8 no
limit points in
{~n.}
,
of
{~n }
A.
x.
Theorem 3.4.13
converges to a
l
~ .
Choose
e > 0,
~D~,.
all but a finite number of n i, ~nl. is defined on which contradicts
has a limit point
~n(t,Xn) = ~(t + tn )"
implies some subsequence
right maximal solution
be less than
x n = ~(t n)
be the right maximal solutions En = 0)
T = sup D~
{tn } C De, -tn § T, {~(t n) }
Assume the theorem is false and that
~n(',Xn)
tn + e
~(t)
We state the result as follows for
Then for any sequence
Proof:
is open
THEOREM
3.4.15
~.
t § T.
A
The same result holds when
is locally compact but the result is stronger now since aA N A
when
tends to the generalized boundary of
leaves every compact subset of
from
~
T = sup D~.
Hence
Theorem 3.4.13 implies that for [0,el,
{~(tn)}
and so
~
is defined at
has no limit points.
The rest of this section deals with invariant and weakly invariant sets though in some problems the connection with invariant sets becomes apparent only after some discussion. uCR
We assume that for the equation
g
is continuous on the open set
n
DEFINITION
3.4.16
A set each
(A),
W~
U
is called positively (negatively) weakly invariant for (A) if for
there exists a maximal solution
x s
t ~ [0,sup Dr
(for all
t ~ (inf Dr
positively and negatively invariant. invariant if for each (for all
x~S
t ~ (inf De, 0]).
and each
W
A set r
r
= r
such that
r
for all
i8 weakly invariant if it is both S~ U
i8 called positively (negatively)
r163
for all
t E [0,sup De)
315
The term "weak" invariance seems to have been first used by Roxin [5]. Yoshizawa
[i0] used the term semi-invariance.
Note that if
S
is positively invariant or weakly positively invariant for (A)
and only if it is negatively invariant or weakly negatively invariant for
(-A)
since
~ = -g(x),
~(t)
is a solution if
~(-t)
is a solution of (-A).
Therefore when we state
results for positive or negative (weak) invariance, "positive" and "negative" may be everywhere substituted for each other and the results will remain true. We shall now give some simple propositions and non-trivial theorems.
Results
on "weak invariance" cannot be strengthened by substituting "invariance" nor can the word "positive" be inserted in these results. S~U,
relative to
3.4.17
PROPOSITION The set
W
U
(SNU)
The "relative" closure of a set
will be denoted
~U
and
is weakly invariant if and only if
w
~Us = U ~ ~S.
is the union of
trajectories 3.4.18
PROPOSITION The set
W
is positively invariant if and only if
U~ W
is negatively
invariant. 3.4. s
PROPOSITION If
t~Dr
W
then
is compact, and
~ is a maximal solution, such that
~(t) ~ W, for
sup De -- + co.
3.4.2 0 PROPOSITION If
W
is positively weakly invariomt, then
~U
is positively weakly invariant.
316
Proof:
Choose
Xn~ W
such that
maximal solutions such that
Xn § x ~ ~Uw
~n(t) ~ W
for
and choose
t ~ D~n.
(uniformly on compact sets) to a right solution
#n = ~(" ,Xn)
By Lemma 3.4.13
~(',x).
Hence
right
~n(t)
converges
~(t,x) ~ U
for
t~D~.
3.4.21
THEOREM The relatively closed set
W
if and only if
g(x)
(or
Proof:
g(x)
is subtangential
Assume
is positively (or negatively) weakly invariant
i8 8ubtangential to
-g(x))
to
W
for all
closed in an open set, it is locally compact. Theorem 3.4.11 says that for each which is maximal for (A) and
~(t n) § XT,
then
letting the domain of however, there is an by (3.4.12iii) invariant.
on
W.
x
- W = ~W~
be
U.
Hence
such that
there is no solution
Similarly
-g(x)
W
Since
W
x ~W.
is relatively
If we restrict the domain of @ = ~(',x)
~U.
Hence
~
in
g W
g(x) ~(',x)
is subtangential
for all ~ = -g(x)
for
W
x ~W
and
W t ~ 0
T = sup D~ < ~,then t n
is also maximal in
is not subtangential remaining in
to
to W
W
at
U,
If, x
then
is not weakly
if and only if
which holds if
W
is
and only if
is negatively weakly invariant for CA).
3.4.22
COROLLARY If each solution
condition to
for all
is positively weakly invariant.
positively weakly invariant for the equation W
x ~W.
x
Theorem 3.4.15 implies that if
~W
xEW
at
there is a solution
xT ~ g
w
W
3.4.23
x
then
for all
s
of
(A) is uniquely determined by the initial
i8 invariant if and only if
g ~)
and
-g ~ ) are subtangential
y~ W.
PROPOSITION If
W
~(',x)
(Roxin)
i8 invariant then 21
and
~ - s
are weakly invariant.
317
Proof:
Let
implies to
J = U - I.
g(x)
By (3.4.20)
and
-g(x)
6 I = J ~ I.
Hence
k I = ~I ~ I
J
and
I
are weakly invariant so
are subtangential to both 6 I
J
is weakly invariant.
and Since
I
x s
and so is subtangential I
is invariant
is also weakly invariant.
The following proposition 3.4.24 is obvious, but Theorem 3.4.25 changes the conditions a little and becomes much tougher.
We use all the machinery we have
developed.
3.4.24 PROPOSITION If
W
is weakly positively invariant and
I i8 positively invariant, then
W ~ I is weakly positively invariant. The following theorem is more significant that it first appears.
We shall later
show that the theorems of Kneser (3.2.14) and F~kuhara (3.2.21) are easy corollaries.
3.4.25
THEOREM Let
such that Proof:
W1 wIu
For any
#2(',x)
and
W 2 = U.
x~ W
defined for
be closed (relative to
W2
Then
W = W I(~ W 2
there exist a t E [0,6]
U),
is positively weakly invariant.
6 = 6(x) > 0
such that
positively weakly invariant sets;
and solutions
~l(t,x) E W 1
and
~l(.,x)
~2(t,x)~W 2
and for
SQ
t ~ [0,6].
We may assume that
6 is sufficiently sm al l~ ha t for
straight line segment
Lt
xtf L t
such that
for
t 6 [0,6]
between
el(t)
and
x t E W 1 ~ W 2.
#2(t) But
t ~ [0,6],
is a subset of
U.
is between
el(t)
xt
SO
d ( W l ~ W2, x + t g(x)) .< d(xt,x + tg(x)) .< sup d(r i = 1,2
+ tg(x))
the Choose and
r
318
But the right-hand for all
x ~ W,
side is
and
W
therefore,
#2(',x)
WI~
W2 = U
be two distinct solutions
i = 1,2,.
Assume further that
W = {x}
x
in theorem 3.4.25,
through
xs
in
is not weakly invariant
WI U
If we let
results
~(.,t,y)
since
then
is a (maximal)
~(',x)
solution
S be a subset of
We say S (or
let
~l(',x)
x
and }
g(x) @ 0
W 2 = {x}.
and
WlO
is not subtangential 2
and let
is a maximal
of
# = f(t,y)
to
for
W
at
x.
there is a convex neighborhood
equations
x = (t,y)
and time varying
and
solution of (A) if
and only if
when we let
Hence the equations
VC
E m+l.
S(t)) is invariant for
x~ D~(.; t , y), ~(x;t,y)~ S(T). S(t)
3.4.28
at
(A) and (B) are equivalent.
DEFINITION Let
and
g(x) x~ W I ~ W
E n = R x E m,
~(T,x) = (T + t, ~(T + t, t,y)).
each
W
W. = {#i(t,x):ts D
carry across to nonautonomous
n = m + i,
g(x) = (i, f(t,y))~ E,
3.4.27
to
W 2.
The previous sets.
Let
is chosen such that
The proof actually uses only that for each x
is subtangential
is weakly invariant.
To see that we need
of
g(x)
Counterexar~le
3.4.26
Then
o(t);
(B)
We will write if for each
S(t) -- {y:(t,y) ~ S}(D Em. (t,y)s S
and each
~(.;t\y)
We will use corresponding definitions for
being weakly and~or positively or negatively invariant for
(B).
DEFINITION
v(-R
We say
m
is subtangential to
S(t)
at
(t,y)
if and only if
d(y + sv, S(t + s)) = o(s)
Note that if subtangential
to
S
v at
is subtangential x-- (t,y)
to
S(t)
at
(t , y),
then
(though the two are not equivalent)
(l,v)
is
S
and
319 since
d((t,y) + (s,sv), S) .< d(y + sv, S(t + s))
Since
(A)
and
(~
9
are equivalent, the following theorems are just forms of
(3.4.11), (3.4.21) and (3.4.25).
