I. D. Chueshov
Title:
Introduction to the Theory
of InfiniteDimensional
Dissipative Systems
«A CTA » 200
Author:
ISBN:
966–7021
966
7021–64
64–5
I. D. Chueshov
I ntroduction to the Theory
of Infinite-Dimensional
D issipative
S ystems
Universitylecturesincontemporarymathematics
This book provides an exhaustive introduction to the scope
of main ideas and methods of the
theory of infinite-dimensional dissipative dynamical systems which
has been rapidly developing in recent years. In the examples
systems generated by nonlinear
partial differential equations
arising in the different problems
of modern mechanics of continua
are considered. The main goal
of the book is to help the reader
to master the basic strategies used
in the study of infinite-dimensional
dissipative systems and to qualify
him/her for an independent scientific research in the given branch.
Experts in nonlinear dynamics will
find many fundamental facts in the
convenient and practical form
in this book.
The core of the book is composed of the courses given by the
author at the Department
of Mechanics and Mathematics
at Kharkov University during
a number of years. This book contains a large number of exercises
which make the main text more
complete. It is sufficient to know
the fundamentals of functional
analysis and ordinary differential
equations to read the book.
Translated by
You can O R D E R this book
while visiting the website
of «ACTA» Scientific Publishing House
http://www.acta.com.ua
www.acta.com.ua/en/
Constantin I. Chueshov
from the Russian edition («ACTA», 1999)
Translation edited by
Maryna B. Khorolska
Chapter
1
Basic Concepts of the Theory
of Infinite-Dimensional Dynamical Systems
Contents
....§1
Notion of Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . 11
....§2
Trajectories and Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . 17
....§3
Definition of Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
....§4
Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . . 24
....§5
Theorems on Existence of Global Attractor . . . . . . . . . . . . . . 28
....§6
On the Structure of Global Attractor . . . . . . . . . . . . . . . . . . . 34
....§7
Stability Properties of Attractor and Reduction Principle . . . 45
....§8
Finite Dimensionality of Invariant Sets . . . . . . . . . . . . . . . . . 52
....§9
Existence and Properties of Attractors of a Class of
Infinite-Dimensional Dissipative Systems . . . . . . . . . . . . . . . 61
....
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
The mathematical theory of dynamical systems is based on the qualitative theory of ordinary differential equations the foundations of which were laid by Henri
Poincaré (1854–1912). An essential role in its development was also played by the
works of A. M. Lyapunov (1857–1918) and A. A. Andronov (1901–1952). At present
the theory of dynamical systems is an intensively developing branch of mathematics
which is closely connected to the theory of differential equations.
In this chapter we present some ideas and approaches of the theory of dynamical systems which are of general-purpose use and applicable to the systems generated by nonlinear partial differential equations.
§1
Notion of Dynamical System
In this book dynamical system is taken to mean the pair of objects ( X , St ) consisting of a complete metric space X and a family St of continuous mappings of the
space X into itself with the properties
St + t = St ° St ,
t, t Î T+ ,
S0 = I ,
(1.1)
where T+ coincides with either a set R+ of nonnegative real numbers or a set
Z+ = { 0 , 1 , 2 , ¼ } . If T+ = R+ , we also assume that y ( t ) = St y is a continuous
function with respect to t for any y Î X . Therewith X is called a phase space , or
a state space, the family St is called an evolutionary operator (or semigroup),
parameter t Î T+ plays the role of time. If T + = Z+ , then dynamical system is
called discrete (or a system with discrete time). If T + = R + , then ( X , St ) is frequently called to be dynamical system with continuous time. If a notion of dimension can be defined for the phase space X (e. g., if X is a lineal), the value dim X is
called a dimension of dynamical system.
Originally a dynamical system was understood as an isolated mechanical system
the motion of which is described by the Newtonian differential equations and which
is characterized by a finite set of generalized coordinates and velocities. Now people
associate any time-dependent process with the notion of dynamical system. These
processes can be of quite different origins. Dynamical systems naturally arise in
physics, chemistry, biology, economics and sociology. The notion of dynamical system is the key and uniting element in synergetics. Its usage enables us to cover
a rather wide spectrum of problems arising in particular sciences and to work out
universal approaches to the description of qualitative picture of real phenomena
in the universe.
12
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
Let us look at the following examples of dynamical systems.
E x a m p l e 1.1
Let f ( x ) be a continuously differentiable function on the real axis posessing the
property x f ( x ) ³ - C ( 1 + x 2 ) , where C is a constant. Consider the Cauchy
problem for an ordinary differential equation
x· ( t ) = -f ( x ( t ) ) ,
t > 0 , x (0) = x .
(1.2)
0
For any x Î R problem (1.2) is uniquely solvable and determines a dynamical
system in R . The evolutionary operator St is given by the formula St x0 = x ( t ) ,
where x ( t ) is a solution to problem (1.2). Semigroup property (1.1) holds
by virtue of the theorem of uniqueness of solutions to problem (1.2). Equations
of the type (1.2) are often used in the modeling of some ecological processes.
For example, if we take f ( x ) = a × x ( x - 1 ) , a > 0 , then we get a logistic equation that describes a growth of a population with competition (the value x ( t )
is the population level; we should take R + for the phase space).
E x a m p l e 1.2
Let f ( x ) and g ( x ) be continuously differentiable functions such that
x
F (x) =
ò f ( x ) dx
³ -c ,
g ( x ) ³ -c
0
with some constant c . Let us consider the Cauchy problem
·
··
ì x + g (x) x + f (x) = 0 , t > 0 ,
í
·
î x ( 0 ) = x0 , x ( 0 ) = x1 .
(1.3)
For any y0 = ( x0 , x1 ) Î R 2 , problem (1.3) is uniquely solvable. It generates
a two-dimensional dynamical system ( R 2 , St ) , provided the evolutionary operator is defined by the formula
S ( x ; x ) = ( x ( t ) ; x· ( t ) ) ,
t
0
1
where x ( t ) is the solution to problem (1.3). It should be noted that equations
of the type (1.3) are known as Liénard equations in literature. The van der Pol
equation:
g ( x ) = e ( x2 - 1 ) , e > 0 ,
f (x) = x
and the Duffing equation:
g (x) = e , e > 0 ,
f ( x) = x3 - a × x - b
which often occur in applications, belong to this class of equations.
Notion of Dynamical System
E x a m p l e 1.3
Let us now consider an autonomous system of ordinary differential equations
k = 1, 2, ¼, N .
x· ( t ) = f ( x , x , ¼ , x ) ,
(1.4)
k
k
1
2
N
Let the Cauchy problem for the system of equations (1.4) be uniquely solvable
over an arbitrary time interval for any initial condition. Assume that a solution
continuously depends on the initial data. Then equations (1.4) generate an N - dimensional dynamical system ( R N , St ) with the evolutionary operator St acting
in accordance with the formula
St y0 = ( x1 ( t ) , ¼ , xN ( t ) ) ,
y 0 = ( x10 , x 20 , ¼ , xN 0 ) ,
where { xi ( t ) } is the solution to the system of equations (1.4) such that
xi ( 0 ) = xi 0 , i = 1 , 2 , ¼ , N . Generally, let X be a linear space and F be
a continuous mapping of X into itself. Then the Cauchy problem
x· ( t ) = F ( x ( t ) ) , t > 0 , x ( 0 ) = x0 Î X
(1.5)
generates a dynamical system ( X , St ) in a natural way provided this problem is
well-posed, i.e. theorems on existence, uniqueness and continuous dependence
of solutions on the initial conditions are valid for (1.5).
E x a m p l e 1.4
Let us consider an ordinary retarded differential equation
x· ( t ) + a x ( t ) = f ( x ( t - 1 ) ) , t > 0 ,
where f is a continuous function on
for (1.6) should be given in the form
R1 ,
(1.6)
a > 0 . Obviously an initial condition
x ( t ) t Î [ -1 , 0 ] = f ( t ) .
(1.7)
Assume that f ( t ) lies in the space C [ - 1 , 0 ] of continuous functions on the
segment [ - 1 , 0 ] . In this case the solution to problem (1.6) and (1.7) can be
constructed by step-by-step integration. For example, if 0 £ t £ 1 , the solution x ( t ) is given by
t
x (t) =
e -a t
f (0) +
òe
-a ( t - t)
f ( f ( t - 1 ) ) dt ,
0
and if t Î [ 1 , 2 ] , then the solution is expressed by the similar formula in terms
of the values of the function x ( t ) for t Î [ 0 , 1 ] and so on. It is clear that the solution is uniquely determined by the initial function f ( t ) . If we now define an
operator St in the space X = C [ - 1 , 0 ] by the formula
( St f ) ( t ) = x ( t + t ) ,
t Î [ -1 , 0 ] ,
where x ( t ) is the solution to problem (1.6) and (1.7), then we obtain an infinite-dimensional dynamical system ( C [ - 1 , 0 ] , St ) .
13
14
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
Now we give several examples of discrete dynamical systems. First of all it should be
noted that any system ( X , St ) with continuous time generates a discrete system if
we take t Î Z+ instead of t Î R + . Furthermore, the evolutionary operator St of
t
a discrete dynamical system is a degree of the mapping S1 , i. e. St = S1 , t Î Z+ .
Thus, a dynamical system with discrete time is determined by a continuous mapping
of the phase space X into itself. Moreover, a discrete dynamical system is very often
defined as a pair ( X , S ) , consisting of the metric space X and the continuous mapping S .
1
E x a m p l e 1.5
Let us consider a one-step difference scheme for problem (1.5):
xn + 1 - xn
------------------------ = F ( xn ) ,
t
n = 0, 1, 2, ¼ ,
t > 0.
There arises a discrete dynamical system ( X , S n ) , where S is the continuous
mapping of X into itself defined by the formula S x = x + t F ( x ) .
E x a m p l e 1.6
Let us consider a nonautonomous ordinary differential equation
x· ( t ) = f ( x , t ) ,
t > 0 , x Î R1 ,
(1.9)
where f ( x , t ) is a continuously differentiable function of its variables and is periodic with respect to t , i. e. f ( x , t ) = f ( x , t + T ) for some T > 0 . It is assumed that the Cauchy problem for (1.9) is uniquely solvable on any time
interval. We define a monodromy operator (a period mapping) by the formula
S x0 = x ( T ) , where x ( t ) is the solution to (1.9) satisfying the initial condition
x ( 0 ) = x0 . It is obvious that this operator possesses the property
Sk x ( t ) = x ( t + k T)
(1.10)
for any solution x ( t ) to equation (1.9) and any k Î Z+ . The arising dynamical
system ( R 1 , S k ) plays an important role in the study of the long-time properties of solutions to problem (1.9).
E x a m p l e 1.7 (Bernoulli shift)
Let X = S 2 be a set of sequences x = { xi , i Î Z } consisting of zeroes and
ones. Let us make this set into a metric space by defining the distance by the
formula
d ( x , y ) = inf { 2 -n : xi = yi ,
i < n}.
Let S be the shift operator on X , i. e. the mapping transforming the sequence
x = { xi } into the element y = { yi } , where yi = x i + 1 . As a result, a dynamical
system ( X , S n ) comes into being. It is used for describing complicated (quasirandom) behaviour in some quite realistic systems.
Notion of Dynamical System
In the example below we describe one of the approaches that enables us to connect
dynamical systems to nonautonomous (and nonperiodic) ordinary differential equations.
E x a m p l e 1.8
Let h ( x , t ) be a continuous bounded function on R 2 . Let us define the hull
Lh of the function h ( x , t ) as the closure of a set
ì
ü
í ht ( x , t ) º h ( x , t + t ) , t Î R ý
î
þ
with respect to the norm
ì
ü
h C = sup í h ( x , t ) : x Î R , t Î R ý .
î
þ
Let g ( x ) be a continuous function. It is assumed that the Cauchy problem
˜
x· ( t ) = g ( x ) + h ( x , t ) , x ( 0 ) = x
(1.11)
0
˜
is uniquely solvable over the interval [ 0 , + ¥ ) for any h Î Lh . Let us define
the evolutionary operator St on the space X = R 1 ´ Lh by the formula
˜
˜
St ( x0 , h ) = ( x ( t ) , h t ) ,
˜
˜
where x ( t ) is the solution to problem (1.11) and ht = h ( x , t + t ) . As a result,
a dynamical system ( R ´ Lh , St ) comes into being. A similar construction is often used when Lh is a compact set in the space C of continuous bounded functions (for example, if h ( x , t ) is a quasiperiodic or almost periodic function).
As the following example shows, this approach also enables us to use naturally
the notion of the dynamical system for the description of the evolution of objects subjected to random influences.
E x a m p l e 1.9
Assume that f0 and f1 are continuous mappings from a metric space Y into itself. Let Y be a state space of a system that evolves as follows: if y is the state of
the system at time k , then its state at time k + 1 is either f0 ( y ) or f1 ( y ) with
probability 1 ¤ 2 , where the choice of f0 or f1 does not depend on time and the
previous states. The state of the system can be defined after a number of steps
in time if we flip a coin and write down the sequence of events from the right to
the left using 0 and 1 . For example, let us assume that after 8 flips we get the
following set of outcomes:
¼ 10 110010 .
Here 1 corresponds to the head falling, whereas 0 corresponds to the tail falling. Therewith the state of the system at time t = 8 will be written in the form:
15
16
C
h
a
p
t
e
r
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
W = ( f1 ° f0 ° f1 ° f1 ° f0 ° f0 ° f1 ° f0 ) ( y ) .
This construction can be formalized as follows. Let S 2 be a set of two-sided sequences consisting of zeroes and ones (as in Example 1.7), i.e. a collection
of elements of the type
w = ( ¼ w-n ¼ w- 1 w0 w1 ¼ wn ¼ ) ,
1
where w i is equal to either 1 or 0 . Let us consider the space X = S 2 ´ Y consisting of pairs x = ( w , y ) , where w Î S 2 , y Î Y . Let us define the mapping
F : X ® X by the formula:
F ( x ) º F ( w , y ) = ( S w , fw ( y ) ) ,
0
where S is the left-shift operator in S 2 (see Example 1.7). It is easy to see that
the n - th degree of the mapping F actcts according to the formula
F n ( w , y ) = ( S n w , ( fw
° ¼ ° fw1 ° fw0 ) ( y ) )
n -1
and it generates a discrete dynamical system ( S 2 ´ Y , F n ) . This system is often
called a universal random (discrete) dynamical system.
Examples of dynamical systems generated by partial differential equations will be given in the chapters to follow.
E x e r c i s e 1.1 Assume that operators S t have a continuous inverse for any t .
ˆ
Show that the family of operators {St : t Î R } defined by the equaˆ
ˆ
lity St = St for t ³ 0 and St = S -t1 for t < 0 form a group, i.e. (1.1)
holds for all t , t Î R .
E x e r c i s e 1.2 Prove the unique solvability of problems (1.2) and (1.3) involved in Examples 1.1 and 1.2.
E x e r c i s e 1.3
Ground formula (1.10) in Example 1.6.
E x e r c i s e 1.4 Show that the mapping S t in Example 1.8 possesses semigroup property (1.1).
E x e r c i s e 1.5 Show that the value d ( x , y ) involved in Example 1.7 is a metric. Prove its equivalence to the metric
d* ( x ,
¥
y) =
å2
i = -¥
-i
xi - yi .
Trajectories and Invariant Sets
§2
Trajectories and Invariant Sets
Let ( X , St ) be a dynamical system with continuous or discrete time. Its trajectory
(or orbit ) is defined as a set of the type
g = {u (t) : t Î T} ,
where u ( t ) is a continuous function with values in X such that St u ( t ) = u ( t + t )
for all t Î T + and t Î T . Positive (negative) semitrajectory is defined as a set
g + = { u ( t ) : t ³ 0 } , ( g – = { u ( t ) : t £ 0 } , respectively), where a continuous on T +
( T– , respectively) function u ( t ) possesses the property St u ( t ) = u ( t + t ) for any
t > 0 , t ³ 0 ( t > 0 , t £ 0 , t + t £ 0 , respectively). It is clear that any positive
semitrajectory g + has the form g + = { St v : t ³ 0 } , i.e. it is uniquely determined by
its initial state v . To emphasize this circumstance, we often write g + = g + ( v ) .
In general, it is impossible to continue this semitrajectory g + ( v ) to a full trajectory
without imposing any additional conditions on the dynamical system.
Assume that an evolutionary operator St is invertible for some
t > 0 . Then it is invertible for all t > 0 and for any v Î X there
exists a negative semitrajectory g – = g – ( v ) ending at the point v .
E x e r c i s e 2.1
A trajectory g = { u ( t ) : t Î T } is called a periodic trajectory (or a cycle ) if
there exists T Î T+ , T > 0 such that u ( t + T ) = u ( t ) . Therewith the minimal
number T > 0 possessing the property mentioned above is called a period of a trajectory. Here T is either R or Z depending on whether the system is a continuous
or a discrete one. An element u0 Î X is called a fixed point of a dynamical system
( X , S t ) if St u0 = u0 for all t ³ 0 (synonyms: equilibrium point , stationary
point ).
E x e r c i s e 2.2 Find all the fixed points of the dynamical system ( R , St ) generated by equation (1.2) with f ( x ) = x ( x - 1 ) . Does there exist
a periodic trajectory of this system?
E x e r c i s e 2.3 Find all the fixed points and periodic trajectories of a dynamical system in R 2 generated by the equations
ì x· = - a y - x [ ( x 2 + y 2 ) 2 - 4 ( x 2 + y 2 ) + 1 ] ,
ï
í
ï y· = a x - y [ ( x 2 + y 2 ) 2 - 4 ( x 2 + y 2 ) + 1 ] .
