Dynamical Systems: Stability Theory and Applications (Lecture Notes in Mathematics)

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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