THEOREM
3.4.29
Let Assume
A
be a locally co.pact subset of
f(t , y)
i8 subtangential to
there exists a
(to, Y o)
and
for all
A(t)
6 = 6(to, X o) > 0
Em+l
f:A § Em
be continuous. Then for each
(t,y) ~ A/~ ~A.
and a solution
~(';to, X o)
for which
[to, t ~ + 6(to, X o ) ) C D~(. ;to,Xo ) . 3.4.30
THEOREM
f:V § E
Let
Assume
m
and assume
v
is subtangential to
f(t,y)
i8 open and S(t)
(or negatively) weakly invariant for
for all
i8 closed relative to (t,y) s S,
then
S
V.
is positively
(B).
THEOREM
3.4.31
If
and
S
V
SI
and
i8 open in
S2
are closed relative to
E m~1,
and if
SIU
V
and are positively weakly invariant
then
S 2 = V,
Sl(~ S 2
is positively weakly
invariant.
We shall also apply other results for
autonomous systems to (B) when it suits
US.
3.4.32
DEEINITION U
S C R m+l such that
will always denote an open subset of is
U
invariant if for each
~(r;t,y)~ U
when
~
between
v,
(t,y) ~ U (] S t
and
T,
the domain of and
~(';t,y)
we have
f.
We shall say
and each
~(T;t,y)~ S.
T This i8
320
equivalent to saying that f
i8 restricted to
U.
S ~ U
i8 positively invariant for (B) when the domain of
We will also use the corresponding terms with "negatively"
or "positively" and~or "weakly." The positive solution funnel through and
~ i8 a solution of
3.4.33
(B) } C Rm+l.
The
(t,y)
is
Ft, y = {(s,~(s;t,y)): s >, t
T-cross section i8
Ft, y(T) CZ/R m.
THEOREM (Fukuhara). Choose
(t,y) 6 V
~{.,t,y) , ~(•
and
tI > t
i8 defined.
If
such that for any maximal solution yl ~ ~(Ft, y(t))
then there i8 a solution
such that
~(T,tl,Y I) 6 ~(Ft,y(T))
T s
tl].
We will prove the slightly more general result: 3.4.34
THEOREM Choose
(t,y) ( V .
Let
U = {(s,w) 6 V :s > t}.
Then
3Ft,y
i8
U-negatively
weakly invariant. Proof:
Note that by definition
is negatively invariant. atlvely
F
is positively invariant.
t,y
Therefore
V - Ft,y
weakly invariant by (3.4.20).
invariant.
and
W I = V kFt,y
By definition
W 2 = Ft,y
Therefore are
(U-)
V~F
t,y
neg-
is U-negatively
W I ~ W 2 = 3Ft,y is U-weakly invarlant.
Since W I ~ W 2 = U, (3.4.31) implies
To see that 3.4.33 implies 3.4.34 we prove a standard result. 3.4.35
PROPOSITION If
then
Proof:
F1
t,y,
and
tI
are chosen a8 in (3.4.33)
and
FI= {(to,Y o) ~ Ft,y:t ~ s [t,tl]}
i8 compact.
If we choose any
{tn,Yn}~F I
and
then (3.4.13) implies there is a subsequence
~n = ~n (';t'y) {~n.} i
such that
~n(tn ) = Yn'
converging to some solution
321
(',t,y)
uniformly on
[t,tl].
tn'l converges to some sequence in
3.4.36
F1
has
Hence
a convergent
(tni,Xni) § (t ,~ (t ,t,y)) ~ F I. subsequence
and
F1
is
Therefore every
compact.
COUNTS
(i)
Theorem 3.4.34 says that for
~(',t2,Y 2) = ~
such that
as in (3.4.33) and (ii)
(iii)
Let
V = R • R
(t2,Y2) ((3Ft,y)(t2)
(T,~(~)) ~ ~Ft,y
t 2 = tl,
Although
in ~Ft,y(t2)) ,
with
t .
We can assume the subsequence was chosen so that
for
Proposition
be the sup {tl:t I
such that
t2 > T
and for each
we have
D~ = (T,~];
~
F
t,y
(T)
defined at
If
implies that
and
in fact it can be shown that T
T(D~[to,t2].
3.4.35
~Ft,y(~))CCSFt,y ) (T)
there is a solution
~(t 2)
tI
is chosen
[to,t 2] C D~
might not have been chosen
~(T) ~ F t , y ( T ) )
for
T ~[t,t 2] (~ D~.
as in Theorem 5,~,33}. Examples can be given
is compact for all t2
T ~ t, T # T
and remaining in ~Ft,y
hence, the restriction on
tI
yet for all
on
D ~ O [ t , t 2]
in (3.4.33) is necessary and
(3.4.34) is in fact more general.
3.4.37
THEOREM (Kneser). If
Proof:
C2
Suppose
and
tI
tI
are chosen as in (3.4.33)~ then
Ft,y(t I)
are non-empty but
(t2,Y 2) ~ V at
t,y,
is not connected.
CI~
C2
and
CI(~C2
such that there is a solution
Then
Ft, y(t I)
Ft,y(t I) = C I ~
are empty.
Let
~ = @(.,t2,Y2)
W1
i8 connected.
C2
where
C1
and
be the set of
which is either not defined
or if defined satisfies
d(~(t I),C I) .< d(~p(t1),C2) We call such a solution a Wl-defining solution for (t2,Y2). reversing the direction of the inequality.
Clearly
Define
W 1 ~ W 2 = V.
using Lemma3.4.13 that each convergent sequence lying, say, in of
W1
so
W1
and
W2
are closed.
Let
W1
W2
similarly
It is immediate converges to a point
U = {(t2,Y 2) ~V: t 2 < tl}.
322
If
(t2,Y2)
is in, say,
T~[t2,tl)~D~,
then
WI
(t,y),~ W I ~ W 2
for
T([t,tl) ~ D~.
is a Wl-defining solution for Hence
W1 ~ W2 = W
is
so there exists a solution By choice of
d(~(t I),C I) = d(~(t l),C 2) Ft,y (tl)
~
(~,~(T))~WI~U.
positively weakly invariant so But
and
is connected.
W1
(and similarly
(t2,Y 2) W 2)
and
are
L~
U-positively weakly invariant. 9(',t,Y)=~
such that
tI, [t,tI] C D~ and ~(t I)
which is a contradiction since
~(r) ~ W
is defined and
~(t I) ~ C1 U C 2 9 Therefore
323
3. 5 Dynamical systems and nonautanomous differential equations In Section 3.1 we have investigated the relationships between the abstract theory of dynamical systems in the Euclidean n-space (Chapter i) and the properties of flows defined by the solutions of an autonomous differential equation.
Conditions
have then been derived under which the flow defined by the solutions of such autonomous differential equations indeed defines a dynamical system.
On the other hand,
in Section 3.1 most of the theorems have been proved for the more general case of the time varying differential equation 2.1.5 for which the flow defined by its solution does not immediately define a dynamical system.
In this section, without
claim of completeness, we shall present the few general results available on time variable flows and in particular on the
~Iow$ defined by the solutions of time-varying
differential equations, having the property of uniqueness and existence in the large. Given the time-varying differential system
3.5.1
i = f(x,t)
we can introduce in the system a new independent variable instead of dt d~
through the relation
--
t,
say
~,
Then the system 3.5.1 can be written in the following
i.
equivalent form (called parametric form)
I dxi
3.5.2
0
3.5.21
Notes and references
{inf[sup(If(x,t)
- g(x,t) l:Ix I + Itl .< T), l/T]}.
Most of the material presented in this section is due to G. R. Sell [5].
329
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. The theorems that we shall prove in the sequel are given in the language
and technique of differential equations.
When not otherwise stated, these theorems
will only apply to strong stability properties. We shall present the Liapunov second method essentially for the case of the autonomous differential equation
3.6.1
where
i = f(x),
f(x)
is defined and continuous for all
x ~ E.
From the material presented
in Section 3.5 it must be obvious to the reader that also the case of the nonautonomous equation 3.5.1 can be included in this framework. From the operational point of view in the second method of Liapunov, the stability properties of closed sets Will be characterized by the relative properties of a pair of functions
v = ~(x)
and
w = ~(x)
connected to the
differential system 3.6.1 through the relation n 3.6.2
~(x) =
For a given
~(x)
=
I i=l
the scalar function
derivative of the scalar function
v = ~(x(t))
~x. 1 ~(x)
fi (x)
is simply the total time
along the solution curves of the
differential system 3.6.1; thus
3.6.3
d__~v dt = @(x)
For a Riven
~(x)
the relation 3.6.2 is a linear partial differential equation,
which will have a solution
~(x)
if ~ntegrability conditions are satisfied.
integrability conditions can be defined in the following way: function
~(x)
and a vector
f(x) # O,
a vector
b(x)
These
given a real-valued
may be chosen such that
330
3.6.4
*(x) = . n(n-1)
is the gradient of a scalar function if the
2
conditions : ~b.(x) x
3.6.5
~b.1(x) =
~xj
(i,j = l,...,n)
~x.
