Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems

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Autonomous dynamical systems 7

b. ϕ(t + τ, u, y) = ϕ(τ, ϕ(t, u, y), σ(t, y)) (t, τ ∈ T

, u ∈ W, y ∈ Y ),

is called

[

6, 290, 291

]

a cocycle on (Y, T

, σ) with ﬁber W.

Deﬁnition 1.20 Let X := W × Y and we deﬁne a mapping π : X × T

→ X

as following: π((u, y), t) := (ϕ(t, u, y), σ(t, y)) (i.e. π = (ϕ, σ)). Then it easy

to see that (X, T

, π) is a dynamical system on X which is called a skew-product

dynamical system

[

2, 292

]

and h = pr

: X → Y is a homomorphism from (X, T

, π)

on (Y, T

, σ) and, consequently, h(X, T

, π), (Y, T

, σ), hi is a non-autonomous

dynamical system.

Thus, if we have a cocycle hW, ϕ, (Y, T

, σ)i on dynamical system (Y, T

, σ) with

the ﬁber W , then it generates a non-autonomous dynamical system h(X, T

, π),

(Y, T

, σ), hi (X := W × Y ), which is called a non-autonomous dynamical system,

generated by cocycle hW, ϕ, (Y, T

, σ)i on (Y, T

, σ).

Non-autonomous dynamical systems (cocycles) play a very important role in the

study of non-autonomous evolutionary diﬀerential equations. Under appropriate

assumptions every non-autonomous diﬀerential equation generates some cocycle

(non-autonomous dynamical system). Below we give an example of this type.

Example 1.5 Let E

be an n-dimensional real or complex Euclidean space. Let

us consider a diﬀerential equation

= f(t, u), (1.5)

where f ∈ C(R × E

, E

). Along with equation (1.5) we consider its H-class

[

32, 137, 238, 300, 302

]

, i.e., the family of equations

= g(t, v), (1.6)

where g ∈ H(f) =

: τ ∈ R}, f

(t, u) = f(t + τ, u) for all (t, u) ∈ R ×E

and by

bar we denote the closure in C(R×E

, E

). We will suppose also that the function

f is regular, i.e. for every equation (1.6) the conditions of existence, uniqueness and

extendability on R

are fulﬁlled. Denote by ϕ(·, v, g) the solution of equation (1.6),

passing through the point v ∈ E

at the initial moment t = 0. Then it is correctly

deﬁned a mapping ϕ : R

× E

× H(f) → E

, verifying the following conditions

(see, for example,

[

32, 290, 291

]

1) ϕ(0, v, g) = v for all v ∈ E

and g ∈ H(f );

2) ϕ(t, ϕ(τ, v, g), g

) = ϕ(t + τ, v, g) for every v ∈ E

, g ∈ H(f) and t, τ ∈ R

;

3) the mapping ϕ : R

×E

×H(f) → E

is continuous.

Denote by Y := H(f) and (Y, R

, σ) a dynamical system of translations

(semigroup system) on Y , induced by dynamical system of translations (C(R ×

, E

), R, σ). The triplet hE

, ϕ, (Y, R

, σ)i is a cocycle on (Y, R

, σ) with the

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

8 Global Attractors of Non-autonomous Dissipative Dynamical Systems

ﬁber E

. Thus the equation (1.5) generates a cocycle hE

, ϕ, (Y, R

, σ)i and a

non-autonomous dynamical system h(X, R

, π), (Y, R

, σ), hi, where X := E

×Y ,

π := (ϕ, σ) and h := pr

: X → Y .

1.2 Limit properties of dynamical systems

Let M ⊆ X and ω(M) be ω–limit set of M. Directly from the deﬁnition of ω(M)

follows

Lemma 1.2 The following statements take place:

(1) the point y ∈ ω(M) if and only if there exist sequences {x

} ⊆ M and {t

} ⊆ T

such that t

→ +∞ and y = lim

n→+∞

;

(2) the set ω(M) is closed and positively invariant;

(3) if A ⊆ B then ω(A) ⊆ ω(B);

(4) ω(A ∪ B) ⊆ ω(A) ∪ ω(B) for any pair of subsets A and B from X;

(5) if set A is positively invariant (negatively invariant, invariant) then ω(A) ⊆

(

A ⊆ ω(A), ω(A) = A);

(6)

∪{ω

| x ∈ M} ⊆ ω(M);

(7) if Σ

(M) is relatively compact, then ω(M) is invariant.

Lemma 1.3 Let B ⊆ X then the following conditions are equivalent:

(1) for every {x

} ⊆ B and t

→ +∞ sequence {x

} is relatively compact;

(2)(a) ω(B) is not empty and is compact;

(b) ω(B) is invariant and the equality below takes place

lim

t→+∞

sup

x∈B

ρ(xt, ω(B)) = 0; (1.7)

(3) there exists a non empty compact K ⊆ X such that

lim

t→+∞

sup

x∈B

ρ(xt, K) = 0.