î
Consider the cases a ¹ 0 and a = 0 .
Hint: use polar coordinates.
E x e r c i s e 2.4 Prove the existence of stationary points and periodic trajectories of any period for the discrete dynamical system described
17
18
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
in Example 1.7. Show that the set of all periodic trajectories is dense
in the phase space of this system. Make sure that there exists a trajectory that passes at a whatever small distance from any point of the
phase space.
The notion of invariant set plays an important role in the theory of dynamical systems. A subset Y of the phase space X is said to be:
a) positively invariant , if St Y Í Y for all t ³ 0 ;
b) negatively invariant , if St Y Ê Y for all t ³ 0 ;
c) invariant , if it is both positively and negatively invariant, i.e. if
St Y = Y for all t ³ 0 .
The simplest examples of invariant sets are trajectories and semitrajectories.
E x e r c i s e 2.5 Show that g + is positively invariant, g – is negatively invariant
and g is invariant.
E x e r c i s e 2.6
Let us define the sets
g + ( A) =
È S (A) º È {v = S u :
t ³ 0
t
t
t³ 0
u Î A}
and
g – ( A) =
ÈS
t ³ 0
-1
t ( A)
º
È {v :
t ³ 0
St v Î A }
for any subset A of the phase space X . Prove that g + ( A) is a positively
invariant set, and if the operator St is invertible for some t > 0 ,
then g – ( A) is a negatively invariant set.
Other important example of invariant set is connected with the notions of w -limit
and a -limit sets that play an essential role in the study of the long-time behaviour
of dynamical systems.
Let A Ì X . Then the w -limit set for A is defined by
w ( A) =
Ç È S (A)
s ³ 0
t ³ s
,
t
X
where St ( A ) = { v = S t u : u Î A } . Hereinafter [ Y ] X is the closure of a set Y in the
space X . The set
a ( A) =
Ç ÈS
s ³ 0
t³ s
-1
t ( A)
,
X
where St-1 ( A) = { v : St v Î A } , is called the a -limit set for A .
Trajectories and Invariant Sets
Lemma 2.1
For an element y to belong to an w -limit set w ( A) , it is necessary and
sufficient that there exist a sequence of elements { yn } Ì A and a sequence of numbers tn , the latter tending to infinity such that
lim d ( St yn , y ) = 0 ,
n
n®¥
where d ( x , y ) is the distance between the elements x and y in the
space X .
Proof.
Let the sequences mentioned above exist. Then it is obvious that for any
t > 0 there exists n0 ³ 0 such that
St yn Î
n
È S (A) ,
t³ t
t
n ³ n0 .
This implies that
È S (A)
y = lim St yn Î
n®¥ n
t ³ t
t
X
for all t > 0 . Hence, the element y belongs to the intersection of these sets,
i.e. y Î w ( A) .
On the contrary, if y Î w ( A) , then for all n = 0 , 1 , 2 , ¼
y Î
È S (A)
t³ n
.
t
X
Hence, for any n there exists an element z n such that
1
d ( y , z n ) £ --zn Î
St ( A) ,
n.
È
t ³ n
Therewith it is obvious that zn = St yn , yn Î A , tn ³ n . This proves the
n
lemma.
It should be noted that this lemma gives us a description of an w -limit set but does
not guarantee its nonemptiness.
E x e r c i s e 2.7 Show that w ( A) is a positively invariant set. If for any t > 0
there exists a continuous inverse to St , then w ( A) is invariant, i.e.
St w ( A) = w ( A) .
E x e r c i s e 2.8 Let S t be an invertible mapping for every t > 0 . Prove the
counterpart of Lemma 2.1 for an a -limit set:
ì
ü
y Î a ( A) Û í ${ yn } Î A , $tn , tn ® + ¥ ; lim d ( St-1 yn , y ) = 0 ý .
n
n®¥
î
þ
Establish the invariance of a ( A) .
19
20
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
E x e r c i s e 2.9 Let g = { u ( t ) : - ¥ < t < ¥ } be a periodic trajectory of a dynamical system. Show that g = w ( u ) = a ( u ) for any u Î g .
E x e r c i s e 2.10 Let us consider the dynamical system ( R , S t ) constructed in
Example 1.1. Let a and b be the roots of the function f ( x ) :
f ( a ) = f ( b ) = 0 , a < b . Then the segment I = { x : a £ x £ b } is
an invariant set. Let F ( x ) be a primitive of the function f ( x )
( F' ( x ) = f ( x ) ). Then the set { x : F ( x ) £ c } is positively invariant
for any c .
E x e r c i s e 2.11 Assume that for a continuous dynamical system ( X , St ) there
exists a continuous scalar function V ( y ) on X such that the value
V ( St y ) is differentiable with respect to t for any y Î X and
d
----- ( V ( St y ) ) + a V ( St y ) £ r ,
dt
(a > 0 , r > 0 , y Î X) .
Then the set { y : V ( y ) £ R } is positively invariant for any R ³
³ r¤ a.
§ 3
Definition of At
A ttractor
Attractor is a central object in the study of the limit regimes of dynamical systems.
Several definitions of this notion are available. Some of them are given below. From
the point of view of infinite-dimensional systems the most convenient concept is that
of the global attractor.
A bounded closed set A1 Ì X is called a global attractor for a dynamical system ( X , St ) , if
1) A1 is an invariant set, i.e. St A1 = A1 for any t > 0 ;
2) the set A1 uniformly attracts all trajectories starting in bounded sets,
i.e. for any bounded set B from X
ì
ü
lim sup í dist ( St y , A1) : y Î B ý = 0 .
t®¥
î
þ
We remind that the distance between an element z and a set A is defined by the
equality:
dist ( z , A) = inf { d ( z , y ) : y Î A } ,
where d ( z , y ) is the distance between the elements z and y in X .
The notion of a weak global attractor is useful for the study of dynamical systems generated by partial differential equations.
Definition of Attractor
Let X be a complete linear metric space. A bounded weakly closed set A2 is
called a global weak attractor if it is invariant ( St A2 = A 2 , t > 0 ) and for any
weak vicinity O of the set A2 and for every bounded set B Ì X there exists
t0 = t0 ( O , B ) such that St B Ì O for t ³ t0 .
We remind that an open set in weak topology of the space X can be described
as finite intersection and subsequent arbitrary union of sets of the form
Ul , c = { x Î X : l ( x ) < c } ,
where c is a real number and l is a continuous linear functional on X .
It is clear that the concepts of global and global weak attractors coincide in the
finite-dimensional case. In general, a global attractor A is also a global weak attractor, provided the set A is weakly closed.
E x e r c i s e 3.1 Let A be a global or global weak attractor of a dynamical system ( X , St ) . Then it is uniquely determined and contains any bounded negatively invariant set. The attractor A also contains the
w - limit set w ( B ) of any bounded B Ì X .
E x e r c i s e 3.2 Assume that a dynamical system ( X , S t ) with continuous
time possesses a global attractor A1 . Let us consider a discrete system ( X , T n ) , where T = St with some t0 > 0 . Prove that A1 is a glo0
bal attractor for the system ( X , T n ) . Give an example which shows
that the converse assertion does not hold in general.
If the global attractor A1 exists, then it contains a global minimal attractor A 3
which is defined as a minimal closed positively invariant set possessing the property
lim dist ( St y , A 3 ) = 0
t®¥
for every
y ÎX.
By definition minimality means that A3 has no proper subset possessing the properties mentioned above. It should be noted that in contrast with the definition of the
global attractor the uniform convergence of trajectories to A 3 is not expected here.
E x e r c i s e 3.3
Show that St A3 = A 3 , provided A3 is a compact set.
E x e r c i s e 3.4
Prove that w ( x ) Î A 3 for any x Î X . Therewith, if A 3 is
a compact, then A3 = È { w ( x ) : x Î X } .
By definition the attractor A3 contains limit regimes of each individual trajectory.
It will be shown below that A3 ¹ A1 in general. Thus, a set of real limit regimes
(states) originating in a dynamical system can appear to be narrower than the global
attractor. Moreover, in some cases some of the states that are unessential from the
point of view of the frequency of their appearance can also be “removed” from A3 ,
for example, such states like absolutely unstable stationary points. The next two
definitions take into account the fact mentioned above. Unfortunately, they require
21
22
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
additional assumptions on the properties of the phase space. Therefore, these definitions are mostly used in the case of finite-dimensional dynamical systems.
Let a Borel measure m such that m ( X ) < ¥ be given on the phase space X of
a dynamical system ( X , S t ) . A bounded set A 4 in X is called a Milnor attractor
(with respect to the measure m ) for ( X , S t ) if A4 is a minimal closed invariant set
possessing the property
1
lim dist ( St y , A 4 ) = 0
t®¥
for almost all elements y Î X with respect to the measure m . The Milnor attractor
is frequently called a probabilistic global minimal attractor.
At last let us introduce the notion of a statistically essential global minimal attractor suggested by Ilyashenko. Let U be an open set in X and let XU ( x ) be its
characteristic function: X U ( x ) = 1 , x Î U ; X U ( x ) = 0 , x Ï U . Let us define the
average time t ( x , U ) which is spent by the semitrajectory g + ( x ) emanating from x
in the set U by the formula
T
--1T®¥ T
t ( x , U ) = lim
òX
U ( S t x ) dt .
0
A set U is said to be unessential with respect to the measure m if
M ( U) º m {x : t (x , U ) > 0} = 0 .
The complement A5 to the maximal unessential open set is called an Ilyashenko
a t tractor (with respect to the measure m ).
It should be noted that the attractors A 4 and A 5 are used in cases when the natural Borel measure is given on the phase space (for example, if X is a closed measurable set in R N and m is the Lebesgue measure).
The relations between the notions introduced above can be illustrated by the
following example.
E x a m p l e 3.1
Let us consider a quasi-Hamiltonian system of equations in R 2 :
ì·
¶H
¶H
,
-mH
ïq =
¶
¶q
p
ï
í
ï
¶H
H
ï p· = - ¶------ -mH
,
¶q
¶p
î
(3.1)
where H ( p , q ) = ( 1 ¤ 2 ) p 2 + q 4 - q 2 and m is a positive number. It is easy
to ascertain that the phase portrait of the dynamical system generated by equations (3.1) has the form represented on Fig. 1.
Definition of Attractor
A separatrix (“eight curve”) separates the domains of the phase plane
with the different qualitative behaviour of the
trajectories. It is given by
the equation H ( p, q ) = 0 .
The points ( p , q ) inside
the separatrix are characterized by the equation
H ( p , q ) < 0 . Therewith
it appears that
Fig. 1. Phase portrait of system (3.1)
A1 = A2 = { ( p , q ) : H ( p , q ) £ 0 } ,
ì
ü
ì
¶ H (p, q) = ¶ H (p, q) = 0 ü ,
A3 = í ( p , q ) : H ( p , q ) = 0 ý È í ( p , q ) :
ý
¶p
¶q
î
þ
î
þ
A4 = { ( p , q ) : H ( p , q ) = 0 } .
Finally, the simple calculations show that A5 = { 0 , 0 } , i.e. the Ilyashenko attractor consists of a single point. Thus,
A1 = A2 É A3 É A 4 É A5 ,
all inclusions being strict.
E x e r c i s e 3.5 Display graphically the attractors A j of the system generated
by equations (3.1) on the phase plane.
Consider the dynamical system from Example 1.1 with
f ( x ) = x ( x 2 - 1 ) . Prove that A1 = { x : - 1 £ x £ 1 } ,
A3 = { x = 0 ; x = ± 1 } , and A4 = A 5 = { x = ± 1 } .
E x e r c i s e 3.6
E x e r c i s e 3.7
Prove that A4 Ì A 3 and A5 Ì A3 in general.
E x e r c i s e 3.8 Show that all positive semitrajectories of a dynamical system
which possesses a global minimal attractor are bounded sets.
In particular, the result of the last exercise shows that the global attractor can exist
only under additional conditions concerning the behaviour of trajectories of the system at infinity. The main condition to be met is the dissipativity discussed in the next
section.
23
24
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
1
§ 4
Dissipativity and Asymptotic
Compactness
From the physical point of view dissipative systems are primarily connected with irreversible processes. They represent a rather wide and important class of the dynamical systems that are intensively studied by modern natural sciences. These
systems (unlike the conservative systems) are characterized by the existence of the
accented direction of time as well as by the energy reallocation and dissipation.
In particular, this means that limit regimes that are stationary in a certain sense can
arise in the system when t ® + ¥ . Mathematically these features of the qualitative
behaviour of the trajectories are connected with the existence of a bounded absorbing set in the phase space of the system.
A set B0 Ì X is said to be absorbing for a dynamical system ( X , St ) if for
any bounded set B in X there exists t0 = t0 ( B ) such that St ( B ) Ì B 0 for every
t ³ t0 . A dynamical system ( X , St ) is said to be dissipative if it possesses a bounded absorbing set. In cases when the phase space X of a dissipative system ( X , St )
is a Banach space a ball of the form { x Î X : x X £ R } can be taken as an absorbing set. Therewith the value R is said to be a radius of dissipativity .
As a rule, dissipativity of a dynamical system can be derived from the existence
of a Lyapunov type function on the phase space. For example, we have the following
assertion.
Theorem 4.1.
Let the phase sp
space of a continuous dynamical system ( X , St ) be a Banach space. Assume that:
(a) there exists a continuous function U ( x ) on X possessing the properties
j1 ( x ) £ U ( x ) £ j 2 ( x ) ,
(4.1)
where jj ( r ) are continuous functions on R+ and j 1( r ) ® + ¥
when r ® ¥ ;
d- U ( S y ) for t ³ 0 and positive numbers
(b) there exist a derivative --t
dt
a and r such that
d U ( S y ) £ -a for
---St y > r .
t
dt
Then the dynamical system ( X , St ) is dissipative.
Proof.
Let us choose R0 ³ r such that j 1( r ) > 0 for r ³ R 0 . Let
l = sup { j2 ( r ) : r £ 1 + R0 }
and R1 > R0 + 1 be such that j 1( r ) > l for r > R1 . Let us show that
(4.2)
Dissipativity and Asymptotic Compactness
St y £ R1 for all t ³ 0
y £ R0 .
and
(4.3)
Assume the contrary, i.e. assume that for some y Î X such that y £ R 0 there
exists a time t > 0 possessing the property S t y > R1 . Then the continuity of St y
implies that there exists 0 < t0 < t such that r < St y £ R0 + 1 . Thus, equation
0
(4.2) implies that
U ( St y ) £ U ( St y ) ,
0
t ³ t0 ,
provided St y > r . It follows that U ( St y ) £ l for all t ³ t0 . Hence, St y £ R1 for
all t ³ t0 . This contradicts the assumption. Let us assume now that B is an arbitrary
bounded set in X that does not lie inside the ball with the radius R 0 . Then equation
(4.2) implies that
U ( St y ) £ U ( y ) - a t £ lB - a t ,
y ÎB,
(4.4)
provided St y > r . Here
lB = sup { U ( x ) : x Î B } .
t*
Let y Î B . If for a time < ( lB - l ) ¤ a the semitrajectory St y enters the ball with
the radius r , then by (4.3) we have St y £ R 1 for all t ³ t * . If that does not take
place, from equation (4.4) it follows that
j 1( St y ) £ U ( St y ) £ l
for
lB - l
t ³ -----------,
a
i.e. St y £ R1 for t ³ a -1 ( lB - l ) . Thus,
St B Ì { x :
x £ R1 } ,
lB - l
t ³ -----------.
a
This and (4.3) imply that the ball with the radius R1 is an absorbing set for the dynamical system ( X , St ) . Thus, Theorem 4.1 is proved.
E x e r c i s e 4.1 Show that hypothesis (4.2) of Theorem 4.1 can be replaced
by the requirement
d U(S y) + g U(S y) £ C ,
---t
t
dt
where g and C are positive constants.
E x e r c i s e 4.2 Show that the dynamical system generated in R by the differential equation x· + f ( x ) = 0 (see Example 1.1) is dissipative, provided the function f ( x ) possesses the property: x f ( x ) ³ d x 2 - C ,
where d > 0 and C are constants (Hint: U ( x ) = x 2 ). Find an upper estimate for the minimal radius of dissipativity.
E x e r c i s e 4.3 Consider a discrete dynamical system ( R , f n ) , where f is
a continuous function on R . Show that the system ( R , f ) is dissipative, provided there exist r > 0 and 0 < a < 1 such that
f ( x ) < a x for x > r .
25
26
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
E x e r c i s e 4.4 Consider a dynamical system ( R 2 , St ) generated (see Example 1.2) by the Duffing equation
x·· + e x· + x 3 - a x = b ,
where a and b are real numbers and e > 0 . Using the properties
of the function
--- x· 2 + 1
--- x 4 - a
--- x 2 + n æ x x· + --e- x 2ö
U ( x , x· ) = 1
è
2
4
2
2 ø
show that the dynamical system ( R 2 , St ) is dissipative for n > 0
small enough. Find an upper estimate for the minimal radius of dissipativity.
E x e r c i s e 4.5 Prove the dissipativity of the dynamical system generated
by (1.4) (see Example 1.3), provided
N
åx
N
k fk ( x1 ,
x2 , ¼ , xN ) £ -d
k=1
åx
2
k
+C,
d > 0.
k=1
E x e r c i s e 4.6 Show that the dynamical system of Example 1.4 is dissipative
if f ( z ) is a bounded function.