1
are satisfied. We shall now first prove a set of theorems which relates the stability properties
of a given compact set
M
with the sign and uniform boundedness
properties
(see Chapter 0) of the real valued functions
v = ~(x)
and
w ffi ~(x).
The same theorem holds for the case of sets with a compact neighborhood.
3.6.6
THEOREM Let
v = r
neighborhood
i)
and
N(M) C
v = r
be real-valued functions defined in an open
of a compact set
E
M.
Ass~ne that
1
ii) v ffi r
is definite for the set
iii) w = ~(x)
M~
is semidefinite for the set
iv) for all V) r
w ffi ~(x)
with
x ~N(M)
and
~(x)
M,
~(x) ~ 0,sign ~(x) # sign ~(x)~
satisfy the relation 3.6.2.
~hen the oompact set
M is (uniformly) stable. Proof.
Since the real-valued
function
~[x)
is definite for the set
Lenmm 0.~.3, it follows that there exists a real number increasing function
3.6.7
Let
~(v)
and
8(v),
~Co(x,M)) .< ~b(x) .< 8 ( p ( x , M ) )
E > 0(E ~ ~)
be given and choose
with
for
6 > 0
n > 0
~(0) = 8(0) = 0
x{~S[M,r[] C N(M)
such that
M,
from
and two strictly such that it is
331
~(~) < ~(~)
3.6.8
that is, such that
0 < a < ~-l(a(d)
3.6.9
where
~-i
denotes the inverse of the function
We claim that
p(x~
~< 6
implies
Obviously ~ < e.
8(v)-
p(x(x~
< e, ts R +.
In fact, in the set
S [M, e] ~(x)
3.6 .i0
:
.< 0
;(x(x~
which gives
~(p(x(x~
3.6.11
.< r176
If there would exist a
t ) ) .< ~(x ~
t I > to
.< ~(p(x~
such that
.< ~(a).
p(x[x~
) : e,
then we
would have
3.6.12
cx(e)
which contradicts the choice of
~< L3(r
~
in 3.6.8 and proves the theorem.
For sake of completeness and for a better understanding
of insta-
bility , we shall now state an obvious corollary regarding negative Liapunov stability of a compact set
COROLLARY
3.6.13
I r a compact set replaced by then
M
3.6.14
M.
M
satisfies Theorem 3.6.6 with the condition ivP
iv$ sign ~(x) = sign ~(x)
for all
x ~E
~(x) # O,
is negatively stable. Remark From the proof of Theorem 3.6.6, it is obvious
a
with
dynamical system, as shown by Theorem 1.5.~)
(as already known for
that a set
M
which satisf;e~
332
Theorem 3.6.6 is positively
invariant.
THEOREM
3.6.15
Let
v = ~(x)
open neighborhood
i)
and
N(M) C E
be real-valued functions, defined in an
w = ~(x)
of a compact set
M.
Assume that
= ~(x)~C I,
v
ii) v -- ~(x)
is definite for the set
M,
iii) w = ~(x)
is definite for the set
M,
iv) sign ~(x) # sign ~(x), v) ~(x) M Proof.
In
and
S[M,~o],
the inequalities
3.6.7 are again satisfied and,
strictly increasing
functions
such that
-~(p(x,M) .< ~(x) . 0 < E
~ ~
it follows that such that for
o
x(x~
is uniformly stable.
8(6 0 ) < a(eo).
t ~ R +,
since
(p (x(x~ such that
~(x(x~
It follows
M
M
Then p(x~
is stable.
To prove the theorem
~ 60
implies that
We assert that
implies that:
lim t§
3.6.17
e ~ > 0,
y(v), ~(0) = y(O) = O,
From Theorem 3.6.6,
For any
satisfy the condition 3.6.2. Then the compact set
there exist two additional
3.6.16
9(x~
~(x)
is (uniformly) asymptotically stable for the system 3.6.1.
furthermore, ~(v)
and
= 0 p(x~
< 6
we set
~(t) = ~(x(x~
= ~(t) .< -y (p (x (x~ , t) ,M) ) ,
then that
~(t) - ~(to) .< - I t t
y(o (x(x~ o
,M))dT
t >~t
o "
We then have
333
Now let
e > 0(e < 60)
8(6) 0(6 < E)
p(x(x~ ~ 6
< e
then
for
p(x(x~
such that
t ~ t I. < e
for
O
t ) to .
If
6 < p(x~
~ 6 ,
then as long as
p(x(x~
> 6
we have
O
t @(t)
- @(t o ) .< -
y(6)dr t
= -(t
- to) Y(6)
0
or
3.6.18
.
0
and
Thus
p(x~
~6.
This
O
completes the proof.
3.6.20
Remarks In the proof of the theorem no use has been made of the left hand part of
the inequality 3.6.16.
By proceeding as before, one can derive the analogue of
inequality 3.6.18 as follows:
3.6.21
Now
~(e) =
T(e)
~(6 o) - S(6) ~(6)
~ t - to
is the minimum time in which the solution of 3.6.1 can cross in the ring
S[M,6 o] ~ S ( M , 6 ) .
By
the same argument as in the above proof of Theorem 3.6.16,
it follows that 3.6.21 does not hold for
t < t
+ T(e).
Thus ~ p(x(x~
O
for
t ~ t
+ T(e) O
for all
t
~ 0 O
and
p(x~
~ 6.
Thus the solutions
> e
334
x(x~
have a uniform rate of approach to From all theorems on asymptotic
M
in
N(M).
stability of compact sets it is
possible to derive trivial corollaries on the complete instability of such sets by reversing the requirements t,
and, therefore,
of the relative sign of the independent variable
inverting the direction of motion on each trajectory.
For
example, from theorem 3.6.15 it can be deduced that
COROLLAR~
3.6.22
If a compact set
satisfies Theorem 3.6.15 with condition iv) replaced by
M
sign @(x) = sign ~(x) ,
then the s e t
M is completely unstable.
We shall now prove the theorem whlch provides sufficient conditions for the instability of a compact set for the differential system 3.6.1.
3.6.2 3
THEOP~R4 Let
v = ~(x)
non-empty set
and
w = ~(x)
B C S(M,n) C E,
be real-valued functions defined in an open n > 0
where
and
M
is a compact set.
Assume
i) ii) @ ( x ) = 0 for
xE[@BOS(M,n)]
, @(x)r
0
for
x([IB
OS(M,n)],
iii) v = ~ ( x ) ~ C l, iv) sign @(x) ffi sign ~(x), v) for all vi) ~(x)
and
for
x ~B/~
x ~ B , l ~ ( x ) I .< {3(p(x,M)) ~(x)
and
S~,n)],
I~(x) l >. y(p(x,M)),
satisfy the condition 3.6.2.~Y-hen the compact set
M
is unstable for the system 3.6.1. Proof. x
o
Assume that
~IB, p(x~
x(t) = x(x~ Integrating v) we obtain
0 such that
in
~(x ~
and the values of ~(x) ffi ~(x)
~.
For a sufficiently small > O.
~(x)
6 > 0
Consider the corresponding
along such solution
there exists solution
~(t) = ~(x(x~
along such solutions and taking into account the condition
335
,(t) - ,(t o ) =
y(O(xC~),M)d~
*(~) d~ ) t
t
o
o
and
~(t) >. y(p(X(to),M)).
If for all
t ~ to, x(x~
the hypothesis (v). X(tl) ~ ~B ~ ~S0~,n).
(t - to) + ,(to)
~ ~B,
then
lim
Kence there exists Since, for all
,(x) >. e(0(x(x~
~(t) = + ~,
t = tI > to
t >~ t o
0; we cannot have
which contradicts
for which
for which
x(t) ~ IB
~(tl) = 0 ~ B
thus
p(x(tl),M) = q
and the theorem is proved.
It must be pointed out that, from the hypothesis of Theorem 3.6.23, the set M.
B
cannot have any compact component which does not contain a component of
In fact, if there would exist such compact component
exist
(3.8.25) at least one point
~(y) = 0
y ~B c
such that
which contradicts the hypothesis iv).
Bc
then there would
grad #(y) = 0.