Proof. Let us show that from 1. follows 2. Let {x

} ⊆ B and t

→ +∞,

then according to 1. the sequence {x

} can be considered convergent. Assume

x = lim

k→+∞

, then x ∈ ω(B) and consequently ω(B) 6= ∅. Let us show ω(B) is

compact. Let ε

↓ 0 and {y

} ⊆ ω(B), then there exist x

∈ B and t

≥ k such

that

ρ(x

, y

) < ε

According to the condition (1), the sequence {x

} is relatively compact and since

↓ 0, {y

} also is relatively compact. From the deﬁnition of ω(B) follows its

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

Autonomous dynamical systems 9

positive invariance and consequently for ω(B) to be invariant it is suﬃcient to

show that it is negatively invariant. Let y ∈ ω(B) and t ∈ T, then there exist

} ⊆ B and t

→ +∞ such that y = lim

k→+∞

= lim

k→+∞

− t + t) =

lim

k→+∞

−t)). As t

−t → +∞ then according to condition (1), the sequence

− t)} can be considered convergent. Assume y

= lim

k→+∞

− t), then

y = π

and y

∈ ω(B), i.e. y ∈ π

ω(B). Like so the invariance of ω(B) is proved.

Now let us show that the equality (1.7) takes place. If we suppose that (1.7) does

not take place, then there will exist ε

> 0, t

→ +∞ and x

∈ B such that

ρ(x

, ω(B)) ≥ ε

. (1.8)

According to the condition (2), the sequence {x

} can be considered convergent.

Let y = lim

k→+∞

, then y ∈ ω(B). On the other hand, passing to the limit in (1.8)

when k → +∞ we will obtain y /∈ ω(B). This contradiction completes the proof of

the implication condition (1) ⇒ condition (2).

It is evident that (2) ⇒ (3) and (3) ⇒ (1). The lemma is proved. 

Corollary 1.2 Let M ⊆ X be nonempty and Σ

(M) be relatively compact, then

ω(M) 6= ∅, is compact, invariant and

lim

t→+∞

sup

x∈M

ρ(xt, ω(M)) = 0. (1.9)

The statement inverse to the corollary 1.2 takes place, but before formulating it

we will review the notion of the measure of non-compactness.

Deﬁnition 1.21 The mapping λ : B(X) → R

, satisfying the following condi-

tions:

(1) λ(A) = 0 if and only if A ∈ B(X) is relatively compact;

(2) λ(A ∪ B) = max(λ(A), λ(B)) for every A, B ∈ B(X).

is called

[

175, 179, 278

]

a measure of non-compactness on X.

Deﬁnition 1.22 The measure of non-compactness of Kuratowsky λ : B(X) →

is deﬁned by the equality λ(A) := inf{ε > 0 | A admits ﬁnite ε-covering }.

Lemma 1.4 Let M ⊆ X be nonempty and relatively compact. If set ω(M) is

nonempty, compact and the equality (1.9) tales place, then Σ

(M) is relatively

compact.

Proof. Let ε > 0, then from the equality (1.9) follows the existence of a positive

number L(ε) such that

[

t≥L(ε)

M ⊆ B(ω(M), ε). (1.10)

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10 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Denote by λ(K) the measure of non-compactness of Kuratowsky of the set K. From

the inclusion (1.10) follows that

λ(Σ

(M)) = λ(π(M, [0, L(ε)]) ∪M

) =

max(λ(π(M, [0, L

]), λ(M

)) = λ(M

) ≤ 2ε.

From this we obtain the equality λ(Σ

(M)) = 0. The lemma is proved. 

Theorem 1.5 Let M ⊆ X be nonempty and relatively compact. For the set

(M) to be compact it is necessary and suﬃcient the fulﬁllment of the following

3 conditions:

(1) ω(M) 6= ∅;

(2) ω(M) is compact;

(3) the equality (1.9) takes place.

The formulated statement follows from the Corollary 1.2 and Lemma 1.4.

1.3 Center of Levinson

Let M be some family of subsets from X.

Deﬁnition 1.23 Dynamical system (X, T, π) will be called M-dissipative if for

every ε > 0 and M ∈ M there exists L(ε, M) > 0 such that π

M ⊆ B(K, ε) for any

t ≥ L(ε, M), where K is a certain ﬁxed subset from X depending only on M. In

this case K we will call the attractor for M.

For the applications the most important ones are the cases when K is bounded or

compact and M = {{x} | x ∈ X} or M = C(X), or M = {B(x, δ

) | x ∈ X, δ

> 0},

or M = B(X).