Consider a cylinder Ц with coordinates ( x , j ) , x Î R ,
j Î [ 0 , 1) and the mapping T of this cylinder which is defined
by the formula T ( x , j ) = ( x ¢, j ¢) , where
E x e r c i s e 4.7
x ¢ = a x + k sin 2 p j ,
j ¢ = j + x ¢ ( mod 1 ) .
Here a and k are positive parameters. Prove that the discrete dynamical system ( Ц , T n ) is dissipative, provided 0 < a < 1 . We note
that if a = 1 , then the mapping T is known as the Chirikov mapping. It appears in some problems of physics of elementary particles.
E x e r c i s e 4.8 Using Theorem 4.1 prove that the dynamical system ( R 2 , St )
generated by equations (3.1) (see Example 3.1) is dissipative.
2
(Hint: U ( x ) = [ H ( p , q ) ] ).
In the proof of the existence of global attractors of infinite-dimensional dissipative
dynamical systems a great role is played by the property of asymptotic compactness.
For the sake of simplicity let us assume that X is a closed subset of a Banach space.
The dynamical system ( X , St ) is said to be asymptotically compact if for any
t > 0 its evolutionary operator St can be expressed by the form
(1)
St = St
where the mappings
St( 1 )
and
St( 2 )
+ St( 2 ) ,
possess the properties:
(4.5)
Dissipativity and Asymptotic Compactness
a) for any bounded set B in X
r B ( t ) = sup St( 1 ) y X ® 0 ,
y ÎB
t ® +¥;
b) for any bounded set B in X there exists t0 such that the set
[ gt( 2 ) ( B ) ] =
0
ÈS
t ³ t0
( 2)
t B
(4.6)
is compact in X , where [ g ] is the closure of the set g .
A dynamical system is said to be compact if it is asymptotically compact and
one can take St( 1 ) º 0 in representation (4.5). It becomes clear that any finite-dimensional dissipative system is compact.
E x e r c i s e 4.9 Show that condition (4.6) is fulfilled if there exists a compact
set K in H such that for any bounded set B the inclusion St( 2 ) B Ì K ,
t ³ t0 ( B ) holds. In particular, a dissipative system is compact if it
possesses a compact absorbing set.
Lemma 4.1.
The dynamical system ( X , St ) is asymptotically compact if there exists
a compact set K such that
lim sup {dist ( St u , K ) : u Î B } = 0
t®¥
(4.7)
for any set B bounded in X .
Proof.
The distance to a compact set is reached on some element. Hence, for any
t > 0 and u Î X there exists an element v º St( 2 ) u Î K such that
dist ( St u , K ) = St u - St( 2 ) u .
Therefore, if we take St( 1 ) u = St u - St( 2 ) u , it is easy to see that in this case decomposition (4.5) satisfies all the requirements of the definition of asymptotic
compactness.
Remark 4.1.
In most applications Lemma 4.1 plays a major role in the proof of the
property of asymptotic compactness. Moreover, in cases when the phase
space X of the dynamical system ( X , St ) does not possess the structure
of a linear space it is convenient to define the notion of the asymptotic
compactness using equation (4.7). Namely, the system ( X , St ) is said
to be asymptotically compact if there exists a compact K possessing
property (4.7) for any bounded set B in X . For one more approach
to the definition of this concept see Exercise 5.1 below.
27
28
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
E x e r c i s e 4.10 Consider the infinite-dimensional dynamical system generated by the retarded equation
x· ( t ) + a x ( t ) = f ( x ( t - 1 ) ) ,
where a > 0 and f ( z ) is bounded (see Example 1.4). Show that
this system is compact.
E x e r c i s e 4.11 Consider the system of Lorentz equations arising as a threemode Galerkin approximation in the problem of convection in a thin
layer of liquid:
ì x· = - s x + s y ,
ï ·
í y = rx -y -xz,
ï ·
î z = -bz + xy.
Here s , r , and b are positive numbers. Prove the dissipativity of
the dynamical system generated by these equations in R 3 .
Hint: Consider the function
2
--- ( x 2 + y 2 + ( z - r - s ) )
V (x, y, z) = 1
2
on the trajectories of the system.
§5
Theorems on Existence
of Global At
A ttractor
For the sake of simplicity it is assumed in this section that the phase space X is
a Banach space, although the main results are valid for a wider class of spaces
(see, e. g., Exercise 5.8). The following assertion is the main result.
Theorem 5.1.
Assume that a dynamical system ( X , St ) is dissipative and asymptotically compact. Let B be a bounded absorbing set of the system ( X , St ) . Then
the set A = w ( B ) is a nonempty compact set and is a global attractor of the
dynamical system ( X , St ) . The attractor A is a connected set in X .
In particular, this theorem is applicable to the dynamical systems from Exercises
4.2–4.11. It should also be noted that Theorem 5.1 along with Lemma 4.1 gives the
following criterion: a dissipative dynamical system possesses a compact global attractor if and only if it is asymptotically compact.
The proof of the theorem is based on the following lemma.
Theorems on Existence of Global Attractor
Lemma 5.1.
Let a dynamical system ( X , St ) be asymptotically compact. Then for
any bounded set B of X the w -limit set w ( B ) is a nonempty compact
invariant set.
Proof.
Let yn Î B . Then for any sequence { tn } tending to infinity the set { St( 2 ) yn ,
n
n = 1 , 2 , ¼ } is relatively compact, i.e. there exist a sequence nk and an element y Î X such that St( 2 ) yn tends to y as k ® ¥ . Hence, the asymptotic
k
n
compactness gives us that k
y - St yn
n
k
£ St( 1 ) yn
n
k
k
k
+ y - St( 2 ) yn
n
k
k
®0
as k ® ¥ .
Thus, y = lim St yn . Due to Lemma 2.1 this indicates that w ( B ) is nonk
k ® ¥ nk
empty.
Let us prove the invariance of w -limit set. Let y Î w ( B ) . Then according
to Lemma 2.1 there exist sequences { tn } , tn ® ¥ , and { z n } Ì B such that
St z n ® y . However, the mapping St is continuous. Therefore,
n
St + t z n = St ° St zn ® St y ,
n
n
n ® ¥.
Lemma 2.1 implies that St y Î w ( B ) . Thus,
St w ( B ) Ì w ( B ) ,
t > 0.
Let us prove the reverse inclusion. Let y Î w ( B ) . Then there exist sequences
{ vn } Ì B and { tn : tn ® ¥ } such that Stn vn ® y . Let us consider the sequence yn = St - t vn , tn ³ t . The asymptotic compactness implies that there
n
exist a subsequence tn and an element z Î X such that
k
z = lim St(n2 ) - t yn .
k®¥
k
k
As stated above, this gives us that
z = lim St - t yn .
k
k ® ¥ nk
Therefore, z Î w ( B ) . Moreover,
St z = lim St ° St - t vn = lim St vn = y .
nk
k
k
k®¥
k ® ¥ nk
Hence, y Î St w ( B ) . Thus, the invariance of the set w ( B ) is proved.
Let us prove the compactness of the set w ( B ) . Assume that { zn } is a sequence in w ( B ) . Then Lemma 2.1 implies that for any n we can find tn ³ n and
yn Î B such that z n - Stn yn £ 1 ¤ n . As said above, the property of asymptotic compactness enables us to find an element z and a sequence { n k } such
that
Stn y n - z ® 0 ,
k
k
k ® ¥.
29
30
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
This implies that z Î w ( B ) and z n ® z . This means that w ( B ) is a closed and
k
compact set in H . Lemma 5.1 is proved completely.
Now we establish Theorem 5.1. Let B be a bounded absorbing set of the dynamical
system. Let us prove that w ( B ) is a global attractor. It is sufficient to verify that
w ( B ) uniformly attracts the absorbing set B . Assume the contrary. Then the value
sup {dist ( St y , w ( B ) ) : y Î B } does not tend to zero as t ® ¥ . This means that
there exist d > 0 and a sequence { tn : tn ® ¥ } such that
ì
ü
sup ídist ( St y , w ( B ) ) : y Î B ý ³ 2 d .
n
î
þ
Therefore, there exists an element yn Î B such that
dist ( Stn yn , w ( B ) ) ³ d ,
n = 1, 2, ¼ .
(5.1)
As before, a convergent subsequence {Stn yn } can be extracted from the sequence
k
k
{ Stn yn } . Therewith Lemma 2.1 implies
z º lim St
k®¥
nk
yn Î w ( B )
k
which contradicts estimate (5.1). Thus, w ( B ) is a global attractor. Its compactness
follows from the easily verifiable relation
A º w (B) =
Ç ÇS
( 2)
t B
t >0
.
t ³ t
Let us prove the connectedness of the attractor by reductio ad absurdum. Assume
that the attractor A is not a connected set. Then there exists a pair of open sets U1
and U2 such that
Ui Ç A ¹ Æ ,
i = 1, 2 ,
A Ì U1 È U2 ,
U1 Ç U2 = Æ .
Let A c = conv ( A) be a convex hull of the set A , i.e.
N
ì
li vi : vi Î A , l i ³ 0 ,
Ac = í
î i=1
å
N
å l = 1,
i
i=1
ü
N = 1, 2, ¼ ý .
þ
It is clear that
is a bounded connected set and A c É A . The continuity of the
mapping St implies that the set St A c is also connected. Therewith A = St A Ì St A c .
Therefore, Ui Ç St A c ¹ Æ , i = 1 , 2 . Hence, for any t > 0 the pair U1 , U2 cannot
cover St A c . It follows that there exists a sequence of points xn = Sn yn Î Sn A c
such that xn Ï U1 È U2 . The asymptotic compactness of the dynamical system
enables us to extract a subsequence { n k } such that xnk = Sn yn tends in X to an
k
k
element y as k ® ¥ . It is clear that y Ï U1 È U2 and y Î w ( A c ) . These equations
contradict one another since w ( A c ) Ì w ( B ) = A Ì U1 È U2 . Therefore, Theorem
5.1 is proved completely.
Ac
Theorems on Existence of Global Attractor
It should be noted that the connectedness of the global attractor can also be proved
without using the linear structure of the phase space (do it yourself).
E x e r c i s e 5.1 Show that the assumption of asymptotic compactness in Theorem 5.1 can be replaced by the Ladyzhenskaya assumption: the sequence { Stn un } contains a convergent subsequence for any
bounded sequence { un } Ì X and for any increasing sequence
{ tn } Ì T + such that tn ® + ¥ . Moreover, the Ladyzhenskaya assumption is equivalent to the condition of asymptotic compactness.
E x e r c i s e 5.2 Assume that a dynamical system ( X , St ) possesses a compact
global attractor A . Let A* be a minimal closed set with the property
lim dist ( St y , A * ) = 0
t®¥
for every
y ÎX.
Then A* Ì A and A* = È { w ( x ) : x Î X } , i.e. A* coincides with the
global minimal attractor (cf. Exercise 3.4).
E x e r c i s e 5.3 Assume that equation (4.7) holds. Prove that the global attractor A possesses the property A = w ( K ) Ì K .
E x e r c i s e 5.4 Assume that a dissipative dynamical system possesses a global attractor A . Show that A = w ( B ) for any bounded absorbing set
B of the system.
The fact that the global attractor A has the form A = w ( B ) , where B is an absorbing set of the system, enables us to state that the set St B not only tends to the attractor A , but is also uniformly distributed over it as t ® ¥ . Namely, the following
assertion holds.
Theorem 5.2.
Assume that a dissipative dynamical system ( X , St ) possesses a compact global attractor A . Let B be a bounded absorbing set for ( X , St ) . Then
lim sup { dist ( a , St B ) : a Î A } = 0 .
t®¥
(5.2)
Proof.
Assume that equation (5.2) does not hold. Then there exist sequences { an } Ì
Ì A and { tn : tn ® ¥ } such that
dist ( an , Stn B ) ³ d
for some
d > 0.
(5.3)
The compactness of A enables us to suppose that { an } converges to an element
a Î A . Therewith (see Exercise 5.4)
a = lim St ym ,
m®¥ m
{ ym } Ì B ,
31
32
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
where { tm } is a sequence such that tm ® ¥ . Let us choose a subsequence { m n }
such that tmn ³ tn + tB for every n = 1 , 2 , ¼ . Here tB is chosen such that St B Ì
Ì B for all t ³ tB . Let zn = Stm - t ym . Then it is clear that { zn } Ì B and
1
n
n
n
a = lim Stm ym = lim Stn z n .
n
n
n®¥
n®¥
Equation (5.3) implies that
dist ( an , Stn z n ) ³ dist ( an , Stn B ) ³ d .
This contradicts the previous equation. Theorem 5.2 is proved.
For a description of convergence of the trajectories to the global attractor it is convenient to use the Hausdorff metric that is defined on subsets of the phase space
by the formula
r ( C , D ) = max { h ( C , D ) ; h ( D , C ) } ,
(5.4)
h ( C , D ) = sup { dist ( c , D ) : c Î C } .
(5.5)
where C , D Î X and
Theorems 5.1 and 5.2 give us the following assertion.
Corollary 5.1.
Let ( X , St ) be an asymptotically compact dissipative system. Then its
global attractor A possesses the property lim r ( St B , A) = 0 for any
t®¥
bounded absorbing set B of the system ( X , St ) .
In particular, this corollary means that for any e > 0 there exists te > 0 such that
for every t > te the set St B gets into the e -vicinity of the global attractor A;
and vice versa, the attractor A lies in the e -vicinity of the set St B . Here B is
a bounded absorbing set.
The following theorem shows that in some cases we can get rid of the requirement of asymptotic compactness if we use the notion of the global weak attractor.
Theorem 5.3.
Let the phase space H of a dynamical system ( H , St ) be a separable
Hilbert space. Assume that the system ( H , St ) is dissipative and its evolutionary operator St is weakly closed, i.e. for all t > 0 the weak convergence
yn ® y and St yn ® z imply that z = St y . Then the dynamical system
( H , St ) possesses a global weak attractor.
attractor.
The proof of this theorem basically repeats the reasonings used in the proof of Theorem 5.1. The weak compactness of bounded sets in a separable Hilbert space plays
the main role instead of the asymptotic compactness.
Theorems on Existence of Global Attractor
Lemma 5.2.
Assume that the hypotheses of Theorem 5.3 hold. For B Ì H we define
the weak w -limit set w w ( B ) by the formula
ww ( B ) =
Ç È S (B)
s ³ 0
t ³ s
,
t
(5.6)
w
where [ Y ] w is the weak closure of the set Y . Then for any bounded set
B Ì H the set w w ( B ) is a nonempty weakly closed bounded invariant
set.
Proof.
s
(B) = [ È
S ( B ) ] w is
The dissipativity implies that each of the sets gw
t³s t
bounded and therefore weakly compact. Then the Cantor theorem on the cols
lection of nested compact sets gives us that w w ( B ) = Çs ³ 0 gw ( B ) is a nonempty weakly closed bounded set. Let us prove its invariance. Let y Î w w ( B ) .
Then there exists a sequence yn Î Èt ³ n St ( B ) such that yn ® y weakly. The
dissipativity property implies that the set { St yn } is bounded when t is large
enough. Therefore, there exist a subsequence { yn } and an element z such
k
that yn ® y and St yn ® z weakly. The weak closedness of St implies that
k
k s
s
( B ) for all s .
z = St y . Since St yn Î gw ( B ) for nk ³ s , we have that z Î gw
k
Hence, z Î w w ( B ) . Therefore, St w w ( B ) Ì ww ( B ) . The proof of the reverse
inclusion is left to the reader as an exercise.
For the proof of Theorem 5.3 it is sufficient to show that the set
Aw = ww ( B ) ,
(5.7)
where B is a bounded absorbing set of the system ( H , St ) , is a global weak attractor
for the system. To do that it is sufficient to verify that the set B is uniformly attracted to Aw = w w ( B ) in the weak topology of the space H . Assume the contrary. Then
there exist a weak vicinity O of the set Aw and sequences { yn } Ì B and { tn : tn ®
® ¥ } such that Stn yn Ï O . However, the set { Stn yn } is weakly compact. Therefore, there exist an element z Ï O and a sequence { nk } such that
z = w - lim St y n .
k
k ® ¥ nk
s
s
( B ) for tnk ³ s . Thus, z Î gw ( B ) for all s ³ 0 and z Î
However, Stn yn Î gw
k
k
Î w w ( B ) , which is impossible. Theorem 5.3 is proved.
E x e r c i s e 5.5 Assume that the hypotheses of Theorem 5.3 hold. Show that
the global weak attractor A w is a connected set in the weak topology
of the phase space H .
Show that the global weak minimal attractor A*w = È { w w ( x ) :
x Î H } is a strictly invariant set.
E x e r c i s e 5.6
33
34
C
h
a
p
t
e
r
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
E x e r c i s e 5.7 Prove the existence and describe the structure of global and
global minimal attractors for the dynamical system generated by
the equations
ì
í
î
1
x· = m x - y - x ( x 2 + y 2 ) ,
y· = x + m y - y ( x 2 + y 2 )
for every real m .
E x e r c i s e 5.8 Assume that X is a metric space and ( X , St ) is an asymptotically compact (in the sense of the definition given in Remark 4.1)
dynamical system. Assume also that the attracting compact K is
contained in some bounded connected set. Prove the validity of the
assertions of Theorem 5.1 in this case.
In conclusion to this section, we give one more assertion on the existence of the global
attractor in the form of exercises. This assertion uses the notion of the asymptotic
smoothness (see [3] and [9]). The dynamical system ( X , St ) is said to be asymptotically smooth if for any bounded positively invariant ( St B Ì B , t ³ 0 ) set
B Ì X there exists a compact K such that h ( St B , K ) ® 0 as t ® ¥ , where the
value h ( . , . ) is defined by formula (5.5).