On the other hand,
Hence B
need not
be a region, but it could be formed by a sequence of sets with non-compact closure which satisfy the conditions of the theorem. From the theorems given it follows that
3.6.24
THEOREM If there exists a pair of real-valued functions
satisfying the condition 3.6.2, in the neighborhood x~M,
N(M) C
E
where and
~(x)
~(x) ~ C I is such that
~(x) = 0
then the additional sign properties of the function
i)
If
~(x)
ii)
If
~(x)
M
M
for all
M. then from the theorem
is asymptotically stable.
is definite and
(3.6.22) it follows that
~(x),
~(x) completely
is definite and sign ~(x) @ sign ~(x),
(3.6.15) it follows that
and
is definite for a compact set M
characterize the stability properties of the compact set Proof.
~(x)
sign
~(x) = sign ~(x),
is completely unstable.
then from Corollary
336
iii)
If
unstable.
is indefinite,
then Theorem 3.6.23 insures that
M
is
Finally
iv) for
~ (x)
If
~(x)
is definite for
M, ~(x)
cannot be semi-definite
M. In fact, if
#(y) = 0,
#(x)
is seml-definite the set
is the absolute minimum of the
that for all
y~G~M,
grad ~ )
= 0
G~M
such that if
#(x), and since
and, thus,
~)
= 0
#(x) E C I,
y~G,
it follows
for some
y
!
M
which contradicts the hypothesis and the theorem is proved.
Notice that Theorem 3.6.24 does not give necessary conditions for the stability of and
~(x)
M.
In fact, there do
not always exist real-valued functions
satisfying 3.6.2 and such that
~(x)
~(x)
is definite for a given (positively)
invariant set.
3.6.25
DEFINITION A real-valued function
v = ~(x)
which 8atisfie$one of the stability
theorems i8 called Lzapunov ~unctzon.I I II
Q
9
Theorem 3.6.15 and Corollary 3.6.22 define only local properties of the compact set
M.
That is, if Theorem 3.6.15 is satisfied, then there exists a
sufficiently small asymptotic theorems,
~ > 0,
such that
stability of the set
S(M,~)C
M.
A(M)
where
A(M)
is the region of
For the practical applications
local properties are not very useful.
It is, therefore,
of the stability
important to
give theorems which provide sufficient conditions for global asymptotic or in the case in which the compact set the exact identification
M
is not globa~asymptotically
of the region of asymptotic
an approximate identification of the set
stability
A(M)
stability stable, allow or at least
~(A(M)).
Our first concern is to derive a theorem which will provide a sufficient condition for the global asymptotic
stability of a compact set
M.
337
3.6.26
THEOREM If the conditions of Theorem 3.6.15 are satisfied in the whole space
E
and, in addition, vl)
tim
~(x) = = .
l[xll §
Then the compact set Proof.
M
is globally asymptotically stable.
Along the solutions of the system 3.6.1, let
;(t) = ~(x(t)) = -X(x(t))
Assume that
X(X)
is positive definite for
M,
For all
t
with
t
.< t ~< t I o
3.6.27
0 .< ~(t) = ~(to) -
We claim that
X(x(~))
i
t X(x(T))dT. t o
is an integrable function in
3.6.2,7 and condition (vi), it follows that if lim X(X(T))dT ~ - ~, t§ We shall o now prove that
3.6.28
lira t§
•
[0, + =).
were not integrable, then
which contradicts the hypothesis on the sign of
x(x~
= 0
for all
(tn,t n + %)
with
p(x(t), M) >. e I
tn § + ~, ~ > 0,
for
tn ~ t ~ tn + %; n = 1,2,...;% > 0 Thus 3.6.28 follows. M
Q.E.D.
and a sequence of
x@ E
n = 1,2,...;% > 0
we have
•
~ e2
which contradicts the integrability of
for •
Since the hypothesis of the theorem obviously implies that
is stable, it follows from 3.6.2B that
stable.
eI > 0
such that
tn .< t .< tn + ~;
But then condition vi) implies that for all
X(X).
x~
In fact, if this were not true, then there would exist a intervals
In fact, from
M
is globally asymptotically
338
Remark
3.6.29
Theorem 3.6.26 would be also true if instead of condition vi), one simply required that condition 3.6.28 ~e satisfied for all that condition
The fact
(vi) is not necessary will be shown by the following theorem which
is a trivial corollary of Theorem 3.6.15 . in practice does, however,
The Liapunov function commonly used
satisfy the condition vi).
THEOREM
3.6.30
Let space
x ( E.
E.
v = ~(x)
and
8 = e(x)
be real-valued functions defined in the whole
Assume that
i) v = ~(x) E C 1, ii) v = #(x) iii)
is definite for a compact set
lira
M,
~(x) = q > 0 ,
llxil§ iV) 0 = 8(x)
v)
~(x)
vi) #(x) M
=
be positive definite for the set
eCx)C~(x)
- n),
~/(x) satisfy the condition 3.6.2. Then the compact set
and
is globally asymptotically stable.
By extending the definition of the function set
B
M,
#(x)
and
@(x)
with noncompact closure one is able to show the existence in
which tend to infinity and have the so-called global
to an open B
of solutions
(but not necessarily complete)
instability.
DEFINITION
3.6.31
A co~pact set
M ~ E
will be called globally unstable (for the flow
defined by the system of differential equations 3.6. I) if there is a sequence {x" }
of points in
each
n.
3.6.32
C(M), x" § M,
such that
IIx(x", t) II ~ + ~
as
t § + ~
for
THEOREM If in Theorem 3.6.23 the vet
~
is nonco,~act, then
M
is globally unstable.
339
Notes and References
3.6.33
The idea of characterizing by means of the sign properties
the stability properties
of differential equations
of a real-valued function is due to Liapunov
[i].
A similar idea in a much more geometrical context, quite near to our point of view 9
is to be found in the work of Poincar~ develops
in
E2
a method *h.%~ e~&f~= ~o ~ e
domair~ of the plane. set
rk~,~e,~
9
Here Polncare
of limit cycles in a certain
This information is derived by analyzing the properties
@(x) = = 0
system, i.e., a real-valued secting,
[i, Vol. i, pg. 73 ff].
(contact curve) where
function such that the curves
v = ~(x)
of the
is a topological
~(x) = const, are noninter-
closed and differentiable. Methods quite close to Liapunov's have also been suggested by ~. Hadamard
and D. C. Lewis
[I ]
[ 2].
We want to emphasize again that Liapunov was originally interested only in the stability properties Let
~ = g(y,t).
Let
of a given motion.
y i = y l(t )
y = y(t)
such that
{lyl(t) - y(t) l I
. t .
~ > O,
In order to
we shall consider solutions
and see if this implies that
This can be easily done by defining a new
O
variable:
x
i =
y
-
y
(t)
Then from the differential equation equation
@ = g~,t)
~ = f(x,t) = g(x + yl(t),t)-g(yl(t),t).
the perturbed motion. for the equation point
.
x = 0
one can obtain a new differential This equation is called equation of
Notice that the stability problem for the motion
@ = g(y,t)
of the equation
y
i
= yl(t)
is now reduced to the stability problem for the equilibrium i = f(x,t).
Theorem 3.6.6 and 3.6.15 are natural extensions
of theorems of Liapunov
[i ].
Theorem 3.6.23 is the extension to compact sets of a theorem due to Chetaev
[2 ].
340
Theorem 3.6.26 is due to E. A. Barbashin and N. N. Krasovskii
[i].
Theorem 3.6.So is an extension of a well known theorem due to Zubov [3,6]. Results for the stability of noncompact sets for differential equations are given in the works of G. P. Szeg~ [3], G. P. Szeg8 and G. R. Geiss [i], and Yoshizawa
[7]. The problem of existence of Liapunov functions for differential equations
(converse problem) has been discussed by many authors, notably J. L. Hassera [5,6], N. N. Krasovskii
[3,6,7,8,9], K. P. Persidski
[2], Vrkoc [i] and J. Kurzwell [1,2]
and Kurzweil and Vrkoc [i]. Stability problems for time-varying differential equations can be found in the excellent review paper by H. A. Antosiewicz and
T. Y ~ h i z a ~ a
[ I0 ].
[3] and in the books by W. Hahn [2]
It has to be noted that most of the results for the
stability of equilibrium points for time-varying differential equations presented in the classical literature can be derived as particular cases of stability theorems for noncompact sets.
341 3.7
New results ~ t h rel=med conditions. Do we really need that a function
definite for a compact set
M~E
v = r
be (locally) positive
to be able to prove that
M
is stable?
Even if this is necessary and sufficient, it may be simpler to use a function which is not definite even if there exists one which is.
The answer is no.
In fact, even indefinite functions may sometime be quite adequate to prove stability, as we shall show by an example.
In what follows we shall restrict
ourselves to the case of a continuum (a compact and connected set)
M~E.