Deﬁnition 1.24 The system (X, T, π) is called:

− point dissipative if there exist K ⊆ X such that for every x ∈ X

lim

t→+∞

ρ(xt, K) = 0; (1.11)

− compact dissipative if the equality (1.11) takes place uniformly w.r.t. x on the

compacts from X;

− locally dissipative if for any point p ∈ X there exist δ

> 0 such that the

equality (1.11) takes place uniformly w.r.t. x ∈ B(p, δ

);

− bounded dissipative if the equality (1.11) takes place uniformly w.r.t. x on

every bounded subset from X.

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Autonomous dynamical systems 11

As we have already mentioned above, for applications our main interest rests on

cases in which K is compact and bounded. According to this the system (X, T, π)

is called point k (b)-dissipative if (X, T, π) is point dissipative and the set K from

(1.11) is compact (bounded). Also like that is deﬁned the notion of the compact k

(b)-dissipative system and of other types of k (b)-dissipativity.

From the deﬁnitions above follows that bounded k (b)-dissipativity implies local

k (b)-dissipativity (see Lemma 1.5), local k (b)-dissipativity implies compact k (b)-

dissipativity, and compact k (b)-dissipativity implies point k(b)-dissipativity.

As it will be shown below in general case introduced by us classes are diﬀerent.

We will take interest in the following tasks:

(1) What connection does exist between diﬀerent types of dissipativity?

(2) Conditions under which a dissipative system admits maximal compact invariant

set.

(3) Structure of the center of Levinson (maximal compact invariant attractor) for

diﬀerent classes of dynamical systems.

(4) Conditions of asymptotic and uniform asymptotic stability of the center of

Levinson.

Lemma 1.5 Let (X, T, π) be local k (b)-dissipative, then it is compact k (b)-

dissipative.

Proof. Let (X, T, π) be local k (b)-dissipative. Then there exist a nonempty

compact (bounded) set K ⊆ X, such that for any ε > 0 and x ∈ X there exist

δ(x) > 0 and l = l(ε, x) > 0 for which

ρ(yt, K) < ε (1.12)

for every t ≥ l and y ∈ B(x, δ(x)).

Let M is a nonempty compact from X. Then for every x ∈ M there exist δ =

δ(x) > 0 and l = l(ε, x) > 0 such that the inequality (1.12) takes place. Consider

the open covering {B(x, δ(x)) | x ∈ M} of the set M . By virtue of the compactness

of M and completeness of the space X, from the constructed covering of the set M

we can extract {B(x

, δ(x

)) | i ∈

1, m}. Assume L(ε, M) = max{l(ε, x

) | i ∈ 1, m},

then from (1.12) follows that ρ(xt, K) < ε for all x ∈ M and t ≥ L(ε, M). The

lemma is proved. 

Lemma 1.6 Let M ⊆ X and the set Σ

(M) be relatively compact. If ω(M) ⊆ M,

then

ω(M) = ∩{π

M | t ∈ T}. (1.13)

Proof. Denote by I(M) := ∩{π

M|t ∈ T}. It is clear that I(M ) ⊆ ω(M). Let us

show that the inverse inclusion also takes place if ω(M) ⊆ M . In fact, according to

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12 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Lemma 1.3 and Corollary 1.2 the set ω(M) is invariant and, consequently, ω(M) =

ω(M) ⊆ π

M for every t ∈ T. From what follows the inclusion ω(M) ⊆ I(M).

The lemma is proved. 

Remark 1.4 Below we will mainly consider k-dissipative dynamical systems.

That’s why the sign k- we will drop out everywhere where it won’t lead to mis-

understanding.

Let (X, T, π) be compact dissipative and K is a nonempty compact set that is

an attractor for compact subsets X. Then for every compact M ⊆ X the equality

lim

t→+∞

sup

x∈M

ρ(xt, K) = 0

holds. Hence ω(K) ⊆ K and, consequently,

J := ω(K) = ∩{π

K | t ∈ T}. (1.14)

Let us show that the set J does not depend on the choice of the set K attracting

all compact subsets of the space X. In fact, if we denote by J(K) := ω(K) and

every other compact set attracting all compacts from X, then there will be

L = L(K, K

, ε) > 0 such that π

J(K) ⊆ K

and π

J(K

) ⊆ K for all t ≥ L. As

J(K) = ω(K) ⊆ K and J(K

) = ω(K

) ⊆ K

then from the invariance of J(K

)

and J(K) follows that J(K) ⊆ K

, J(K

) ⊆ K, J(K) ⊆ π

and J(K

) ⊆ π

for all t ∈ T and consequently J(K) = J(K

So, the set J deﬁned by the equality (1.14) does not depend on the choice of

the attractor K but is deﬁned only by inner properties of the dynamical system

(X, T, π).