E x e r c i s e 5.9 Prove that every asymptotically compact system is asymptotically smooth.
E x e r c i s e 5.10 Let ( X , St ) be an asymptotically smooth dynamical system.
Assume that for any bounded set B Ì X the set g +(B) =
=
t ³ 0 St ( B ) is bounded. Show that the system ( X , St ) possesses a global attractor A of the form
È
A=
È {w (B) :
B Ì X , B is bounded } .
E x e r c i s e 5.11 In addition to the assumptions of Exercise 5.10 assume that
( X , St ) is pointwise dissipative, i.e. there exists a bounded set
B 0 Ì X such that dist X ( St y , B 0 ) ® 0 as t ® ¥ for every point
y Î X . Prove that the global attractor A is compact.
§ 6
On the Structure of Global Attractor
The study of the structure of global attractor of a dynamical system is an important
problem from the point of view of applications. There are no universal approaches to
this problem. Even in finite-dimensional cases the attractor can be of complicated
structure. However, some sets that undoubtedly belong to the attractor can be poin-
On the Structure of Global Attractor
ted out. It should be first noted that every stationary point of the semigroup St belongs to the attractor of the system. We also have the following assertion.
Lemma 6.1.
Assume that an element z lies in the global attractor A of a dynamical
system ( X , St ) . Then the point z belongs to some trajectory g that lies
in A wholly.
Proof.
Since St A = A and z Î A , then there exists a sequence { z n } Ì A such
that z0 = z , S1 z n = zn - 1 , n = 1 , 2 , ¼ . Therewith for discrete time the required trajectory is g = { u n : n Î Z } , where un = Sn z for n ³ 0 and un =
= z-n for n £ 0 . For continuous time let us consider the value
t ³ 0,
ì St z ,
u ( t) = í
î St + n zn , - n £ t £ - n + 1 , n = 1 , 2 , ¼
Then it is clear that u ( t ) Î A for all t Î R and St u ( t ) = u ( t + t ) for t ³ 0 ,
t Î R . Therewith u ( 0 ) = z . Thus, the required trajectory is also built in the
continuous case.
E x e r c i s e 6.1 Show that an element z belongs to a global attractor if and
only if there exists a bounded trajectory g = { u ( t ) : - ¥ < t < ¥ }
such that u ( 0 ) = z .
Unstable sets also belong to the global attractor. Let Y be a subset of the phase
space X of the dynamical system ( X , St ) . Then the unstable set emanating
from Y is defined as the set M + (Y ) of points z Î X for every of which there exists
a trajectory g = { u ( t ) : t Î T } such that
u (0) = z,
E x e r c i s e 6.2
t > 0.
lim dist ( u ( t ) , Y ) = 0 .
t ® -¥
Prove that M+(Y ) is invariant, i.e. St M+ (Y ) = M +(Y ) for all
Lemma 6.2.
Let N be a set of stationary points of the dynamical system ( X , St )
possessing a global attractor A . Then M + (N ) Ì A .
Proof.
It is obvious that the set N = { z : St z = z , t > 0 } lies in the attractor
of the system and thus it is bounded. Let z Î M+(N ) . Then there exists a trajectory g z = { u ( t ) , t ÎT } such that u ( 0 ) = z and
dist ( u ( t ) , N ) ® 0 ,
t ® -¥ .
35
36
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
Therefore, the set Bs = { u ( t ) : t £ - s } is bounded when s > 0 is large
enough. Hence, the set St Bs tends to the attractor of the system as t ® + ¥ .
However, z Î St Bs for t ³ s . Therefore,
1
This implies that z Î A . The lemma is proved.
dist ( z , A) £ sup { dist ( St y , A) : y Î Bs } ® 0 ,
E x e r c i s e 6.3
that
t ® +¥.
Assume that the set N of stationary points is finite. Show
l
M+(N ) =
È M (z ) ,
k=1
+
k
where z k are the stationary points of St (the set M+( z k ) is called
an unstable manifold emanating from the stationary point z k ).
Thus, the global attractor A includes the unstable set M +(N ) . It turns out that under certain conditions the attractor includes nothing else. We give the following definition. Let Y be a positively invariant set of a semigroup St : St Y Ì Y , t > 0 . The
continuous functional F ( y ) defined on Y is called the Lyapunov function of the
dynamical system ( X , St ) on Y if the following conditions hold:
a) for any y Î Y the function F ( St y ) is a nonincreasing function with respect to t ³ 0 ;
b) if for some t0 > 0 and y Î X the equation F ( y ) = F ( St y ) holds, then
0
y = St y for all t ³ 0 , i.e. y is a stationary point of the semigroup St .
Theorem 6.1.
Let a dynamical system ( X , St ) possess a compact attractor A . Assume
also that the Lyapunov function F ( y ) exists on A . Then A = M+(N ) , where
N is the set of stationary points of the dynamical system.
Proof.
Let y Î A . Let us consider a trajectory g passing through y (its existence follows from Lemma 6.1). Let
g = {u (t) : t Î T}
and
gt– = { u ( t ) : t £ t } .
Since gt– Ì A , the closure [ g t– ] is a compact set in X . This implies that the a -limit
set
a (g) =
Ç [g ]
t< 0
–
t
of the trajectory g is nonempty. It is easy to verify that the set a ( g ) is invariant:
St a ( g ) = a ( g ) . Let us show that the Lyapunov function F( y ) is constant on a ( g ) .
Indeed, if u Î a ( g ) , then there exists a sequence { tn } tending to - ¥ such that
On the Structure of Global Attractor
lim u ( tn ) = u .
tn ® - ¥
Consequently,
F( u ) = lim F( u ( tn ) ) .
n®¥
By virtue of monotonicity of the function F ( u ) along the trajectory we have
F ( u ) = sup { F( u ( t ) ) : t < 0 } .
Therefore, the function F( u ) is constant on a ( g ) . Hence, the invariance of the set
a ( g ) gives us that F( St u ) = F( u ) , t > 0 for all u Î a ( g ) . This means that a ( g )
lies in the set N of stationary points. Therewith (verify it yourself)
lim dist ( u ( t ) , a ( g ) ) = 0 .
t ® -¥
Hence, y Î M+(N ) . Theorem 6.1 is proved.
E x e r c i s e 6.4 Assume that the hypotheses of Theorem 6.1 hold. Then for
any element y Î A its w -limit set w ( y ) consists of stationary points
of the system.
Thus, the global attractor coincides with the set of all full trajectories connecting the
stationary points.
E x e r c i s e 6.5 Assume that the system ( X , St ) possesses a compact global
attractor and there exists a Lyapunov function on X . Assume that
the Lyapunov function is bounded below. Show that any semitrajectory of the system tends to the set N of stationary points of the system as t ® + ¥ , i.e. the global minimal attractor coincides with the
set N .
In particular, this exercise confirms the fact realized by many investigators that the
global attractor is a too wide object for description of actually observed limit regimes
of a dynamical system.
E x e r c i s e 6.6 Assume that ( R , St ) is a dynamical system generated by the
logistic equation (see Example 1.1): x· + a x ( x - 1 ) = 0 , a > 0 .
Show that V ( x ) = x 3 ¤ 3 - x 2 ¤ 2 is a Lyapunov function for this system.
E x e r c i s e 6.7
Show that the total energy
--- x 2 - b x
--- x· 2 + 1
--- x 4 - a
E ( x , x· ) = 1
2
2
4
is a Lyapunov function for the dynamical system generated (see
Example 1.2) by the Duffing equation
x·· + e x· + x 3 - a x = b , e > 0 .
37
38
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
If in the definition of a Lyapunov functional we omit the second requirement, then
a minor modification of the proof of Theorem 6.1 enables us to get the following assertion.
Theorem 6.2.
1
Assume that a dynamical system ( X , St ) possesses a compact global attractor A and there exists a continuous function Y ( y ) on X such that
Y ( St y ) does not increase with respect to t for any y Î X . Let L be a set of
elements u Î A such that Y ( u ( t ) ) = Y ( u ) for all - ¥ < t < ¥ . Here { u ( t ) } is
a trajectory of the system passing through u ( u ( 0 ) = u ).. Then M +( L ) = A
and L contains the global minimal attractor A* = Èx Î X w ( x ) .
Proof.
In fact, the property M+( L ) = A was established in the proof of Theorem 6.1.
As to the property A* Ì L , it follows from the constancy of the function Y ( u ) on
the w -limit set w ( x ) of any element x Î X .
E x e r c i s e 6.8 Apply Theorem 6.2 to justify the results of Example 3.1 (see
also Exercise 4.8).
If the set N of stationary points of a dynamical system ( X , St ) is finite, then Theorem 6.1 can be extended a little. This extension is described below in Exercises 6.9–
6.12. In these exercises it is assumed that the dynamical system ( X , St ) is continuous and possesses the following properties:
(a) there exists a compact global attractor A ;
(b) there exists a Lyapunov function F ( x ) on A ;
(c) the set N = { z 1 , ¼ zN } of stationary points is finite, therewith F ( z i ) ¹
¹ F ( zj ) for i ¹ j and the indexing of zj possesses the property
F ( z1 ) < F ( z2 ) < ¼ < F ( zN ) .
(6.1)
We denote
j
Aj =
E x e r c i s e 6.9
È M (z ) ,
k=1
+
k
j = 1, 2, ¼, N ,
A0 = Æ .
Show that St Aj = Aj for all j = 1 , 2 , ¼ N .
E x e r c i s e 6.10 Assume that B Ì Aj \ { zj } . Then
lim sup { dist ( St y , Aj - 1 ) : y Î B } = 0 .
t®¥
(6.2)
E x e r c i s e 6.11 Assume that the function F is defined on the whole X . Then
(6.2) holds for any bounded set B Ì { x : F ( x ) < F ( zj ) - d } ,
where d is a positive number.
On the Structure of Global Attractor
E x e r c i s e 6.12 Assume that [ M +( z j ) ] is the closure of the set M+( zj ) and
¶M+( zj ) = [ M+( zj) ] \ M+( zj ) is its boundary. Show that ¶ M+ zj Ì
Ì A j - 1 and
St [ M+( z j ) ] = [ M+( z j ) ] ,
St ¶ M+( z j ) = ¶ M+( zj ) .
It can also be shown (see the book by A. V. Babin and M. I. Vishik [1]) that under
some additional conditions on the evolutionary operator St the unstable manifolds
M+( zj ) are surfaces of the class C 1 , therewith the facts given in Exercises 6.9–6.12
remain true if strict inequalities are substituted by nonstrict ones in (6.1). It should
be noted that a global attractor possessing the properties mentioned above is frequently called regular .
Let us give without proof one more result on the attractor of a system with a finite number of stationary points and a Lyapunov function. This result is important
for applications.
At first let us remind several definitions. Let S be an operator acting in a Banach space X . The operator S is called Frechét differentiable at a point
x Î X provided that there exists a linear bounded operator S ¢ ( x ) : X ® X such
that
S (y) - S (x) - S ¢(x) (y - x)
£ g( x -y ) x -y
for all y from some vicinity of the point x, where g ( x ) ® 0 as x ® 0 . Therewith,
the operator S is said to belong to the class C 1 + a , 0 < a < 1 , on a set Y if it is
differentiable at every point x Î Y and
S ¢( x ) - S ¢ ( y ) L (X, X) £ C x - y
a
for all y from some vicinity of the point x Î Y . A stationary point z of the mapping
S is called hyperbolic if S Î C 1+ a in some vicinity of the point z , the spectrum
of the linear operator S ¢( z ) does not cross the unit circle { l : l = 1 } and the spectral subspace of the operator corresponding to the set { l : l > 1 } is finite-dimensional.
Theorem 6.3.
Let X be a Banach space and let a continuous dynamical system
( X , St ) possess the properties:
1) there exists a global attractor A ;
2) there exists a vicinity W of the attractor A such that
St x - St y £ C e a ( t - t ) St x - St y
for all t ³ t ³ 0 , provided St x and St y belong to W for all t ³ 0 ;
3) there exists a Lyapunov function continuous on X ;
4) the set N = { z 1 , ¼ , z N } of stationary points is finite and all the
points are hyperbolic;
39
40
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
5) the mapping ( t , u ) ® St u is continuous.
Then for any compact set B in X the estimate
1
holds for all t ³ 0 , where h > 0 does not depend on B .
ì
ü
sup ídist ( St y , A ) : y Î B ý £ CB e - h t
î
þ
(6.3)
The proof of this theorem as well as other interesting results on the asymptotic behaviour of a dynamical system possessing a Lyapunov function can be found in the
book by A. V. Babin and M. I. Vishik [1].
To conclude this section, we consider a finite-dimensional example that shows
how the Lyapunov function method can be used to prove the existence of periodic
trajectories in the attractor.
E x a m p l e 6.1 (on the theme by E. Hopf)
Studying Galerkin approximations in a model suggested by E. Hopf for the description of possible mechanisms of turbulence appearence, we obtain the following system of ordinary differential equations
ì u· + m u + v 2 + w 2 = 0 ,
ï ·
í v + nv -vu -bw = 0,
ï ·
î w + nw -wu + bv = 0.
(6.4)
(6.5)
(6.6)
Here m is a positive parameter, n and b are real parameters. It is clear that the
Cauchy problem for (6.4)–(6.6) is solvable, at least locally for any initial condition. Let us show that the dynamical system generated by equations (6.4)–(6.6)
is dissipative. It will also be sufficient for the proof of global solvability. Let us
introduce a new unknown function u* = u + m ¤ 2 - n . Then equations (6.4)–
(6.6) can be rewritten in the form
ì ·
m
ï u* + m u* + v 2 + w 2 = m æè --- - nöø ,
2
ï
ï
í v· + 1
--- m v - v u* - b w = 0 ,
2
ï
ï
·
ïw + 1
--- m w - w u* + b v = 0 .
2
î
These equations imply that
m
m
2
2
d
2
2
2
2
1
--- ---( u* + v + w ) + m u* + --- ( v + w ) = m æ --- - nö u*
è2
ø
2 dt
2
on any interval of existence of solutions. Hence,
2
m
2
2
d
2
2
2
2
---- æ ( u* + v + w )ö + m ( u* + v + w ) £ m æ --- - nö .
ø
è2
ø
dt è
On the Structure of Global Attractor
Thus,
2
2
u* ( t ) + v ( t ) + w ( t )
2
£
m
2
2
2
£ æ u* ( 0 ) + v ( 0 ) + w ( 0 ) ö e -m t + æ --- - nö ( 1 -e -m t ) .
è
ø
è2
ø
2
Firstly, this equation enables us to prove the global solvability of problem (6.4)–
(6.6) for any initial condition and, secondly, it means that the set
2
2ü
ì
m
m
B0 = í ( u , v , w ) : æ u + --- - nö + v 2 + w 2 £ 1 + æ --- - nö ý
è
ø
è2
ø
2
î
þ
is absorbing for the dynamical system ( R 3 , St ) generated by the Cauchy problem for equations (6.4)–(6.6). Thus, Theorem 5.1 guarantees the existence of
a global attractor A. It is a connected compact set in R 3 .
E x e r c i s e 6.13 Verify that B0 is a positively invariant set for ( R 3 , St ) .
In order to describe the structure of the global attractor A we introduce the polar
coordinates
v ( t ) = r ( t ) cos j ( t ) ,
w ( t ) = r ( t ) sin j ( t )
on the plane of the variables { v ; w } . As a result, equations (6.4)–(6.6) are transformed into the system
ì u· + m u + r 2 = 0 ,
í·
î r + nr -ur = 0,
(6.7)
(6.8)
therewith, j ( t ) = - b t + j 0 . System (6.7) and (6.8) has a stationary point { u = 0 ,
r = 0 } for all m > 0 and n Î R . If n < 0 , then one more stationary point { u = n,
r = -m n } occurs in system (6.7) and (6.8). It corresponds to a periodic trajectory
of the original problem (6.4)–(6.6).
E x e r c i s e 6.14 Show that the point ( 0 ; 0 ) is a stable node of system (6.7)
and (6.8) when n > 0 and it is a saddle when n < 0 .
E x e r c i s e 6.15 Show that the stationary point { u = n , r = - m n } is stable
m
( n < 0 ) being a node if - m ¤ 8 < n < 0 and a focus if n < - --- .
8
If n > 0 , then (6.7) and (6.8) imply that
d (u 2 + 2) +
1
--- ---min ( m , n ) ( u 2 + r 2 ) £ 0 .
r
2 dt
Therefore,
2
u (t) + r (t)
2
2
£ u (0) + r (0)
2
e -2 min ( m , n ) t .
41
42
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
Hence, for n > 0 the global attractor A of the system ( R 3 , St ) consists of the single
stationary exponentially attracting point
{u = 0, v = 0, w = 0} .
E x e r c i s e 6.16 Prove that for n = 0 the global attractor of problem (6.4)–
(6.6) consists of the single stationary point { u = 0 , v = 0 , w = 0 } .
Show that it is not exponentially attracting.
1
Now we consider the case n < 0 . Let us again refer to problem (6.7) and (6.8). It is
clear that the line r = 0 is a stable manifold of the stationary point { u = 0 , r = 0 } .