This is not a restriction with respect to compact sets, since we know that if a compact set is stable all its components are stable.
This stronger stability
theorem for the differential equation
3.7.1
i = f(x)
is based upon the following lemma whose proof is obvious.
3.7.2
/~/~MA
Let
v = r
and
open neighborhood
N(M) ~
i ) vffi r
EC 1,
ii) ,(x) .< 0,
E
be real-valued functions defined on an
w ffi ~(x)
of a continuum
M.
Assume that
xEN(~),
iii) ~(x) = ,
iv) Qk = {x: ~(x) .< k} Then for every real
k,
k
real.
every compact component of
Qk
which is contained in
N(M)
is (strongly) positively invariant for the flow defined by the solution of the differential equation 3.7. i. Let now which contains x = x(t,x~ properties of
r M.
x~ M
= 0
for
x~M
Clearly then if
and let ~
~
(k > 0)
be the component of
Qk
is compact then all solutions
of the differential equation 3.7.1 are bounded.
The stability
are then clearly related to the geometrical properties of
~.
342
3.7.3
THEOREM Let
v = @(x)
open neighborhood
and
N(M)~E
3.7.~
w = ~(x)
be real-valued functions defined on an
of a continuum
IIHkll:
M.
Let
sup {p(~,M):x ~ H k } ,
i) v = @(x) ECl , ii) @(x) = 0, x e S ,
iii;
k~-limo+ I1~11 -~ o,
iv) ~(x) = .< 0, Then the continu~
M
x & r.
is (positively strongly Liapunov) stable for the differential
equation 3.7. i. Proof:
The condition ill) of the theorem is equivalent to the following
given any
one.
there exists a
From the continuity of
e > 0
such that
q(e) > 0 x~
n > 0,
S(M,e) ~ ~ .
such that
S(M,e)~ ~
~
k > 0
such that
II~I I< ~
condition:
The proof is the usual
and condition ii) it follows that there exist
r
From Condition (iii) we have the existence of
S(M,q)CN(M).
From Lennna 3.7.2 we have that
implies that all solutions
3.7.1 have the property that for all
x = x(t,x ~
of the differential equation
t >~ to, x(t, x ~
which
is (positive strong Liapunov) stability and completes the proof.
3.7.5
Remark If
v = r
is positive definite and continuous in
condition (iii) of Theorem 3.7.3 is satisfied.
N(M),
then
However, there do exist semi-deflnite
and even indefinite functions which satisfy condition (iii) in Theorem 3.7.3.
Thus the
above theorem seems stronger than the classical theorem of Liapunov on stability (3.6.6).
3.7.6
Ezc~ple Consider the second order differential equation:
343
+ r 2 (r S l n r 1-
cos
s +x
where
= 0,
r = x
2
+ y
2
or the equivalent system
= y,
~ = -x - r 2 (r sin i - cos ~) y,
where
r = x 2 + y2
We take
#(x,y) = (x 2 + y2) sin (
2 x
The function
~x,y)
=
For the above system
-2 y2r2 (r sin --i COS i) r
+ r cos l_r 4- ~ ) r
= -2y 2 r (r sin l _ r ~(x,y) ~ 0,
origin is thus $(x,y)
S i n -1 r
(-2y2r 2 (r sin _lr - cos ~))
i = -2y 2 (r sin ~ -
Notice that
2)
+y
is indefinite in any neighborhood of the origin, but satisfies
the condition (ii) in Theorem 3.7.3.
r
i
i i I cos ~) (r2 sin---r r cos ~)
cos l) 2 r
and all conditions of Theorem 3.7.3 are satisfied.
The
stable for the given differential system, although the function
is not even semi-definite.
We shall now present a very general Theorem (3.7.11) which gives sufficient conditions for asymptotic stability and attraction of compact sets under much less stringent requirements then those of the classical theorems in Section 3.6.
In
particular, we shall relax condition iii) of Theorem 3.6.15. Our main reason for relaxing condition (iii) of Theorem 3.6.15 is practical convenience.
In fact, from the theoretical point of view, if a compact set is
asymptotically stable, then there always exists a Liapunov functlon~i.e., a function
344
which satisfies all the requirements
of Theorem 3.6.15.
This fact has been proved
for a dynamical system in Section 1.7 and can be proved for the special case of a differential function,
system.
While the theory assures us of the existence of such a scalar
in practical cases it may be rather difficult to find one which satisfies
all requirements.
The enlargement of the class of Liapunov functions may be extremely
helpful for the solution of stability problems.
The severity of condition
(iii)
of Theorem 3.6.15 can be quite well illustrated by the following example.
3.7.7
EzampZe Consider the second order differential system
~ffiX
3.7.8 = o(•
,o(o,o)
:
0
which is derived from the second order differential equation
eCfi,n)
=
We are interested in establishing X = n = 0.
the stability properties of the equilibrium point
For this consider the real-valued function 2
3.7.9
~(X,q) = all(X ) + a22(n)q
where the real-valued functions
aii(X,~ )
are defined in the whole plane
X,~.
Consider then the total time derivative of 3.7.9 along the solutions of 3.7.8.
$ = q,(x)
=
8all x
+
~a22 8(X,n)q + 2a22(q)8(X,~)] %-6--
This scalar function vanishes identically on the axis
definite for
M ffi {0}
for a_ll differential
q = 0.
Thus
and the condition (iii) is never satisfied.
~(x)
is not
This means that
systems of the type 3.7.8 no scalar function of the class 3.7.9
can be used to prove either asymptotic
stability or complete instability of the
345
critical point
X = n = 0.1t is immediate that this is the case for all
real-valued functions
~(x)
whose level curves are orthogonal to the axis
X.
In
fact, all solutions of systems of the form 3.7.8 have have this property. Thus all systems of the type 3.7.8 have solution curves which are tangent to the level curves of the function 3.7.9 on the axis of the function is, i n m o s t
#(x)
cases
n = 0.
Thus this particular property
with respect to the solution curves of the differential system
and in particular in the case of Example 3.7.8, not a property
of the norm of the solutions and
therefore
is not a stability property.
It seems
obvious that, at least in some cases, it should still be possible to use such a real-valued function properties of sets.
v = ~(x)
for the characterization of the asymptotic stability
This will be done in the next theorem.
The key of the whole problem is in the particular properties of the set
P = { x ~ E: ~(x) = 0}. In this set we can distinguish 3 different components
i)
PI = { x ~ E :
grad ~(x) = 0}
ii)
P2 = { x E E :
f(x) = 0}
iii)
P3 = {xe E: grad #(x)
or, which is the same,
P3
orthogonal to
is the set of all points
f(x)} o x , in
which at least one of the corresponding solutions of the differential equation 3.7.1 is tangent to the level surface
~(x) = C,
defined by
~-l(c) = x ~
Along these lines the
following theorem is of interest.
3.7. i0
THEORE~ Let
neighborhood
v = ~(x)
N(M)C
E
and
w = ~(x)
of a compact set
be real-valued functions defined in an open M.
Ass~ne that:
i) v = ~(x)~C I, ii) ~(X) = .< 0
~ x~N(M),
346
iii)
There exists
x~
such that at least one solution
of 3.7.1 is b o n d e d and such that
Proof:
Let
Yl
and
§ Yl
~(x(t,x~
and
§ Y2"
=
Since
lim ~(X(Zn,X~
oo
n
-+
,
~(x) - 0
on
A+(x~
By definition, then, there
{~n } ~ R +, 9n + + ~ ,such that x(t,x~
Then from the hypothesis (iii)
lim r176 -+
X(Tn'X~
+ 0~ and
7hen
A+(x ~
is a non-increasing function of
continuity.
~
N(M).
be two points in
{in}6 R +, tn §
exist sequences X(tn 'x~
Y2
A+(x~
x = x(t,x ~
t
N(M),
by hypothesis (ii),
which is bounded from below because of lira ~(x(t,x~
exists and is such that
which proves the theorem.
oo
We shall now prove the main theorem on asymptotical stability and attraction.
This will be done for the special case of a differential equation which
defines a dynamical system.
3.7. ii
THEOREM Let
~
be a compact, positively invariant set for the flow described by
the differential equation 3.7.1, and
w = ~(x)
which defines a d y n ~ i c a l
be real-valued function defined on
i) v = r
I,
x~,
ii) w = ~(x) .< O,
x~,
iii)
~(x) = .
Consider the following sets I) S II) Q III) P IV) M V) U
= largest invariant set in = {xf~: ~(x) = {xs
~
is a minim~, on
~}
~(x) = 0},
= largest invariant set in
P
(M ~ S ) ,
= largest invariant set contained in
Q. Then:
Let
v = r
347
a)
~ll these sets are closed,
b)
Q~P
c)
M
i8 attracting relative to
d)
S
is asymptotic stable relative to
e)
~M N 3 S ~ r
~)
S = D+(M) = {y: ~{x } ~ , { t
and
Q
i8 stable relative to
where
to
and
and
~ ,
such that
n
g)
S
h)
~f for all
x
§
M
relative to
-> y}
x t
n
i8 the first positive prolongation of
D~(M)
~
~ 9
}~R +
n
~ ,
n
n
~,
is the smallest relatively asymptotically stable set containing x E ~M, ~(x) = const~ M
M,
i8 asymptotically stable relative
M = S i)
M
minimal implies
j)
M = Q
k)
if either
implies
M = S = U~
M
is asymptotically stable relative to
M=S~
~, S
or
M
are homeaaorphic to the unit ball, then
M
contains a rest point, l)
if either
M~I~
or
S~I~,
then the words "relative to
~"
may
be deleted from the above statements.
Proof:
a)
This is clear, for if a set is invariant, so is its closure.
b)
For all
x s Q, grad #(x) = 0
implies
@(x) = 0
which implies
Q ~P.
Stability then follows from the usual theorem. c)
Since for each A+(y) ~
d)
e)
M
as
ys ~, A + ( y ) ~ A+(y)
S ~ M,
S
is
largest
invariant
For if
@(x) = 0
is attracting relative to
~M N ~S = ~,
A+(x) # ~
and
set
then
in
~,
f~,
asia
and A+(x) N M = ~,
attracting relative to
for
x s
we get
is invariant.
Since the
~
since
D (S)~
~.
Stability follows as
a,
so that
Da (S) = S.
being invariant, we have M~
S,
contradicting (c).
S
we conclude that
x6~S, M
is not
348
f)
Follows from proof of (d).
g)
Obvious.
h)
We need only show that
then there is a sequence
{xn}
Xnt n + y ~ M .
such that
Indeed
invariant and compact.
Since
However,
we get
if
z EA+(y),
#(x) ~ ~(y) ~ ~(z) = #(x), yRr-~ that
and
#(x)
yRCM,
in
M
xn § x~M,
~,
y~
is stable relative to
~(Xn) ~ ~(Xntn) , #(z) ~ ~(y),
showing that
is constant on
and a sequence
x n[0'tn] ~ n
and
~.
we get
and since
~(y) = #(x).
,yR. Consequently
If not, {tn},
tn ~ 0,
is positively
as
~(x) ~ #(y) z ~ ~M,
by continuity.
we have
This shows, however, ~(x) = 0
on
yR,
that
showing
a contradiction. i)
If
M
is minimal,
then
follows from (h) and, the fact that j)
#(x)
is constant on
and the result
U CS.
Follows from (h) and (c).
Q.E.D.
We consider now the problem of the identification means of real-valued functions
M
v = r
of a region
defined in a neighborhood
of
~ M.
by This can
be easily done with the help of Lemma 3.7.2 for pairs of real-valued functions v = ~(x)
and
w = ~(x)
which satisfy the requirements
of Theorem 3.7.11.
In
particular
3.7.12
Remark Let
some
v = ~(x)
k > 0 we have l l~II
and < ~,
3.7.13
~
be as in Theorem 3.7.11.
Ass~e
that for
then we may take
ffi~ ,
Remark
3.7.14
Let N(M)
w = ~(x)
of
M,
v = r
where
Theorem 3.7.11 to
w = $(x)
~(x) ~ 0. ~.
be continuous functions defined in a neighborhood
If for any
If further
S~1%),
k > O,
~
is compact, then w e , m y
then notice that
M,
and
S
apply
are
349
respectively an attractor and an asymptotically stable set, and region of k
attraction.
the same set
S
Moreover
k > 0
is the largest invariant set in
(region of attraction of O (~) = E,
if for each
~ ~,
S)~ and indeed in this case also
~
is compact then
is in their and for each
U~)~A(S)
[J%)~A(M).
then we can detect a globally asymptotically stable set
Lastly if
by means of
Theorem 3.7. ii.
3.7.15
Instability Theorems. Such theorems can easily be derived from Theorem 3.7.11.
fl is compactgnegatively invariant~and
~(x) >. 0,
will be reversed in the sense that the set completely unstable with respect to
M
Note that if
then conclusion of Theorem 3.7.11
will be negatively attracting, i.e.,
ft. This observation can be used to derive the
classical instability theorem of Chetaev for example and many others.
3.7.16
Notes and References T h e extension of the Liapunov T h e o r y presented in T h e o r e m $. 7. $
is due to A. Strauss [5]. T h e extension of the Liapunov theorem for asymptotic stability of the rest point x=o
invariant
allowing ~(x) to be semidefinite while the largest
s e t c o n t a i n e d i n the s e t [ x s
V(X) = o} is to}
, i s due to E . A .
Barbashin
and N . N . Krasovskii [ I]. A n extension of Liapunov's theorem for attraction of compact sets leading to the approach stressed by the theorem 3.7. ii w a s originated byJ.P,
e a Salle i3].
350
3.8
Yhe e~tens~on theorem. The "classical" theorems 3.6.15,
defined behavior of both
~(x)
and
3.6.22 and 3.6.23 require a very well-
~(x).
hand, connected by the equation 3.6.2.
Those two functions are, on the other
We shall show that, if the given system
3.6.1 satisfies certain conditions, then from the global properties of the properties of
~(x)
in many applications where the function space, while the behavior of ~(x)
and
~(x)
~(x).
has, therefore, three steps:
grad ~(x)
M,
one
~(x)
This fact will be of extreme help
has known properties in the whole
is not known for large values of
~(x)
i)
p(x,M).
The
is given by equation 3.6.2 which contains
information about the gradient of the real-valued function
global properties of
and
in an arbitrarily small neighborhood of the set
is able to deduce the global properties of
connection between
~(x)
~(x).
This problem
deduce from the global properties of
~(x)
the
grad ~(x); ii) deduce from the global properties of
and the local properties of
~(x)
the global properties of
iii) deduce from the local stability properties of a compact set
M~E
~(x); in the flow
defined by the solutions of the ordinary differential equation 3.6.1 and the global properties of
~(x), the global stability properties of
M.
The theorems that we shall
present are called extension theorems because they give conditions under which the local stability properties of compact sets can be extended to the whole space.
This
problem is essentially an investigation of the relationships between topological and analytical properties in differentiable function
E
of the level lines of the real-valued, continuously
v = ~(x)
and the stability properties of the ordinary
differential equation
3.8.1
~ = grad ~(x)
or
3.8.2
gr~d ~ (x)
351
In general, we will be interested in characterizing of a compact set
M~E
the stability properties
in the flow defined b F the solutions of the ordinary
differential equation
~ ffi f(x)
3.8.3
through the analytical and topological properties of the level lines of the realvalued, continuously
differentiable
function
v = ~(x),
which has the property that the
real-valued function
3.8.~
(x) =
is definite in a suitable open set
N(M).
The results that we shall obtain are related
to various problems of differential geometry and topology.
In the sequel we shall use
the notion of critical points of a real-valued function.
3.8.5
DEFINITION Let
v = r
~ C1
be a real-valuedfunction defined in
E.
A point
c
x EE
for which
grad r
function
v=
sequence
{xn}: I lxnll § ~,
r
r
c) = 0
is called a critical point of
r
The real-valued
is said to have an infinite critical point if there exists a such that
grad r
does not have any critical points in
n) § 0
E ~ {~}
as
l lxnll § ~.
we mean that
r
By saying that has neither
finite nor infinite critical points.
We shall prove next the "extension theorems" for strong stability properties of compact sets.
The first theorem is an extension theorem for asymptotic stability for
the case in which the differential equation 3.8.3 defines a dynamical system.
This
theorem will then be used for the proof of a stronger result (Theorem 3.8.13) on the analytical properties of Liapunov functions.
352
3.8.6
THEOREM Let
space
E.
v = ~(x)
Let
i)
and
be a compact set. Assume that
M~E
v=
be real-valued functions defined in the whole
w = ~(x)
~(x)~C I
ii) ~(x) = 0
,
for
x ~S
iii) there exists iv) ~(x) = 0
n > 0
for
, such that
x~M,~(x)
# 0
~(x) # 0
for
x~M,
for
x E H(M,~)~
does not have zeros at
~(x)
infinity~ v) sign ~(x) # sign ~(x)
for
~ E S ( M , n ) \ M~
vi) ~(x) = , vii) the differential system Then the compact set Proof:
M
~ = f(x) defines a dynamical system.
is globally asym~otically 8table.