Moreover, it is obvious that if r ( t0 ) > 0 , then the value r ( t ) remains positive for all
t > t0 . Therefore, the function
2
--- ( u - n ) + 1
--- r 2 + m n ln r
V (u, r) = 1
2
2
(6.9)
is defined on all the trajectories, the initial point of which does not lie on the line
{ r = 0 } . Simple calculations show that
d
2
---- ( V ( u ( t ) , r ( t ) ) ) + m ( u ( t ) - n ) = 0
dt
(6.10)
and
V(u, r) ³ V(n,
2
--- æ u - n 2 + r - -m n ö ;
-m n ) + 1
ø
2è
(6.11)
therewith, V ( n , - m n ) = ( 1 ¤ 2 ) m n ln ( e ¤ ( m n ) ) . Equation (6.10) implies that
the function V ( u , r ) does not increase along the trajectories. Therefore, any semitrajectory { ( u ( t ) ; r ( t ) ) , t Î R + } emanating from the point { u0 , r 0 ; r 0 ¹ 0 } of
the system ( R ´ R + , St ) generated by equations (6.7) and (6.8) possesses the
property V ( u ( t ) , r ( t ) ) £ V ( u0 , r 0 ) for t ³ 0 . Therewith, equation (6.9) implies
that this semitrajectory can not approach the line { r = 0 } at a distance less then
exp { [ 1 ¤ ( m n ) ] × V ( u0 , r0 ) } . Hence, this semitrajectory tends to y = { u = n,
r = -m n } . Moreover, for any x Î R the set
Bx = { y = ( u , r ) : V ( u , r ) £ x }
is uniformly attracted to y , i.e. for any e > 0 there exists t0 = t0 ( x , e ) such that
St Bx Ì { y : y - y £ e } .
Indeed, if it is not true, then there exist e0 > 0 , a sequence tn ® + ¥ , and zn Î Bx
such that St z n - y > e 0 . The monotonicity of V ( y ) and property (6.11) imply that
n
V ( St z n ) ³ V ( St z n ) ³ V ( n ,
n
1 2
- m n ) + --- e 0
2
for all 0 £ t £ tn . Let z be a limit point of the sequence { zn } . Then after passing
to the limit we find out that
On the Structure of Global Attractor
Fig. 2. Qualitative behaviour of solutions to problem (6.7), (6.8):
a) - m ¤ 8 < n < 0,
b) n < - m ¤ 8
V ( St z ) ³ V ( n ,
1 2
-m n ) + --- e 0 ,
2
t ³ 0
with z Ï { r = 0 } . Thus, the last inequality is impossible since St z ® y = { u = n ,
r = -m n } . Hence
lim sup { dist ( St y , y ) : y Î B x } = 0 .
(6.12)
t®¥
The qualitative behaviour of solutions to problem (6.7) and (6.8) on the semiplane
is shown on Fig. 2.
In particular, the observations above mean that the global minimal attractor
Amin of the dynamical system ( R 3 , St ) generated by equations (6.4)–(6.6) consists
of the saddle point { u = 0 , v = 0 , w = 0 } and the stable limit cycle
Cn = { u = n ,
v2 + w 2 = -m n }
(6.13)
for n < 0 . Therewith, equation (6.12) implies that the cycle Cn uniformly attracts
all bounded sets B in R 3 possessing the property
d º inf { v 2 + w 2 : ( u , v , w ) Î B } > 0 ,
(6.14)
i.e. which lie at a positive distance from the line { v = 0 , w = 0 } .
E x e r c i s e 6.17 Using the structure of equations (6.7) and (6.8) near the stationary point { u = n , r = - m n } , prove that a bounded set B possessing property (6.14) is uniformly and exponentially attracted to
the cycle Cn , i.e.
sup { dist ( St y , Cn ) , y Î B } £ C e
for t ³ tB , where g is a positive constant.
-g ( t - tB )
43
44
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
Now let y0 = ( u0 , v0 , w0 ) lie in the global attractor A of the system ( R 3 , St ) .
Assume that r 0 ¹ 0 and r 02 = v02 + w02 ¹ - m n . Then (see Lemma 6.1) there exists
a trajectory g = { y ( t ) = ( u ( t ) ; v ( t ) ; w ( t ) ) , t Î R } lying in A such that y (0) = y0 .
The analysis given above shows that y ( t ) ® Cn as t ® + ¥ . Let us show that
y ( t ) ® 0 when t ® -¥ . Indeed, the function V ( u ( t ) , r ( t ) ) is monotonely nondecreasing as t ® - ¥ . If we argue by contradiction and use the fact that y ( t ) is
bounded we can easily find out that
1
lim V ( u ( t ) , r ( t ) ) = ¥
t ® -¥
and therefore
2
2
r (t) = æ v (t) + w(t) ö
è
ø
1/2
®0
t ® -¥ .
as
(6.16)
Equation (6.7) gives us that
t
u ( t ) = e -m ( t - t ) u ( s ) -
òe
-m ( t - t )
2
[ r ( t ) ] dt .
(6.17)
s
Since u ( s ) is bounded for all s Î R , we can get the equation
t
u ( t) = -
òe
-m ( t - t )
2
[ r ( t ) ] dt
-¥
by tending s ® – ¥ in (6.17). Therefore, by virtue of (6.16) we find that u ( t ) ® 0
as t ® – ¥ . Thus, y ( t ) ® 0 as t ® – ¥ . Hence, for n < 0 the global attractor A
Fig. 3. Attractor of the system (6.4)–(6.6);
a) - m ¤ 8 < n < 0 , b) n < - m ¤ 8
Stability Properties of Attractor and Reduction Principle
of the system ( R 3 , St ) coincides with the union of the unstable manifold M+( 0 )
emanating from the point { u = 0 , v = 0 , w = 0 } and the limit cycle (6.13). The attractor is shown on Fig. 3.
§ 7
Stability Properties of Attractor
and Reduction Principle
A positively invariant set M in the phase space of a dynamical system ( X , St ) is said
to be stable (in Lyapunov’s sense) in X if its every vicinity O contains some
vicinity O ¢ such that St ( O ¢) Ì O for all t ³ 0 . Therewith, M is said to be asymptotically stable if it is stable and St y ® M as t ® ¥ for every y Î O ¢ . A set
M is called uniformly asymptotically stable if it is stable and
lim sup { dist ( St y , M ) : y Î O ¢} = 0 .
t®¥
(7.1)
The following simple assertion takes place.
Theorem 7.1.
Let A be the compact global attractor of a continuous dynamical system ( X , St ) . Assume that there exists its bounded vicinity U such that the
mapping ( t , u ) ® St u is continuous on R+ ´ U . Then A is a stable set.
Proof.
Assume that O is a vicinity of A . Then there exists T > 0 such that St U Ì O
for t ³ T . Let us show that there exists a vicinity O ¢ of the attractor A such that
St O ¢ Ì O for all t Î [ 0 , T ] . Assume the contrary. Then there exist sequences { un }
and { tn } such that dist ( un , A) ® 0 , { tn } Î[ 0 , T ] and Stnu n Ï O . The set A being
compact, we can choose a subsequence { n k } such that u n ® u Î A as tn ®
k
k
® t Î [ 0 , T ] . Therefore, the continuity property of the function ( t , u ) ® St u
gives us that St nku n ® St u Î A . This contradicts the equation Stnun ÏO . Thus,
k
there exists O ¢ such that St O ¢ Ì O for t Î [ 0 , T ] . We can choose T such that
St ( O ¢ Ç U ) Ì O for all t ³ 0 . Therefore, the attractor A is stable. Theorem 7.1
is proved.
It is clear that the stability of the global attractor implies its uniform asymptotic
stability.
Assume that M is a positively invariant set of a system
( X , St ) . Prove that if there exists an element y Ï M such that its
a -limit set a ( y ) possesses the property a ( y ) Ç M ¹ Æ , then M
is not stable.
E x e r c i s e 7.1
45
46
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
In particular, the result of this exercise shows that the global minimal attractor can
appear to be an unstable set.
1
E x e r c i s e 7.2 Let us return to Example 3.1 (see also Exercises 4.8 and 6.8).
Show that:
(a) the global attractor A1 and the Milnor attractor A4 are
stable;
(b) the global minimal attractor A3 and the Ilyashenko attractor A5 are unstable.
Now let us consider the question concerning the stability of the attractor with respect to perturbations of a dynamical system. Assume that we have a family of dynamical systems ( X , Stl ) with the same phase space X and with an evolutionary
operator Stl depending on a parameter l which varies in a complete metric space
L . The following assertion was proved by L. V. Kapitansky and I. N. Kostin [6].
Theorem 7.2.
Assume that a dynamical system ( X , Stl ) possesses a compact global attractor A l for every l Î L . Assume that the following conditions hold:
(a) there exists a compact K Ì X such that A l Ì K for all l Î L ;
l
l
(b) if lk ® l0 , xk Î A k and xk ® x0 , then St k xk ® St x0 for some
0
0
t0 > 0 .
Then the family of att
attractors
ttractors A l is upper semicontinuous at the point l 0 ,
i.e.
l
l
l
l
h ( A k , A 0 ) º sup { dist ( y , A 0 ) : y Î A k } ® 0
(7.2)
as lk ® l 0 .
Proof.
Assume that equation (7.2) does not hold. Then there exist a sequence lk ®
l
l
® l0 and a sequence xk Î A k such that dist ( xk , A 0 ) ³ d for some d > 0 . But
the sequence xk lies in the compact K . Therefore, without loss of generality we can
l
assume that xk ® x0 Î K for some x0 Î K and x0 Ï A 0 . Let us show that this result leads to contradiction. Let gk = { uk ( t ) : – ¥ < t < ¥ } be a trajectory of the dyl
namical system ( X , St k) passing through the element xk ( uk ( 0 ) = xk ). Using the
standard diagonal process it is easy to find that there exist a subsequence { k ( n ) }
and a sequence of elements { um } Ì K such that
lim uk ( n ) ( -m t0 ) = u m
n®¥
for all
m = 0, 1, 2, ¼ ,
where u0 = x0 . Here t0 > 0 is a fixed number. Sequential application of condition
(b) gives us that
Stability Properties of Attractor and Reduction Principle
um - l =
l
l
lim uk ( n ) ( -( m - l ) t0 ) = lim Sl tk ( n ) uk ( n ) ( -m t0 ) = Sl t0 u m
0
0
n®¥
n®¥
for all m = 1 , 2 , ¼ and l = 1 , 2 , ¼ , m . It follows that the function
ì S l0 u ,
0
ï t
u (t) = í l
ïS 0
u ,
î t + t0 m m
t ³ 0,
-t0 m £ t < -t0 ( m - 1 ) , m = 1 , 2 , ¼
gives a full trajectory g passing through the point x0 . It is obvious that the trajectory
g is bounded. Therefore (see Exercise 6.1), it wholly belongs to Al0 , but that contradicts the equation x0 Ï Al0 . Theorem 7.2 is proved.
E x e r c i s e 7.3 Following L. V. Kapitansky and I. N. Kostin [6], for l ® l0
define the upper limit A ( l0 ; L ) of the attractors Al along L by
the equality
A ( l0 , L ) =
Ç È{A
l:
l Î L , 0 < dist ( l , l 0 ) < d } ,
d >0
where [ . ] denotes the closure operation. Prove that if the hypotheses of Theorem 7.2 hold, then A ( l0 , L) is a nonempty compact invariant set lying in the attractor Al0 .
Theorem 7.2 embraces only the upper semicontinuity of the family of attractors
{ A l } . In order to prove their continuity (in the Hausdorff metric defined by equation
(5.4)), additional conditions should be imposed on the family of dynamical systems
( X , Stl ) . For example, the following assertion proved by A. V. Babin and M. I. Vishik
concerning the power estimate of the deviation of the attractors A l and Al0 in the
Hausdorff metric holds.
Theorem 7.3.
Al
Assume that a dynamical system ( X , Stl ) possesses a global attractor
for every l Î L . Let the following conditions hold:
(a) there exists a bounded set B0 Ì X such that A l Ì B0 for all l Î L
and
h ( Stl B0 , A l ) £ C0 e -h t ,
l ÎL,
(7.3)
with constants C0 > 0 and h > 0 independent of l and with
h ( B , A) = sup { dist ( b , A) : b Î B } ;
(b) for any l 1 , l2 Î L and u 1 , u 2 Î B0 the estimate
l
l
dist ( St 1 u 1 , St 2 u2 ) £ C1 e a t ( dist ( u1 , u2 ) + dist ( l1 , l 2 ) )
(7.4)
47
48
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
holds, with constants C1 and a independent of l .
Then there exists C2 > 0 such that
l
l
h
q = --------------- .
h+a
q
r ( A 1 , A 2 ) £ C 2 [ dist ( l 1 , l 2 ) ] ,
(7.5)
Here r ( . , . ) is the Hausdorff metric defined by the formula
r ( B , A) = max { h ( B , A) ; h ( A , B ) } .
Proof.
By virtue of the symmetry of (7.5) it is sufficient to find out that
l
l
q
h ( A 1 , A 2 ) £ C2 [ dist ( l 1 , l 2 ) ] .
(7.6)
Equation (7.3) implies that for any e > 0
Stl B0 Ì O e ( A l )
l ÎL
for all
h -1 ( ln 1 ¤ e
t* ( e ,
when t ³
+ ln C0 ) . Here O e
C0 ) º
A l . It follows from equation (7.4) that
l
l
h ( St 1 B0 , St 2 B0 ) = sup
( Al )
(7.7)
is an e -vicinity of the set
l
l
inf dist ( St 1 x , St 2 y ) £
x Î B0 y Î B0
l
l
£ sup dist ( St 1 x , St 2 x ) £ C1 e a t dist ( l1 , l 2 ) .
x Î B0
(7.8)
Since A l Ì B0 , we have A l = S tl A l Ì Stl B 0 . Therefore, with t ³ t * ( e , C 0 ) , equation (7.7) gives us that
A l Ì Stl B0 Ì O e ( A l ) .
For any x , z Î X the estimate
(7.9)
dist ( x , A l ) £ dist ( x , z ) + dist ( z , A l )
holds. Hence, we can find that
dist ( x , A l ) £ dist ( x , z ) + e
for all x Î X and z Î O e ( A l ) . Consequently, equation (7.9) implies that
dist ( x , A l ) £ dist ( x , Stl B0 ) + e ,
for t ³ t * ( e , C 0 ) . It means that
l
l
h (A 1, A 2) =
£
x ÎX
l
sup dist ( x , A 2 ) £
l
x ÎA 1
l
l
l
sup dist ( x , St 2 B0 ) + e £ h ( St 1 B0 , St 2 B0 ) + e .
l
x ÎA 1
Thus, equation (7.8) gives us that for any e > 0
Stability Properties of Attractor and Reduction Principle
l
l
h ( A 1 , A 2 ) £ C1 e a t dist ( l 1 , l2 ) + e
h
q
-------------for t ³ t* ( e , C 0 ) . By taking e = [ dist ( l1 , l 2 ) ] , q = h
+ a- and t = t * ( e , C0 ) º
º h -1 ( ln 1 ¤ e + ln C0 ) in this formula we find estimate (7.6). Theorem 7.3 is proved.
It should be noted that condition (7.3) in Theorem 7.3 is quite strong. It can be verified only for a definite class of systems possessing the Lyapunov function (see Theorem 6.3).
In the theory of dynamical systems an important role is also played by the notion of the Poisson stability. A trajectory g = { u ( t ) : - ¥ < t < ¥ } of a dynamical
system ( X , St ) is said to be Poisson stable if it belongs to its w -limit set w ( g ) .
It is clear that stationary points and periodic trajectories of the system are Poisson
stable.
E x e r c i s e 7.4 Show that any Poisson stable trajectory is contained in the
global minimal attractor if the latter exists.
E x e r c i s e 7.5 A trajectory g is Poisson stable if and only if any point x
of this trajectory is recurrent, i.e. for any vicinity O ' x there exists
t > 0 such that St x Î O .
The following exercise testifies to the fact that not only periodic (and stationary) trajectories can be Poisson stable.
E x e r c i s e 7.6 Let Cb ( R ) be a Banach space of continuous functions bounded on the real axis. Let us consider a dynamical system ( Cb ( R ) , St )
with the evolutionary operator defined by the formula
( St f ) ( x ) = f ( x + t ) ,
f ( x ) Î Cb ( R ) .
Show that the element f0 ( x ) = sin w1 x + sin w 2 x is recurrent for
any real w1 and w 2 (in particular, when w1 ¤ w 2 is an irrational
number). Therewith the trajectory g = { f0 ( x + t ) : - ¥ < t < ¥ }
is Poisson stable.
In conclusion to this section we consider a theorem that is traditionally associated
with the stability theory. Sometimes this theorem enables us to significantly decrease
the dimension of the phase space, this fact being very important for the study of infinite-dimensional systems.
Theorem 7.4. (reduction principle).
Assume that in a dissipative dynamical system ( X , St ) there exists
a positively invariant locally compact set M possessing the property of
uniform attraction, i.e. for any bounded set B Ì X the equation
49
50
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
lim
sup dist ( St y , M ) = 0
t ® ¥ y ÎB
(7.10)
holds. Let A be a global attractor of the dynamical system ( M , St ) . Then A
is also a global attractor of ( X , St ) .
Proof.
It is sufficient to verify that
lim sup dist ( St y , A ) = 0
t ® ¥ y ÎB
(7.11)
for any bounded set B Ì X . Assume that there exists a set B such that (7.11) does
not hold. Then there exist sequences { yn } Ì B and { tn : tn ® ¥ } such that
dist ( St y n , A ) ³ d
n
(7.12)
for some d > 0 . Let B 0 be a bounded absorbing set of ( X , St ) . We choose a moment
t0 such that
ì
ü d
sup í dist ( St y , A ) : y Î M Ç B0 ý £ --- .
0
î
þ 2
(7.13)
This choice is possible because A is a global attractor of ( M , St ) . Equation (7.10)
implies that
dist ( St - t yn , M ) ® 0 ,
n
0
tn ® ¥ .
The dissipativity property of ( X , St ) gives us that St - t yn Î B0 when n is large
n
0
enough. Therefore, local compactness of the set M guarantees the existence of an
element z Î M Ç B0 and a subsequence { n k } such that
z = lim St - t yn .
k
k ® ¥ nk 0
This implies that Stn yn ® St0 z . Therefore, equation (7.12) gives us that
k
k
dist ( St0 z , A) ³ d . By virtue of the fact that z Î M Ç B0 this contradicts equation
(7.13). Theorem 7.4 is proved.
E x a m p l e 7.1
We consider a system of ordinary differential equations
ì y· + y 3 - y = y z 2 , y
= y0 ,
ï
t=0
(7.14)
í ·
ï z + z ( 1 + y 2 ) = 0 , z t = 0 = z0 .
î
It is obvious that for any initial condition ( y0 , z 0 ) problem (7.14) is uniquely
solvable over some interval ( 0 , t * ( y 0 , z 0 ) ) . If we multiply the first equation by
y and the second equation by z and if we sum the results obtained, then we get
that
Stability Properties of Attractor and Reduction Principle
d 2
1
--- ---( y + z2 ) + y4 - y2 + z2 = 0 ,
2 dt
t Î ( 0 , t * ( y0 , z0 ) ) .
This implies that the function V ( y , z ) = y 2 + z 2 possesses the property
d
---- V ( y ( t ) , z ( t ) ) + 2 V ( y ( t ) , z ( t ) ) £ 2 ,
dt
t Î [ 0 , t * ( y0 , z0 ) ) .
Therefore,
V ( y ( t ) , z ( t ) ) £ V ( y0 , z0 ) e -2 t + 1 ,
t Î [ 0 , t * ( y0 , z0 ) ) .
This implies that any solution to problem (7.14) can be extended to the whole
semiaxis R+ and the dynamical system ( R 2 , St ) generated by equation (7.14)
is dissipative. Obviously, the set M = { ( y , 0 ) : y Î R } is positively invariant.
Therewith the second equation in (7.14) implies that
d
1
--- ----- z 2 + z 2 £ 0 ,
2 dt
t > 0.
2
Hence, z ( t ) £ z 02 e -2 t . Thus, the set M exponentially attracts all the bounded sets in R 2 . Consequently, Theorem 7.4 gives us that the global attractor of
the dynamical system ( M , St ) is also the attractor of the system ( R 2 , St ) . But
on the set M system of equations (7.14) is reduced to the differential equation
y· + y 3 - y = 0 ,
y t = 0 = y0 .
(7.15)
Thus, the global attractors of the dynamical systems generated by equations
(7.14) and (7.15) coincide. Therewith the study of dynamics on the plane is reduced to the investigation of the properties of the one-dimensional dynamical
system.
Show that the global attractor A of the dynamical system
( R 2 , St ) generated by equations (7.14) has the form
E x e r c i s e 7.7
A = { ( y , z) : -1 £ y £ 1 , z = 0 } .
Figure the qualitative behaviour of the trajectories on the plane.
E x e r c i s e 7.8
Consider the system of ordinary differential equations
ì y· - y 5 + y 3 ( 1 + 2 z ) - y ( 1 + z 2 ) = 0
ï
í
ï z· + z ( 1 + 4 y 4 ) - 2 y 2 ( z 2 + y 4 - y 2 + 3 ¤ 2 ) = 0 .
î
(7.16)
Show that these equations generate a dissipative dynamical system
in R 2 . Verify that the set M = { ( y , z ) : z = y 2 , y Î R } is invariant
and exponentially attracting. Using Theorem 7.4, prove that the global attractor A of problem (7.16) has the form
A = { ( y , z ) : z = y 2 , -1 £ y £ 1 } .
Hint: Consider the variable w = z - y 2 instead of the variable z .
51
52
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
1
§8
Finite Dimensionality
of Invariant Sets
Finite dimensionality is an important property of the global attractor which can be
established in many situations interesting for applications. There are several approaches to the proof of this property. The simplest of them seems to be the one
based on Ladyzhenskaya’s theorem on the finite dimensionality of the invariant set.
However, it should be kept in mind that the estimates of dimension based on Ladyzhenskaya’s theorem usually turn out to be too overstated. Stronger estimates can
be obtained on the basis of the approaches developed in the books by A. V. Babin
and M. I. Vishik, and by R. Temam (see the references at the end of the chapter).
Let M be a compact set in a metric space X . Then its fractal dimension
is defined by
ln n ( M , e )
dimf M = lim --------------------------- ,
e ® 0 ln ( 1 ¤ e )
where n ( M , e ) is the minimal number of closed balls of the radius e which cover
the set M .
Let us illustrate this definition with the following examples.
E x a m p l e 8.1
Let M be a segment of the length l . It is evident that
l- + 1 .
l - - 1 £ n ( M , e ) £ ---------2e
2e
Therefore,
-2e
+ 2 el---------------- + ln l-------------£ ln n ( M , e ) £ ln --1
ln 1
e
e + ln 2 .
2
Hence, dimf M = 1 , i.e. the fractal dimension coincides with the value of the
standard geometric dimension.
E x a m p l e 8.2
Let M be the Cantor set obtained from the segment [ 0 , 1 ] by the sequentual
removal of the centre thirds. First we remove all the points between 1 ¤ 3 and
2 ¤ 3 . Then the centre thirds ( 1 ¤ 9 , 2 ¤ 9 ) and ( 7 ¤ 9 , 8 ¤ 9 ) of the two remaining
segments [ 0 , 1 ¤ 3 ] and [ 2 ¤ 3 , 1 ] are deleted. After that the centre parts
( 1 ¤ 27 , 2 ¤ 27 ) , ( 7 ¤ 27 , 8 ¤ 27 ) , ( 19 ¤ 27 , 20 ¤ 27 ) and ( 25 ¤ 27 , 26 ¤ 27 ) of the four
remaining segments [ 0 , 1 ¤ 9 ] , [ 2 ¤ 9 , 1 ¤ 3 ] , [ 2 ¤ 3 , 7 ¤ 9 ] and [ 8 ¤ 9 , 1 ] , respectively, are deleted. If we continue this process to infinity, we obtain the Cantor
set M . Let us calculate its fractal dimension. First of all it should be noted that
Finite Dimensionality of Invariant Sets
M=
¥
ÇJ
k,
k=0
J0 = [ 0 , 1 ] ,
J1 = [ 0 , 1 ¤ 3 ] È [ 2 ¤ 3 , 1 ] ,
J2 = [ 0 , 1 ¤ 9 ] È [ 2 ¤ 9 , 1 ¤ 3 ] È [ 2 ¤ 3 , 7 ¤ 9 ] È [ 8 ¤ 9 , 1 ]
and so on. Each set Jk can be considered as a union of 2 k segments of the
length 3 -k . In particular, the cardinality of the covering of the set M with the
segment of the length 3 -k equals to 2 k . Therefore,
ln 2 k
ln 2
= -------- .
dimf M = lim ---------------------k
ln 3
k ® ¥ ln ( 2 × 3 )
Thus, the fractal dimension of the Cantor set is not an integer (if a set possesses
this property, it is called fractal).
It should be noted that the fractal dimension is often referred to as the metric order
of a compact. This notion was first introduced by L. S. Pontryagin and L. G. Shnirelman in 1932. It can be shown that any compact set with the finite fractal dimension
is homeomorphic to a subset of the space R d when d > 0 is large enough.
To obtain the estimates of the fractal dimension the following simple assertion
is useful.
Lemma 8.1.
The following equality holds:
ln N ( M , e )
dimf M = lim --------------------------- ,
e ® 0 ln ( 1 ¤ e )
where N ( M , e ) is the cardinality of the minimal covering of the compact M with closed sets diameter of which does not exceed 2 e (the diameter of a set X is defined by the value d ( X ) = sup { x - y : x , y Î X } ).
Proof.
It is evident that N ( M , e ) £ n ( M , e ) . Since any set of the diameter d lies
in a ball of the radius d , we have that n ( M , 2 e ) £ N ( M , e ) . These two inequalities provide us with the assertion of the lemma.
All the sets are expected to be compact in Exercises 8.1–8.4 given below.
E x e r c i s e 8.1
Prove that if M1 Í M2 , then dimf M1 £ dimf M 2 .
E x e r c i s e 8.2
Verify that dimf ( M1 È M 2 ) £ max { dimf M1 ; dimf M2 } .
E x e r c i s e 8.3
Assume that M1 ´ M2 is a direct product of two sets. Then
dimf ( M1 ´ M2 ) £ dimf M1 + dimf M2 .
53
54
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
E x e r c i s e 8.4 Let g be a Lipschitzian mapping of one metric space into
another. Then dimf g ( M ) £ dimf M .
The notion of the dimension by Hausdorff is frequently used in the theory of dynamical systems along with the fractal dimension. This notion can be defined as follows.
Let M be a compact set in X . For positive d and e we introduce the value
m ( M , d , e ) = inf
å (r )
j
d
,
where the infimum is taken over all the coverings of the set M with the balls of the
radius r j £ e . It is evident that m ( M , d , e ) is a monotone function with respect
to e . Therefore, there exists
m ( M , d ) = lim m ( M , d , e ) = sup m ( M , d , e ) .
e > 0
e®0
The Hausdorff dimension of the set M is defined by the value
dimH M = inf { d : m ( M , d ) = 0 } .
E x e r c i s e 8.5 Show that the Hausdorff dimension does not exceed the fractal one.
E x e r c i s e 8.6 Show that the fractal dimension coincides with the Hausdorff
one in Example 8.1, the same is true for Example 8.2.
E x e r c i s e 8.7 Assume that M = { an }¥
Ì R , where an monotonically
n=1
tends to zero. Prove that dimH M = 0 (Hint: m ( M , d , e ) £
£ and +1 + n 2 -d n when an +1 £ e £ an ).
E x e r c i s e 8.8
Let M = { 1 ¤ n }n¥= 1 Ì R . Show that dimf M = 1 ¤ 2 .
1
1 - when --------------------------------æ Hint: n < n ( M , e ) < n + 1 + ------------------- £ e<
è
(n + 1) e
(n + 1) (n + 2)
1 -ö.
< -------------------n (n + 1) ø
¥
E x e r c i s e 8.9
ì 1 ü
-ý
Ì R . Prove that dimf M = 1 .
Let M = í -------î ln n þn = 2
E x e r c i s e 8.10 Find the fractal and Hausdorff dimensions of the global minimal attractor of the dynamical system in R generated by the differential equation
1- = 0 .
y· + y sin ----y
The facts presented in Exercises 8.7–8.9 show that the notions of the fractal and
Hausdorff dimensions do not coincide. The result of Exercise 8.5 enables us to restrict ourselves to the estimates of the fractal dimension when proving the finite dimensionality of a set.
Finite Dimensionality of Invariant Sets
The main assertion of this section is the following variant of Ladyzhenskaya’s
theorem. It will be used below in the proof of the finite dimensionality of global attractors of a number of infinite-dimensional systems generated by partial differential
equations.
Theorem 8.1.
Assume that M is a compact set in a Hilbert space H . Let V be a continuous mapping in H such that V ( M ) É M . Assume that there exists a finitedimensional projector P in the space H such that
P ( V v1 - V v2 ) £ l v1 - v2 ,
v1 , v2 Î M ,
( 1 - P ) ( V v1 - V v2 ) £ d v1 - v2 ,
(8.1)
v1 , v2 Î M ,
(8.2)
where d < 1 . We also assume that l ³ 1 - d . Then the compact M possesses
a finite fractal dimension and
-1
2
9l
dimf M £ dim P × ln ----------- × ln -----------.
1+ d
1- d
(8.3)
We remind that a projector in a space H is defined as a bounded operator P with the
property P 2 = P . A projector P is said to be finite-dimensional if the image PH is
a finite-dimensional subspace. The dimension of a projector P is defined as a number dim P º dim PH .
The following lemmata are used in the proof of Theorem 8.1.
Lemma 8.2.
d
Let BR be a ball of the radius R in R . Then
Rö d .
------N ( BR , e ) £ n ( BR , e ) £ æ 1 + 2
e ø
è
(8.4)
Proof.
Estimate (8.4) is self-evident when e ³ R . Assume that e < R . Let
{ x1 , ¼ , x l } be a maximal set in BR with the property xi - xj > e , i ¹ j .
By virtue of its maximality for every x Î BR there exists x i such that
x - x i £ e . Hence, n ( BR , e ) £ l . It is clear that
Be ¤ 2 ( xi ) Ì BR + e ¤ 2 ,
Be ¤ 2 ( xi ) Ç ( Be ¤ 2 ( x j ) ) = Æ ,
Here B r ( x ) is a ball of the radius r centred at x . Therefore,
l
l Vol ( Be ¤ 2 ) =
å Vol (B
e ¤ 2 ( xi ) )
£ Vol ( BR + e ¤ 2 ) .
i=1
This implies the assertion of the lemma.
E x e r c i s e 8.11 Show that
R d
n ( BR , e ) ³ æ ---ö ,
è eø
dimf B R = d .
i ¹ j.
55
56
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
Lemma 8.3.
Let F be a closed subset in H such that equations (8.1) and (8.2) hold
for all its elements. Then for any q > 0 and e > 0 the following estimate holds:
4 lö n
N ( V F , e ( q + d ) ) £ æ 1 + ----q ø N ( F, e ) ,
è
1
(8.5)
where n = dim P is the dimension of the projector P .
Proof.
Let { Fi } be a minimal covering of the set F with its closed subsets the diameter of which does not exceed 2 e . Equation (8.1) implies that in PH there
exist balls B i with radius 2 l e such that P V Fi Ì B i . By virtue of Lemma 8.1
N
there exists a covering { B i j }j =i 1 of the set P V Fi with the balls of the diameter
2 q e , where N i £ ( 1 + ( 4 l ¤ q ) ) n . Therefore, the collection
ì
ü
í G i j = B i j + ( 1 - P ) V Fi : i = 1 , 2 , ¼ , N ( F, e ) , j = 1 , 2 , ¼ , N i ý
î
þ
is a covering of the set V F . Here the sum of two sets A and B is defined by the
equality
A + B = {a + b : a Î A, b Î B} .
It is evident that
diam Gi j £ diam B i j + diam ( 1 - P ) V Fi .
Equation (8.2) implies that diam ( 1 - P ) V Fi £ 2 d e . Therefore, diam Gi j £
£ 2 ( q + d ) e . Hence, estimate (8.5) is valid. Lemma 8.3 is proved.
Let us return to the proof of Theorem 8.1. Since M Ì VM , Lemma 8.3 gives us
that
4 lö n
N ( M , e ( q + d ) ) £ N ( M , e ) × æ 1 + ----qø .
è
It follows that
4 lö n m
m
,
N ( M , ( q + d ) ) £ N ( M , 1 ) × æ 1 + ----qø
è
m = 1, 2, ¼ .
We choose q and m = m ( e ) such that
d + q < 1,
(d + q)
m
£ e,
where 0 < e < 1 . Then
4 lö
m
N ( M , e ) £ N ( M , ( d + q ) ) £ N ( M , 1 ) × æ 1 + ----qø
è
Consequently,
n m (e)
.
ln N ( M , e )
m (e)
4 lö
× lim ------------------- .
dimf M = lim ---------------------------- £ n × ln æ 1 + ----q
è
ø
(
¤
e
)
(1 ¤ e)
1
ln
ln
e®0
e®0
Finite Dimensionality of Invariant Sets
Obviously, the choice of m ( e ) can be made to fulfil the condition
ln e
m ( e ) £ ----------------------- + 1 .
(
ln q + d )
Thus,
4 lö
1 - -1 .
-----------dimf M £ n ln æ 1 + ----q ø ln q + d
è
By taking q = 1 ¤ 2 ( 1 - d ) we obtain estimate (8.3). Theorem 8.1 is proved.
E x e r c i s e 8.12 Assume that the hypotheses of Theorem 8.1 hold and
l < 1 - d . Prove that dimf M = 0 .
Of course, in the proof of Theorem 8.1 a principal role is played by equations (8.1)
and (8.2). Roughly speaking, they mean that the mapping V squeezes sets along the
space ( 1 - P ) H while it does not stretch them too much along PH . Negative invariance of M gives us that M Ì V k M for all k = 1 , 2 , ¼ . Therefore, the set M should
be initially squeezed. This property is expressed by the assertion of its finite dimensionality. As to positively invariant sets, their finite dimensionality is not guaranteed
by conditions (8.1) and (8.2). However, as the next theorem states, they are attracted to finite-dimensional compacts at an exponential velocity.
Theorem 8.2.
Let V be a continuous mapping defined on a compact set M in a HilHilbert space H such that VM Ì M . Assume that there exists a finite-dimensional projector P such that equations (8.1) and (8.2) hold with 0 < d < 1 ¤ 2
and l + d ³ 1 . Then for any q Î ( d , 1 ) there exists a positively invariant
closed set A q Ì M such that
sup { dist ( V k y , Aq ) : y Î M } £ q k ,
k = 1, 2, ¼
(8.6)
and
4l ö
4 lö ü
ì ln æ 1 + -----------ln æ 1 + ----ï è
qøï
è
q - dø
dimf Aq £ dim P × max í ---------------------------------- , --------------------------- ý ,
1
ï
ï
ln --1ln --------------------q
2 (q + d) þ
î
(8.7)
where q is an arbitrary number from the interval ( 0 , 1 ¤ 2 - d ) .