The conditions of the theorem imply that
i.e., the set
P = {x:x~S~,n)~
~(x) ~ 0
can have only isolated points, because at a point
P
one has
grad ~(y) = 0,
get a contradiction to (iv). N
of
z,
N CS(M,n)
extr,mal point of contradicting asymptotically
(iv).
invariant set.
then
Thus
If
P
which is a limit point of
~(y)=0
has an isolated point
such that
~(x) ~ 0
and
therefore
P
is empty.
for
z,
and since
we
then there is a neighborhq
x ~N, x # z.
grad ~(z) = O:
y~M,
But then
z
is an
also, by (vi)~ ~(z) = 0
This shows that the set
M
is (locally)
stable.
Let then
aA(M) = ~.
k M,
~(x)
y~P
which implies by (vi) that
x E S(M,n) \ M,
To see this notice first that
M, ~(x) = O} is empty.
P
for
A(M)
be the region of attraction of
We will show that
In fact, if
A(M) / E,
@ = A(M) ~ C(A(M))
union of two nonempty, a contradiction.
.A(M) = E.
and hence
then
M;
A(M)
is an open
This is equivalent to proving that
aA(M) = A(M) ~ C(A(M)),
E = A(M) U
C(A(M))
but if
which implies that
disjoint closed sets and, therefore, not connected.
aA(M) = ~, E
is the
This is
353
Now by asymptotic stability of S[M,6]C_A(M).
(x):xs
We claim that for all
xCSA(M),
~(x)
~(x(x~
it follows that Now, if for some
find an is a
x~A(M),
9 > 0
~(x(x~
~(x) > 0
Assume for simplicity that
= min{~
x~
there is a
M,
such that
such that
for
and let
In fact, from the hypothesis made on
it were
~(y) < ~
#(x ~
x(x~
~(x) ~ 9
xES[M,n]~M,
is a strictly decreasing
Then
is strictly decreasing. Since
for
such that
.
~(x) ~ ~.
yESA(M),
x~S[M,~]
6, 0 < 6 $ q,
< ~.
function of
t
for all
it would be possible As
x~
to there
~ S ~(x(x~
< #(x ~
< ~
for
This is absurd.
xESA(M),
we have by
(v)
that
~(x) < 0
for
xESA(M).
Let -p = sup{~(x):x By (iv) ~ > 0
since
x(x~ t)ESA(M)
for
~(x(x~
~A(M)
.
is bounded away from
t >. 0,
= ~(x ~
8A(M)}
since
+
it t
8A(M)
~(x(x~
M.
Let now
is invariant.
~ ~(x ~
-
Then
[tp t
o
then
x~
dT = ~(x ~
- p(t-t o)
o
which shows that
lim ~(x(x~ = - ~, which is absurd, since we have t+ + ~ proved that for all xESA(M), ~(x) ~ ~. This contradiction shows that 3A(M) = and proves the theorem. 3.8.7
Remark Theorem 3.8.6 still holds if one replaces either condition
iii'): M and
is (locally)
asymptotically
stable, or if one replaces conditions
(lii) with
iii''):
M
(iii) with
is invariant and (locally)
asymptotically
stable.
(ii)
354
We shall now prove an extension theorem for the case that equation 3.6.1 does not define a dynamical system.
The proof of this extension theorem is based upon
the following fundamental l~,~a on a property of real-valued functions.
3.8.8
LEM~4 Let
compact set.
v
~(x),
=
be a real-valued function defined in
E.
F,et
M~E
be a
A s s ~ e that
i) v =
E c 2,
r
ii) @(x)
0
for
x GM ,
iii) @(x) > 0
for
xGH(M,6),
=
iV) for
{xn}cE
~ > O~
, grad 4p(xn) § 0
~plie8
x
Then there exist two strictly increasing functions
n
+M.
~ (~ )
and
B(~), (x(o) = B(o) = o,
such that (o (x,M)) .< ~ (x) .< S (~x,M) )
3.8.9
and, furthermore, a.8.1o
lira ~(~)
=
+
=.
In addition to this, if = mi= {r
then Proof:
@(x) >
for
x~a(M,~)},
x s (s[M,~]).
Consider the differential system defined by
i = f(x) = -
3.8.11
grad ~(x) i + [[grad ~(x) ll
It is well known that the differential system 3.8.11 defines a dynamical system. Conditions
3.8.12
(ii),
(iii) and (iv) above imply that the function
$(x) = ffi -
~(x)
ljq[grdd ad Cx, Li 2 ~(x)[[
i +
and
355
satisfy the conditions of the Theorem 3.8.6.
Thus the set
asymptotically
Notice now that for any
X~
stable for the system 3.8.11.
E ~ S(M,~),
Since
there is a
#(x(x~
T > 0
for any
conclude that
~(x ~
for
x~M,
and
a(n)
and
8(n)
x~
x(x~
is globally
~H[M,~).
Thus
~(x(x~
E \ M , is a strictly decreasing function of
> #(x(x~
#(x)
such that
M
>~ 9.
Lastly as
#(x) = 0,
for
x~M,
>. ~. t,
we
~(x) > 0
is continuous, we can define two continuous increasing functions
by
a(n) = min (~(x) :x~E(M,n) } and
S(n) = max {~(x) : x E ~(~,n) } .
Notice that
a(0) = 8(0) = 0
and
8(n) >~ a(n) > 0
strictly increasing continuous functions
With these
a(n)
S(n)
.< a ( n )
and
8(n)
.< S ( n )
a(n)
and
for ~(n)
n > 0.
There are thus
such that
.< ~ ( n ) .
we have
(p (x,M)) .< + (x) .< ~ (p (x,M)) . It remains to be proved that there exists one function condition 3.8.10. t § - m.
We shall prove first that for all
In fact, notice that for any
,(x(x~
= ~(x ~
+
x~
x~
C~M)
~(x(x~ o
For
T .< 0
we h a v e
P(x(x~
>. rl > O.
Thus
~(x(xO,T)) .< max(~(x):x/S(M,~)} where
6 > 0
Hence for
is
t .< 0
such that
x(x~
= -X < 0
for
"~ .< O.
a(n)
C(M), $(x(x~
which satisfies § + =
as
356
o
r176
>, ~(x ~
+
xdT = ~(x ~
- Xt .
Jt Thus
~(x(x~
§ + ~
as
t § - ~.
Now the existence q + + ~
is equivalent
of our
~(q)
to the property
that the last assertion is not true. r P
= k
are compact for
k < h
with the property
that
Then there is a
and noncompact
for
as
~(n)
O(x,M)
h > 0 k >, h.
§ +
+ + ~.
"
as
Assome
such that the surfaces Consider the open set
defined by
P = {x: 0 < k < r
< h}.
This open set is bounded away from the set ~(x) = k
and
#(x) = h.
egress point of
P
For, otherwise,
and the surface
t) E {x: #(x) = h},
r
= k x
o
o
for every
x(x~
x ,
~(x(x~
~(x(x~
§ + ~
as
By applying
~(x) = h
x~
~(x ~
with
= k
is a strict
of strict ingress points
such that
= k
r
x(x~ t) E P
for
we will have a unique
set
< h t § - ~
~(x) = h
for
t ~ 0
for all
the fundamental
which is impossible. which contradicts
x~
lemma
only.
t < O.
t < 0
such that set
Notice that for such
the fact that
The theorem is completely proved. 3.8.8 we are now in the position of proving
systems which may not define dynamical systems.
THEOREM Let
space
consists
= k,
~(X ~
the extension theorem for differential
3.8.13
and is bounded by the surfaces
so that there will be continuous map of the compact
onto the noncompact
we have
M
Notice that each point of the surface
We claim that there is a point
a
#(x) + + ~
that
E.
v-- r
Let
M
i) v =
r
iv) r (x)
w = ~(x)
be a compact set.
be real-valued functions defined in the whole Ass~ne that
E C 2,
ii) ~(x) = 0
iii) r
and
# 0
for
xs
for
x6H(M,6),
~ > 0,
does not have (finite or infinite) critical points for
x
357
V) ~(x)
is semidefinite for M
vi) sign r
in
# sign ~(x), x E S(M,6),
vii) ~(x) = .
is globally asymptotically stable, for the differential
system 3.6.1. Proof: 8(r)
By Lemma 3.8.B there are strictly increasing continuous functions such that
a(p(x,M))
We notice now that stability
(3.6.26).
r
~ r
~ 8(0(x,M)).
Further
a(q) + + ~
as
a(r), q + ~.
satisfies conditions of usual theorem of global asymptotic
Hence the compact set
M
is globally asymptotically
stable.
Similarly to what is done for Theorem 3.8.6 one has
3.8.14
COROLLARY Theorem 3.8.13 holds if condition Ciii) i8 replaced by the condition: iii')
the compact set M
is (locally) asymptotically stable.