Proof.
The pair ( M , V k ) is a discrete dynamical system. Since M is compact, Theorem 5.1 gives us that there exists a global attractor M0 = Çk ³ 0 V kM with the properties V M0 = M0 and
h ( V k M , M0 ) º sup { dist ( V k y , M0 ) : y Î M } ® 0 .
(8.8)
57
58
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
We construct a set Aq as an extension of M0 . Let E j be a maximal set in V j M possessing the property dist ( a , b ) ³ q j for a , b Î Ej , a ¹ b . The existence of such
a set follows from the compactness of V j M . It is obvious that
1
--- q jö .
--- q jö £ N æ V j M , 1
Lj º Card Ej = N æ Ej , 1
è
è
3 ø
3 ø
Lemma 8.3 with F = M , q = q - d , and e = ( 1 ¤ 3 ) q j - 1 gives us that
4 l -ö n N æ V j - 1 M , 1
--- q jö £ æ 1 + ------------- q j - 1ö
N æè V j M , 1
è
è
ø
3 ø
3
q - dø
with q > d . Hereinafter n = dim P . Therefore,
4l nj
---ö ,
Lj º Card Ej £ æ 1 + ------------ö N æ M , 1
è
è
3ø
q - dø
q > d.
(8.9)
Let us prove that the set
A q = M0 È { È { V k Ej : j = 1 , 2 , ¼ , k = 0 , 1 , 2 , ¼ } }
(8.10)
possesses the properties required. It is evident that VAq Ì Aq . Since V k Ej Ì
Ì V k + j M , by virtue of (8.8) all the limit points of the set
È{V E :
k
j
j = 1, 2, ¼, k = 0, 1, 2, ¼}
lie in M0 . Thus, A q is a closed subset in M . The evident inequality
h ( V k M , Aq ) £ h ( V k M , Ek ) £ q k
(8.11)
implies (8.6). Here and below h ( X , Y ) = sup { dist ( x , Y ) : x Î X } . Let us prove
(8.7). It is clear that
Aq = V Aq È { È { E j : j = 1 , 2 , ¼ }} .
(8.12)
Let { Fi } be a minimal covering of the set Aq with the closed sets the diameter
of which is not greater than 2 e . By virtue of Lemma 8.3 there exists a covering { G i }
of the set VAq with closed subsets of the diameter 2 e ( q + d ) . The cardinality of this
covering can be estimated as follows
lön
N ( e , q , d ) º N ( VAq , e ( q + d ) ) £ æ 1 + 4 -qø N ( Aq , e ) .
è
(8.13)
Using the covering { G i } , we can construct a covering of the same cardinality of the
set VAq with the balls B ( xi , 2 e ( q + d ) ) of the radius 2 e ( q + d ) centered at the
points xi , i = 1 , 2 , ¼ , N ( e , q , d ) . We increase the radius of every ball up to the
value 2 e ( q + d + g ) . The parameter g > 0 will be chosen below. Thus, we consider
the covering
{ B ( xi , 2 e ( q + d + g ) ) ,
i = 1, 2, ¼, N (e, q, d) }
of the set VAq . It is evident that every point x Î VAq belongs to this covering together with the ball B ( x , 2 g e ) . If j ³ 2 , the inequalities
Finite Dimensionality of Invariant Sets
h ( Ej , VAq ) £ h ( V j M , VA q ) £ h ( V jM , V Ej -1 )
hold. By virtue of equation (8.11) with the help of (8.1) and (8.2) we have that
h ( V j M , VEj - 1 ) £ ( l + d ) h ( V j - 1 M , Ej - 1 ) £ ( l + d ) q j - 1 .
Therefore, h ( Ej , VA q ) £ 2 g e , provided 2 g e £ ( l + d ) q j - 1 , i.e. if
l + dln ---------2ge
j ³ j0 º 2 + ----------------- .
ln --1q
Here [ z ] is an integer part of the number z . Consequently,
ì
ü
VAq È í
Ej ý Ì
îj³j
þ
0
È
N (e , q , d)
È
i=1
B ( xi , 2 e ( q + d + g ) ) .
Therefore, equation (8.12) gives us that
j0 - 1
N ( Aq , e x ) £ N ( e , q , d ) +
å Card E ,
j
j=0
where x = 2 ( q + d + g ) . Using (8.9) and (8.13) we find that
---ö
N ( Aq , e x ) £ h N ( Aq , e ) + N æ M , 1
è
3ø
j0 - 1
4 l -ö
å æè1 + -----------q - dø
nj
j=1
4l n
for q > d . Here and further h = æ 1 + -----ö . Since
qø
è
--ln 1
e
j0 £ --------- + C ( l , d , g , q ) ,
1
ln --q
it is easy to find that
j0 - 1
4l - ö
å æè1 + -----------q -d ø
j=0
nj
a
--- ö
£ b × æè 1
eø
for
n
4l - ö ,
a = -------------------- × ln æè 1 + -----------q -d ø
ln ( 1 ¤ q )
where the constant b > 0 does not depend on e (its value is unessential further).
Therefore,
N ( Aq , e x ) £ h N ( Aq , e x ) + b e -a .
If we take e = x m - 1 , then after iterations we get
N ( Aq , x m ) £ h N ( Aq , x m - 1 ) + b x -( m - 1 ) a £
£
hm N ( A
q,
1)
+ b hm - 1
m-1
å (x
i=0
a
h)
-i
.
59
60
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
Let us fix d Î ( 0 , 1 ¤ 2 ) , q Î ( d , 1 ) and q Î ( 0 , 1 ¤ 2 - d ) and choose g > 0 such
that x = 2 ( q + d + g ) < 1 and x a h ¹ 1 . Then summarizing the geometric progression we obtain
1
x -a m - h m
- £
N ( Aq , x m ) £ h m N ( A q , 1 ) + b ----------------------------x -a - h
b
b
-ö + --------------------- x -a m .
£ h m æ N ( A q , 1 ) + ---------------------a
-a
è
ø
x -h
x -h
(8.14)
Let e > 0 be small enough and
ln ( 1 ¤ e )
m ( e ) = 1 + ------------------- ,
ln ( 1 ¤ x )
where, as mentioned above, [ z ] is an integer part of the number z . Since e £ x m( e ) ,
equation (8.14) gives us that
-----lö
N ( Aq , e ) £ N ( Aq , x m ) £ a1 æ 1 + 4
qø
è
n m (e)
1 ö m (e) ,
+ a2 æ -----è x aø
where a1 and a2 are positive numbers which do not depend on e . Therefore,
ln N ( A q , e )
dimf Aq = lim ----------------------------- £
e®0
--ln 1
e
nm
m (e)
1 ömü .
1- ì a ( 1 + 4
æ -----------e )
£ lim ------------- lim ---+
a
í
1
2
q
è aø ý
e ® 0 ln 1
--- m ® ¥ m î
x
þ
e
Simple calculations give us that
n
ì
-a ü
1
------eö , x ö ý .
dimf Aq £ --------- × ln ímax æè æè 1 + 4
qø
ø
1
î
þ
ln --x
This easily implies estimate (8.7). Thus, Theorem 8.2 is proved.
E x e r c i s e 8.13 Show that for d < q < 1 ¤ 2 formula (8.7) for the dimension
of the set A q can be rewritten in the form
4l
ln æ 1 + ------------ö
è
q - dø
dimf Aq = dim P × --------------------------------- .
1ln -----2q
(8.15)
If the hypotheses of Theorem 8.2 hold, then the discrete dynamical system ( M , V k )
possesses a finite-dimensional global attractor M0 . This attractor uniformly attracts
all the trajectories of the system. Unfortunately, the speed of its convergence to the
attractor cannot be estimated in general. This speed can appear to be small. However,
Theorem 8.2 implies that the global attractor is contained in a finite-dimensional po-
Existence and Properties of Attractors …
sitively invariant set possessing the property of uniform exponential attraction. From
the applied point of view the most interesting corollary of this fact is that the dynamics of a system becomes finite-dimensional exponentially fast independent of
the speed of convergence of the trajectories to the global attractor. Moreover, the reduction principle (see Theorem 7.4) is applicable in this case. Thus, finite-dimensional invariant exponentially attracting sets can be used to describe the qualitative
behaviour of infinite-dimensional systems. These sets are frequently referred to as
inertial sets , or fractal exponential attractors . In some cases they turn out
to be surfaces in the phase space. In contrast with the global attractor, the inertial
set of a dynamical system can not be uniquely determined. The construction in the
proof of Theorem 8.2 shows it.
§9
Existence and Properties of Attractors
of a Class of Infinite-Dimensional
Dissipative Systems
The considerations given in the previous sections are mainly of general character.
They are related to a dissipative dynamical system of the generic structure. Therewith, we inevitably make additional assumptions on the behaviour of trajectories of
these systems (e.g., the asymptotic compactness, the existence of a Lyapunov function, the squeezing property along a subspace, etc.). Thereby it is natural to ask
what properties of the original objects of a particular dynamical system guarantee
the fulfilment of the assumptions mentioned above. In this section we discuss this
question in terms of the dynamical system generated by a differential equation of
the form
d
---y + A y = B (y) ,
dt
y t = 0 = y0
(9.1)
in a separable Hilbert space H , where A is a linear operator and B is a nonlinear
mapping which is coordinated with A in some sense. Our main goal is to demonstrate the generic line of arguments as well as to describe those properties of the
operators A and B which provide the applicability of general theorems proved in
the previous sections. The main attention is paid to the questions of existence and finite dimensionality of a global attractor. Nowadays the presented line of arguments
(or a modification of it) is one of the main components of a great number of works
on global attractors.
It is assumed below that the following conditions are fulfilled.
61
62
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
(A) There exists a strongly continuous semigroup S t of continuous map-
pings in H such that y ( t ) = St y0 is a solution to problem (9.1) in the
sense that the following identity holds:
t
St y0 = Tt y0 + G ( t , y0 ) º Tt y0 +
òT
t -t
B ( St y0 ) dt ,
(9.2)
0
where Tt = exp ( - A t ) (see condition (B) below). The semigroup St is
dissipative, i.e. there exists R > 0 such that for any B from the collection B (H) of all bounded subsets of the space H the estimate St y <
< R holds when y Î B and t ³ t0 ( B ) . We also assume that the set
g +( B ) = Èt ³ 0 St B is bounded for any B ÎB (H ) .
(B) The linear closed operator A generates a semigroup Tt = exp ( - A t )
which admits the estimate Tt £ L1 exp ( w t ) ( L1 and w are some
constants). There exists a sequence of finite-dimensional projectors
{ Pn } which strongly converges to the identity operator such that
1) A commutes with Pn , i.e. Pn A Ì A Pn for any n ;
2) there exists n 0 such that Tt ( 1 - Pn ) £ L 2 exp ( - e t ) for n > n0 ,
where e , L 2 > 0 ;
3) rn = A -1 ( 1 - Pn ) ® 0 as n ® ¥ .
(C) For any R ¢ > 0 the nonlinear operator B ( u ) possesses the properties:
1) B ( u1 ) - B ( u2 ) £ C1 ( R ¢) u1 - u2 if ui £ R ¢, i = 1 , 2 ;
2) for n ³ n0 , u i £ R ¢, i = 1 , 2 , and for some s > 0 the following equations hold:
As ( 1 - Pn ) B ( u1 ) £ C2 ( R ¢) ,
As ( 1 - Pn ) ( B ( u1 ) - B ( u2 ) ) £ C 3 ( R ¢) u1 - u2
(the existence of the operator A s ( 1 - Pn ) follows from (B2)).
It should be noted that although conditions (A)–(C) seem a little too lengthy, they are
valid for a class of problems of the theory of nonlinear oscillations as well as for
a number of systems generated by parabolic partial differential equations.
The following assertion should be mainly interpreted as a principal result which
testifies to the fact that the asymptotic behaviour of the system is determined by
a finite set of parameters.
Theorem 9.1.
If conditions (A)–(C) are fulfilled, then the semigroup St possesses a
compact global attractor A . The attractor has a finite fractal dimension
which can be estimated as follows:
dimf A = a1 ( 1 + ln Pn L1 L2 ) ( 1 + D ( R ) e -1 ) dim Pn ,
(9.3)
Existence and Properties of Attractors …
where D ( R ) = w + L1 C1 ( R ) and n ³ n 0 is determined from the condition
rns £ a2 D ( R ) [ L1 L 2 C3 ( R ) ]
-1
× exp { -a3 D ( R ) ( 1 + ln L2 ) e -1 } .
(9.4)
Here a1 , a2 , and a3 are some absolute constants.
constants.
When proving the theorem, we mainly rely on decomposition (9.2) and the lemmata
below.
Lemma 9.1.
Let Kn be a set of elements which for some t ³ 0 have the form v = u 0 +
+ ( 1 - Pn ) G ( t , u ) , where u 0 Î Pn H , u0 £ c 0 R with the constant c0 determined by the condition Pn £ c 0 for n = 1 , 2 , ¼ . Here the value
G ( t , u ) is the same as in (9.2) with the element u Î H being such that
St u £ R for all t ³ 0 . Then the set Kn is precompact in H for
n ³ n0 .
Proof.
Properties (B2) and (C2) imply that
t
As ( 1 - Pn ) G ( t , u )
£ L2
ò
C2 ( R ) L2
e -e ( t - t ) As ( 1 - Pn ) B ( St u ) dt £ ---------------------e
0
when St u £ R for t ³ 0 . Therefore, the set
{ v : v = ( 1 - Pn ) G ( t , u ) , t > 0 } ,
(9.5)
where St u £ R for all t ³ 0 , is bounded in the space D ( As
with the norm As
. . The symbol
( 1 - Pn ) H )
denotes the restriction of an operator on
a subspace. However, property (B3) implies that
lim
m®¥
Pm A -1 ( 1 - Pn ) - A -1 ( 1 - Pn ) = 0 .
Therefore, the operator A-1 ( 1 - Pn )H is compact. Hence, D ( A s ( 1 - Pn ) H )
is compactly embedded into ( 1 - Pn ) H . It means that the set (9.5) is precompact in ( 1 - Pn ) H . This implies the precompactness of Kn .
Lemma 9.2.
There exists a compact set K in the space H such that
h ( St B , K ) º sup { dist ( St y , K ) : y Î B } £ L2 R e
- e ( t - t0 )
(9.6)
for any bounded set B Ì H and t ³ t0 º t0 ( B ) .
Proof.
Let u Î B , where B is a bounded set in H . Then St u £ R for t ³ t0 =
= t0 ( B ) . By virtue of (9.2) we have that
63
64
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
where
1
It is evident that Wn ( t , t0 , u ) Î Kn for t ³ t0 . Therefore,
St u = ( 1 - Pn ) Tt - t St u + Wn ( t , t0 , u ) ,
0 0
Wn ( t , t0 , u ) = Pn St u + ( 1 - Pn ) G ( t - t0 , St u ) .
0
dist ( St u , Kn ) £ ( 1 - Pn ) Tt - t St u £ L2 R e
0 0
-e ( t - t0 )
.
This implies (9.6) with K = [ K n ] , where [ K n ] is the closure in H of the set
K n described in Lemma 9.1.
Show that K = [ Kn ] lies in the set
E x e r c i s e 9.1
K n , s = { v1 + v2 : v1 Î Pn H , v2 Î ( 1 - Pn ) H ,
v1 £ C 1 ,
A s v2 £ C2 } ,
(9.7)
where C1 and C2 are some constants.
In particular, Lemma 9.2 means that the system ( H , St ) is asymptotically compact.
Therefore, we can use Theorem 5.1 (see also Exercise 5.3) to guarantee the existence of the global attractor A lying in K = [ Kn ] .
Let us use Theorem 8.1 to prove the finite dimensionality of the attractor. Verification of the hypotheses of the theorem is based on the following assertion.
Lemma 9.3.
Let St u i £ R ,
t ³ 0 , i = 1 , 2 . Then
St u 1 - St u2 £ L1 exp ( D ( R ) t ) u1 - u2
(9.8)
and
( 1 - Pn ) ( St u 1 - St u 2 )
L1 C3 ( R ) a tö
£ L 2 e -e t × æ 1 + C 0 r ns --------------------e
u - u2
è
ø 1
D(R)
(9.9)
for n ³ n0 and a = e + D ( R ) .
Proof.
Decomposition (9.2) and condition (C1) imply that
St u 1 - St u 2
æ
£ L1 ç u1 - u2 + C1 ( R )
ç
è
ö
e -w t St u 1 - St u2 dt÷ e w t .
÷
ø
0
t
ò
With the help of Gronwall’s lemma we obtain (9.8).
To prove (9.9) it should be kept in mind that decomposition (9.2) and equations (B2) and (C2) imply that for n ³ n0
Existence and Properties of Attractors …
æ
( 1 - Pn ) ( St u1 - St u 2 ) £ L2 e -e t ç u1 - u 2 +
ç
è
ö
e e t S t u 1 - S t u 2 dt ÷ .
÷
ø
0
t
+ C 0 r ns C 3 ( R )
ò
(9.10)
Here the inequality A -s ( 1 - Pn ) £ C0 r ns is used . If we put (9.8) in the righthand side of formula (9.10), we obtain estimate (9.9).
The following simple argument completes the proof of Theorem 9.1. Let us fix
an arbitrary number 0 < d < 1 and choose t0 and n such that
L2 e
-e t0
d
= --2
and
C 3( R ) a t
- e 0 £ 1.
C0 rns L1 -------------D(R)
Then the hypotheses of Theorem 8.1 with M = A , V = St , P = Pn , and l =
0
= L Pn exp ( D ( R ) t0 ) hold for the attractor A . Hence, it is finite-dimensional with
estimate (9.3) holding for its fractal dimension. Theorem 9.1 is proved.
E x e r c i s e 9.2 Prove that the global attractor A of problem (9.1) is stable
(Hint: verify that the hypotheses of Theorem 7.1 hold).
Properties (A)–(C) also enable us to prove that the system generated by equation
(9.1) possesses an inertial set. A compact set A exp in the phase space H is said to
be an inertial set (or a fractal exponential attractor) if it is positively invariant
( St A exp Ì Aexp ) , its fractal dimension is finite ( dimf Aexp < ¥ ) and it possesses
the property
h ( St B , A exp ) º sup { dist ( St y , A exp ) : y Î B } £ CB e
- g ( t - t0 )
(9.11)
for any bounded set B Ì H and for t ³ t0 ³ t0 ( B ) , where C B and g are positive
numbers. (The importance of this notion for the theory of infinite-dimensional dynamical systems has been discussed at the end of Section 8).
Lemma 9.4.
Assume that properties (A)–(C) hold. Then the dynamical system
( H , St ) generated by equation (9.1) possesses the following properties:
1) there exist a compact positively invariant set K and constants
C , g > 0 such that
sup { dist ( St y , K ) : y Î B } £ C e
- g ( t - tB )
for any bounded set B in H and for t ³ tB > 0 ;
(9.12)
65
66
C
h
a
p
t
e
r
1
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
2) there exist a vicinity O of the compact K and numbers D1 and
a 1 > 0 such that
St y1 - St y2 £ D1 e
a1 t
y1 - y2 ,
(9.13)
provided that for all t ³ 0 the semitrajectories St yi lie in the closure [O ] of the set O ;
3) there exist a sequence of finite-dimensional projectors { Pn } in the
space H , constants D2 , a2 , b > 0 , and a sequence of positive
numbers { rn } tending to zero as n ® ¥ such that
( 1 - Pn ) ( St y1 - St y2 ) £ D2 e -b t ( 1 + rn e
a2 t
) y1 - y2
(9.14)
for any y1 , y2 Î K .
Proof.
Let K be a compact set from Lemma 9.2. Let
K * = g+ ( K ) º
ÈS K.
t
t³ 0
It is clear that St K * Ì K * and equation (9.12) holds for K = K * with C = L2 R
and g = e . Let us prove that K * is a compact set. Let { zn } be a sequence of
elements of K * = g + ( K ) . Then z n = Stn yn for some tn > 0 and yn Î K .
If there exists an infinitely increasing subsequence { tnk } , then equation (9.6)
gives us that
lim dist æ Stn yn , K ö = 0 .
è k k
ø
k®¥
Therefore, the sequence { zn } possesses a limit point in K Ì K * . If { tn } is
a bounded sequence, then by virtue of the compactness of K there exist a number t0 > 0 , an element y Î K and a sequence { nk } such that yn ® y and
k
tn ® t . Therewith
k
St
nk
yn - St y £ St y - St y + St yn - St y .
n
nk
nk
k
k
0
0
The first term in the right-hand side of this inequality evidently tends to zero.
As for the second term, our argument is the same as in the proof of formula
(9.8). We use the boundedness of the set g + ( K ) (see property (A)) and properties (B) and (C2) to obtain the estimate
St y1 - St y2 £ C e
CK t
y1 - y2 ,
y1 , y2 Î K .
It follows that
lim St yn - St y = 0 .
nk
nk
k
k®¥
(9.15)
Existence and Properties of Attractors …
Therefore,
St
y
nk nk
® St y Î K * = g + ( K ) .
0
K*
The closedness of the set
can be established with the help of similar arguments. Thus, property (9.12) is proved for K = K * . Now we suppose that
K = St K * , where t0 is chosen such that y < R for all y Î K . It is obvious
0
that K is a compact positively invariant set. As it is proved above, it is easy to
find the estimate of form (9.15) for all y1 and y2 from an arbitrary bounded set
B . Here an important role is played by the boundedness of the set g + ( B ) (see
property (A)). Therefore, for any B Î B (H) there exists a constant C0 =
= C ( B , K *, t0 ) > 0 such that
St y1 - St y2 £ C 0 y1 - y2 ,
0
0
y1 , y2 Î B È K * .
Hence, for y Î B we have that
dist ( St y , K ) = dist ( St × St - t y , St K * ) £ C0 dist ( St - t y, K * )
0
0
0
0
for t > t0 . This implies estimate (9.12) with the constant C depending on K
and B . However, if we change the moment tB in equation (9.12), we can presume that, for example, C = 1 . Therewith g = e . Thus, the first assertion of the
lemma is proved.
Since the set K lies in the ball of dissipativity {z Î H : z £ R } , estimates
(9.13) and (9.14) follow from Lemma 9.3. Moreover,
D1 = L1 , a1 = D ( R ) ,
b = g = e,
D2 = L2 ,
a2 = e + D ( R ) ,
rn = C0 r ns C 3 ( R ) × D ( R )
-1
.
(9.16)
Thus, Lemma 9.4 is proved.
Lemma 9.4 along with the theorem given below enables us to verify the existence of
an inertial set for the dynamical system generated by equation (9.1).
Theorem 9.2.
Let the phase space H of a dynamical system ( H , St ) be a Hilbert
space. Assume that in H there exists a compact positively invariant set K
possessing properties (9.12)–
–(9.14).. Then for any n > ln 2 there exists an
inertial set Anexp of the dynamical system ( H , St ) such that
g + a1 ö
ì
ü
- ( t - tB ) ý
h ( St B , Anexp ) £ C ( B, n ) × exp í -g æ 1 - ------------------------è
ø
n
+
g
+
a
1
î
þ
(9.17)
for any bounded set B and t ³ tB . Here, as above, h ( X , Y ) = sup { dist ( x , Y ) :
x Î X } . Moreover,
dimf Anexp £ C0 × ( 1 + ln Pn ) dim Pn ,
(9.18)
67
68
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
where the number n is determined from the condition
1
rn £ ( 4 D 2
a2
------) b × exp
ì a2 ü
í -n ------ ý
bþ
î
(9.19)
and constant C 0 does not depend on n and n .
The proof of the theorem is based on the following preliminary assertions.
Lemma 9.5.
Let ( K , St ) be a dynamical system, its phase space being a compact in
a Hilbert space H . Assume that for all ( y1 , y2 ) Î K equations (9.13)
and (9.14) are valid. Then for any n > ln 2 there exists an inertial set
Anexp of the system ( K , St ) such that
h ( St K , Anexp ) = sup { dist ( St y , Anexp ) : y Î K } £ Cn e -n t .
Moreover, estimate (9.18) holds for the value dimf Anexp .
(9.20)
Proof.
--- e -n , where t0 and
We use Theorem 8.2 with M = K , V = St , and d = 1
2
0
n are chosen to fulfil
D2 e
-b t0
--- = 1
--- e -n
=d
2 4
and
rn e
a2 t0
£ 1.
--- e -n and
In this case conditions (8.1) and (8.2) are valid for V = St0 with d = 1
2
a 1 t0
. Therefore, there exists a bounded closed positively invariant
l = Pn D1 e
set A q with d < q < 1 ¤ 2 such that (see (8.6) and (8.15))
sup { dist ( V m y , Aq ) : y Î K } £ q m ,
m = 1, 2, ¼
(9.21)
and
4l
1
dimf A q £ ln æ1 + ------------ö ln -----è q - dø 2 q
-1
× dim Pn
(9.22)
Assume that q = 2 d = e -n and consider the set
Anexp =
È {S A
t
q:
0 £ 0 £ t0 } .
Here n = ln --1- > ln 2 . It is easy to see that
q
dimf Anexp £ 1 + dimf Anexp .
Therefore, equations (9.20) and (9.18) follow from (9.21) and (9.20) after some
simple calculations.
Lemma 9.6.
Assume that in the phase space H of a dynamical system ( H , St )
there exist compact sets K and K0 such that (a) K0 Ì K ; (b) properties
Existence and Properties of Attractors …
(9.12) and (9.13) are valid for K ; and (c) the set K0 possesses the property
h ( St K , K0 ) £ C e
-g0 t
(9.23)
,
where h ( X , Y ) = sup { dist ( x , Y ) : x Î X } . Then for any bounded set B Ì H
and t ³ tB the following inequality holds
g g0
ì
ü
- tý .
h ( St B , K0 ) £ CB exp í - --------------------------g
+
g
+
a
0
1 þ
î
(9.24)
Proof.
By virtue of (9.12) every bounded set B reaches the vicinity O in finite
time and stays in it. Therefore, it is sufficient to prove the lemma for a set
B Î B (H) such that St B Ì [O ] for t ³ 0 , where [O ] denotes the closure of O .
Let k0 Î K 0 and y Î B . Evidently,
St y - k0 £ Sš t S( 1 - š ) t y - Sš t k + Sš t k - k0
for any 0 £ š £ 1 and k Î K . With the help of (9.13) we have that
St y - k0 £ D1 e
a1 š t
S( 1 - š ) t y - k + Sš t k - k0 .
Therefore, for any 0 < š < 1 and k Î K we have that
dist ( St y , K 0 ) £ D1 e
£ D1 e
a1 š t
a1 š t
S( 1 - š ) t y - k + dist ( Sš t k , K0 ) £
S( 1 - š ) t y - k + h ( Sš t K , K0 ) .
If we take an infimum over k Î K and a supremum over y Î B , we find that
h ( St B , K 0 ) £ D1 e
a1 š t
h ( S( 1 - š ) t B , K ) + h ( Sš t K , K 0 )
for all 0 < š < 1 . Hence, equations (9.12) and (9.23) give us that
h ( St B , K0 ) £ CB e
( a1 š - g ( 1 - š ) ) t
for t ³ tB . If we choose š = g ( g + g0 + a1 )
is proved.
-1
+ CK e
-š g0 t
, we obtain (9.24). Lemma 9.6
If we now use Lemma 9.6 with K 0 = Anexp and estimate (9.20), we get equation
(9.17). This completes the proof of Theorem 9.2.
Thus, by virtue of Lemma 9.4 and Theorem 9.2 the dynamical system (H , St ) generated by equation (9.1) possesses an inertial set Anexp for which equations (9.17)–
(9.19) hold with relations (9.16).
It should be noted that a slightly different approach to the construction of inertial sets is developed in the book by A. Eden, C. Foias, B. Nicolaenko, and R. Temam
(see the list of references). This book contains further developments and applications of the theory of inertial sets.
69
70
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
To conclude this section, we outline the results on the behaviour of the projection onto the finite-dimensional subspace Pn H of the trajectories of the system
(H , St ) generated by equation (9.1).
Assume that an element y0 belongs to the global attractor A of a dynamical
system (H , St ) . Lemma 6.1 implies that there exists a trajectory g = { y ( t ) , t Î R }
lying in A wholly such that y ( 0 ) = y0 . Therewith the following assertion is valid.
1
Lemma 9.7.
Assume that properties (A)–(C) are fulfilled and let y0 Î A . Then the
following equation holds:
0
( 1 - Pn ) y0 =
ò (1- P ) T
n
-t
B ( y ( t ) ) dt ,
n ³ n0 ,
(9.25)
-¥
where { y ( t ) } is a trajectory passing through y0 , the number n0 can be
found from (B2) and the integral in (9.25) converges in the norm of the
space H .
Proof.
Since y0 = St y ( - t ) , equation (9.2) gives us that
0
æ
ö
ç
( 1 - Pn ) y0 = ( 1 - Pn ) Tt y ( -t ) + ( 1 - Pn ) T- t B ( y ( t ) ) dt÷ .
ç
÷
è
ø
-t
ò
(9.26)
A trajectory in the attractor possesses the property y ( t ) £ R , t Î ( - ¥ , ¥ ) .
Therefore, property (B2) implies that
( 1 - Pn ) Tt y ( -t ) £ L 2 × R e - e t
and
( 1 - Pn ) T-t B ( y ( t ) ) £ L2 CR e -e t .
These estimates enable us to pass to the limit in (9.26) as t ® - ¥ . Thereupon
we obtain (9.25).
The following assertion is valid under the hypotheses of Theorem 9.1.
Theorem 9.3.
There exists N0 ³ n 0 such that for all N ³ N0 the following assertions
are valid:
1) for any two trajectories y1 ( t ) and y2 ( t ) lying in the attractor of the
system generated by equation (9.1) the equality PN y1 ( t ) = PN y2 ( t )
for all t Î R implies that y1 ( t ) º y2 ( t ) ;
2) for any two solutions u 1 ( t ) and u 2 ( t ) of the system (9.1) the equation
lim PN ( u1 ( t ) - u2 ( t ) ) = 0
t ® -¥
implies that u1 ( t ) - u2 ( t ) ® 0 as t ® ¥ .
(9.27)
Existence and Properties of Attractors …
We can also obtain an upper estimate of the number N 0 from the inequali-1
s
ty rN
£ a0 e ( L2 C3 ( R ) ) .
0
Proof.
Equation (9.25) implies that for any trajectory yi ( t ) lying in the attractor
of system (9.1) the equation
t
( 1 - PN ) yi ( t ) =
ò (1- P ) T
N
t -t
B ( y i ( t ) ) dt ,
i = 1, 2 ,
-¥
holds. Therefore, if PN y1 ( t ) = PN y2 ( t ) , then properties (B2), (B3), and (C2) give us
that
t
s
C0 rN L 2 C3 ( R )
y1 ( t ) - y2 ( t ) £
òe
-e ( t - t )
y1 ( t ) - y2 ( t ) dt .
-¥
It follows that the estimate
t0
y1 ( t ) - y2 ( t ) £ AN exp { - e t + AN ( t - t0 ) }
s
òe
et
y1 ( t ) - y2 ( t ) dt
-¥
holds for t ³ t0 , where AN = C0 r N L2 C 3 ( R ) . If we tend t0 ® - ¥ , we obtain the
first assertion, provided A N < e .
Now let us prove the second assertion of the theorem. Let
a N ( t ) = PN ( u1 ( t ) - u2 ( t ) ) .
Then
u1 ( t ) - u2 ( t )
£ a N ( t ) + ( 1 - PN ) ( u1 ( t ) - u2 ( t ) ) .
Therefore, equation (9.10) for the function y ( t ) = u1 - u2 exp ( e t ) gives us that
t
y ( t ) £ aN ( t )
ee t
+ L 2 u1 ( 0 ) - u 2 ( 0 ) + A N
ò y ( t ) dt .
0
This and Gronwall’s lemma imply that
u1 ( t ) - u2 ( t )
£ aN ( t ) + L2 exp ( -( e - A N ) t ) u1 ( 0 ) - u2 ( 0 ) +
t
+ AN
òa
N ( t ) exp ( - ( e - A N ) ( t - t ) ) dt .
0
Therefore, if AN < e , then equation (9.27) gives us that u1 ( t ) - u2 ( t ) ® 0 . Thus,
the second assertion of Theorem 9.3 is proved.
71
72
Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
C
h
a
p
t
e
r
Theorem 9.3 can be presented in another form. Let { ek : k = 1 , ¼ , dN } be a basis
in the space PN H . Let us define linear functionals lj ( u ) = ( u , e j ) on H , j =
= 1 , ¼ , dN . Theorem 9.3 implies that the asymptotic behaviour of trajectories of
the system (H , St ) is uniquely determined by its values on the functionals lj .
Therefore, it is natural that the family of functionals { lj } is said to be the determining collection. At present some general approaches have been worked out which
enable us to define whether a particular set of functionals is determining. Chapter 5
is devoted to the exposition of these approaches. It should be noted that for the first
time Theorem 9.3 was proved for the two-dimensional Navier-Stokes system by
C. Foias and D. Prodi (the second assertion) and by O. A. Ladyzhenskaya (the first
assertion).
Concluding the chapter, we would like to note that the list of references given
below does not claim to be full. It contains only references to some monographs and
reviews devoted to the developments of the questions touched on here and comprising intensive bibliography.
1
References
1. BABIN A. V., VISHIK M. I. Attractors of evolution equations. –Amsterdam:
North-Holland, 1992.
2. TEMAM R. Infinite-dimensional dynamical systems in Mechanics and
Physics. –New York: Springer, 1988.
3. HALE J. K. Asymptotic behavior of dissipative systems. –Providence: Amer.
Math. Soc., 1988.
4. EDEN A., FOIAS C., NICOLAENKO B., TEMAM R. Exponential attractors for dissipative evolution equations. –New York: Willey, 1994.
5. LADYZHENSKAYA O. A. On the determination of minimal global attractors for the
Navier-Stokes equations and other partial differential equations // Russian
Math. Surveys. –1987. –Vol. 42. –# 6. –P. 27–73.
6. KAPITANSKY L. V., KOSTIN I. N. Attractors of nonlinear evolution equations
and their approximations // Leningrad Math. J. –1991. –Vol. 2. –P. 97–117.
7. CHUESHOV I. D. Global attractors for nonlinear problems of mathematical
physics // Russian Math. Surveys. –1993. –Vol. 48. –# 3. –P. 133–161.
8. PLISS V. A. Integral sets of periodic systems of differential equations.
–Moskow: Nauka, 1977 (in Russian).
9. HALE J. K. Theory of functional differential equations. –Berlin; Heidelberg;
New York: Springer, 1977.