In what follows we shall prove a stronger version of Theorem 3.8.13 which is based upon the following Lpmma 3.8.15. 3.8.8.
This l~mma is an improvement of Lemma
Its proof, which was suggested by C. Olech in a private communication,
based upon Theorem 3.8.6.
An alternative possible proof of this theorem is based upon
an improved version of Theorem 3.8.6 for flows without uniqueness.
3.8.15
LEM~4
Lemma ~.8.8 still holds if condition i) is replaced by: i') v = r
Proof:
Ass,-,e that
C I.
r
is not
> 0
= rain { r
for
x~H(M,6).
Let
>
0 .
358 Consider the sets N(~) ffi {x~E:~(X) < p}
and B(~) = ( x e Z : r
The set N(vl2)
N(v/2)
.< ~ }
is obviously open, in addition every component
Ni(v/2)
of
such that
3.8.16
is bounded.
This is due to the fact that
sN(~) 0 E~,~)
3.8.17
=
r
Thus there exists at least one component has the properties 3.8.16 and 3.8.17.
N~(~/2)
Let now
of
N(vl2)
N (6)
which is bounded and
8 ~v/2
be that component
C
of
N(6)
with the property
N~(v/2)C N C (6)
3.8.19
Notice that if
Nc(8)
is bounded for
a)
Nc(8 o) = Bc(8 o),
b)
~/2 .< ~(x) .< 8~
c)
there exists
where
for
8 = 8 o, Bc(6 o)
then is the analogous component of
B(8o),
X~Bc(8 o) \ Nc(~/2).
e > 0,
such that
N (6)
is bounded, for
c
8
o
.- k
for
> 0
x~A~
Ns
.
Consider next the differential equation 3.8.22
~ =
- grad ~(x[ i + [[grad ~(X)[l
which has global extendability (Theorem 3.1.62), but not necessarily uniqueness. Let
x(x~
be a solution of the differential equation 3.8.22 with
then the function
~(x(x~
is a strictly decreasing function of
x~ t
A \ N~(~/2); and, in
addition
d__
dt
if
x(x~
x(x~
< -L < 0
\ N~(,/2)
or if
~(x(x~
>. ,/2.
of the differential equation 3.8.22 with
T : (a - v/2) / L each point of is bounded. U
r176
A
such that
6 > 0
x~
for
is at a finite distance from
Then for all
Nc(6) = E.
x(x~
the set
Then for each solution A \ N~(,/2) 9 >. T
N~(,/2),
N C (6)
and
x~
there exists A ~N~(,/2).
which is bounded; also
Hence A
is bounded and
Thus
lira
r (x) § +
llxll + which proves the most important part of the l~mma.
The remaining statements can be
proved in exactly the same way as in Lemma 3.8.8 . We can now apply L~mma 3.8.15 to the proof of the following result.
360
3.8.2 3
THEOREM Theorem 3.8.13 and Corollary 3.8.14 still hold if condition i)
is
replaced by
i')
v = r
E C l.
The proof of this theorem is exactly the same as the one of Theorem 3.8.13 when instead of L~mma 3.8.8, we use L~mma 3.8.15.
3.8.24
Remark With obvious variations, theorems similar to 3.8.6, 3.8.7, 3.8.13, 3.8.14
and 3.8.23 can be proved also for the case of complete instability.
We shall now prove a theorem similar to 3.8.23 for the case of instability. This theorem is based upon two l~,,,as which have rather simple proofs.
3.8.25
LEMMA
Let compact set.
be a real-valued function defined in
v = ~(x)
E.
Let
be a
M~E
Assume that
i) v = ~(x) ec I, ii) v -- ~(x)
is indefinite for
iii) there exists
n > ~
Then there exists a point Proof:
M
such that
xC~ S(M,n)\ M
To fix the ideas assume that
in r
# 0
r
= 0}.
Furthermore,
that
for
x~r~
By continuity
~ > O,
for
x~H(M,n).
which i8 a critical point of
~[x) > 0
Z = {x ~S(M,~):
S(M, 6),
for
x ~H~M,~.
~(x).
Then there exists a set
there exists an open set
r-~S(M,~)
such
B
~(x) < 0
least upper and greatest lower bounds to
~F ,
in
since then it would follow that
3F C r .
Z.
The function
~(x)
has then its
Obviously the extremals cannot both belon~
~[x) E 0
for
x~
=,
and then
~(x)
m
~9
not be indefinite.
Thus
~(x)
has one extremal in ~r
which is the critical point.
361
In the same fashion one may now prove the following lemma .
3.8.26
LEnA Let
on
E
be a compact set, and let
M~E
v = ~(x)
be a
C1
+ 0 ,
then
Xn§
with the following properties: i)
if for any sequence
ii)
{Xn} , grad r
there is an open connected set for
iii)
xEarO
ar O M
S(M,n),
and
r
and an
~(x) ~ 0
such that
~(x) = 0
for
x Car .
The above lemma
xEr
0S(M,n),
r
= 0
and
and
~(x) ~ o
such that
r*
for
x~r
r*O
S(M,n) - r ~ S(M,n),
.
M.
THEOREM Let
~
for
such that
is useful in deriving results on global instability
(Def. 3.6.30) of a compact set
let
n > 0
M,
~ ~.
Then there exists an u~bounded open connected set
3.8.27
function defined
v = ~(x)
and
be compact set.
be real-valued functions defined in E ,
w = ~(x)
If
i) v = ~(x) EC I, ii)
there i8 an open set
iii) ~ ( x ) = 0
for
such that
x~arOs(M,n),
iv) sign ~(x) = sign ~(x) v) for any sequence
r
for
arO
and
aM ~ 4,
~(x)~
o
for
x~rOS(M,~),
x E F ~ S(M,n),
{Xn} , ~(x n) + 0
implies
Xn § M,
vi) ~(x) = . ~hen the compact set
M
is globally unstable.
The following theorem summarizes the results obtained above :
and
362
3.8.28
Theorem (Extension Theorem) Let
MC E
v = ~(x)
be campact.
and
w = ~(x)
be real-valued functions defined on
E.
Let
A s s ~ e that
i) v = ~ ( x ) E C 1 ii) ~(x) = 0
for
xEM,
iii) for any sequence
~plie8
{x }, @(Xn) § 0 m
x
§ M,
n
iV) ~(x) = .
Then whatever the local stability properties of
M
for the system 3.8.3, these
properties are global. 3.8.29
THEOREM Let
M C E
v-- ~(x)
and
be a compact set.
w = ~(x)
be real-valued functions defined on
E.
Let
A s s ~ e that
i) v -- ~(x)E c 1 ii) ~(x) -- o iii) for
{Xn } C E "
iv) ~(x) v)
for
x~ M ,
is semidefinite for
E,
M
~(x) = O,
Then ,whatever the
m a ~ e ,they are global.
THEOREM Let
implies
in
is the largest invariant set in the set
local stability properties of
Ass~e
M
Xn ->M,
~(x) = ,
vi) M
3.8.30
implies
grad ~(Xn) + 0
v = ~(x)
that for some x n + M,
and
w = ~(x)
l lHk[ j
k
the set
~
be real-valued function8 defined on
0,
$l(y) ,< 0
such that
x(t,z){~K k x(t,z)
Hk
contains a co~pact invariant subset.
Then there exists a point
where
Let
~(x)~C 2
ii) for all
iv) ~
{x ~ E: r
E.
for all
t
> 0
is the solution of the differential equation
3.9.1
with
x(0,z)
=
z.
If in Theorem 3.9.4 conditions i) and ii) are satisfied and instead of (iii) and (iv) we assume that iii9 the set Then the set n-
3.9.6
~l(y) > 0}
{z E E : x ( t , z ) ('H k
is none~pty and nonconnected. for all
t > O}
has dimension at least
i~
THEOREM Let
that
{y ~
M C. E
be a compact invariant set and let
v = r
2
be such
367
i)
it)
= o
for all
x~M ,
~(x) >. o
for all
xEE,
~(x)
as
~(x)
+ |
iii) ~l(X) = 0
implies
iv) ~l(X) = 0
P2(x) > 0
for all
zI
and
z2
in
E \ M,
v) x ( t , z 1) +M §
|
vi) x ( t , z 2) + M [x(t,Z2) [ vii) For all
I~(t,z)
x~ E \ M~
for some
Then there exist points
Ix(t, z l) [
ilxli+ |
§
as
t§
as
t§
as
t+ -~,
=
z 6E ~ M
l § |
such that
either
behaves as in v) or vi) or
x(t,z)
a8
Itl
+
|
368
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