Deﬁnition 1.25 The set J deﬁned by the equality (1.14), according to

[

332

]

we will call the center of Levinson of the compact dissipative dynamical system

(X, T, π).

Theorem 1.6

[

102, 175, 232

]

Let (X, T, π) be compact dissipative system and let

J be its center of Levinson. Then:

(1) J is compact invariant set;

(2) J is orbitally stable;

(3) J is the attractor of the family of all compacts of X;

(4) J is the maximal compact invariant set in (X, T, π).

Proof. The ﬁrst statement of the Theorem directly follows from the deﬁnition of J

and from Lemma 1.3. Let us show that the set J is orbitally stable. If we suppose

the contrary we will obtain the existence of ε

> 0, δ

→ 0 (δ

> 0), x

∈ B(J, δ

)

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Autonomous dynamical systems 13

and t

→ +∞ such that

ρ(x

, J) ≥ ε

. (1.15)

As x

∈ B(J, δ

), δ

→ 0 and J is compact, without loss of generality the sequence

} can be considered convergent. By virtue of the compact dissipativity of the

system (X, T, π) the set Σ

({x

}) is relatively compact. Note that together with the

set K the set K

= K ∪H

({x

}) is the attractor for the family of all compacts from

X and consequently ω(K

) = ω(K) = J. In particular, ω(H

({x

})) ⊆ ω(K) = J.

By compactness of H

({x

}) the sequence {x

} can be considered convergent.

Let p = lim

n→+∞

, then p ∈ ω(H

({x

})). On the other hand from (1.15) follows

that p /∈ J. The obtained contradiction proves the second statement of the theorem.

Now let M ∈ C(X). Then by virtue of the compact dissipativity of (X, T, π) the

set Σ

(M) is relatively compact and according to Theorem 1.5 the conditions (1)-

(2) are fulﬁlled. In particular, for every ε > 0 there exists L(ε) > 0 such that π

M ⊆

B(ω(M), ε) for all t ≥ L(ε). Together with the set K the set K

= K ∪ω(M) also is

the attractor of the compact subsets from X and consequently ω(K

) = ω(K) = J.

So, ω(M) ⊆ ω(K

) = J, and hence

β(π

M, J) ≤ β(π

M, ω(M)) < ε,

i.e. lim

t→+∞

β(π

M, J) = 0 for all M ⊆ C(X) where β(A, B) is semi-deviation of the

set A from the set B.

At last let us prove the 4th statement of the theorem. Let J

be compact

invariant subset from X. Then according to the 3rd statement of the theorem we

have

lim

t→+∞

β(π

, J) = 0. (1.16)

By virtue of the invariance of J

the equality π

= J

holds for all t ∈ T. From

this and from the equality (1.16) we obtain J

⊆ J. The theorem is completely

proved. 

Denote by {K

|λ ∈ Λ} the family of all nonempty compact positive invariant

sets that attract all compacts from X.

The following theorem holds.

Theorem 1.7 Let (X, T, π) be compact dissipative dynamical system and let J be

its Levinson center.

J = J

:= ∩{K

|λ ∈ Λ},

i.e. J is the least compact positive invariant set attracting all compacts from X.

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14 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. Assume

K := ∩{K

|λ ∈ Λ}.

First of all note that J ⊆ K and consequently K 6= ∅. In fact, for every λ ∈ Λ we

have J = ω(K

) ⊆ K

, i.e. J ⊆ K.

Let us show now that the reverse inclusion also holds. As J attracts all com-

pacts from X and is nonempty and positive invariant, then J ∈ {K

|λ ∈ Λ} and

consequently K ⊆ J. The theorem is proved. 

Lemma 1.7 Let (X, T, π) be compact dissipative dynamical system, J be its Levin-

son’s center and K be a nonempty compact attracting all compacts from X, then

J =

t∈T

Proof. As K is the attractor of all compacts from X, then ω(K) ⊆ K and according

to Lemma 1.6

J := ω(K) =

The lemma is proved. 

Lemma 1.8 Let M be compact positive invariant asymptotically stable set, then

the following statements hold:

(1) the domain of attraction W

(M) of the set M is open;

(2) the equality

lim

t→+∞

β(π

K, M) = 0 (1.17)

takes place for every compact K from W

(M).

Proof. Show that W

(M) is open. Really, as M is attracting set, then there

exists δ > 0 such that B(M, δ) ⊂ W

(M). Let us show that for every point p ∈

(M)\B(M, δ) there exists η > 0 such that B(p, η) ⊂ W

(M). As p ∈ W

(M),

there will be t

> 0 such that pt

∈ B(M, δ). By openness of B(M, δ) there is γ > 0

for which B(pt

, γ) ⊂ B(M, δ). According to continuity of the mapping π(t

, ·) :

X −→ X there exists η > 0 such that the inclusion π

B(p, η) ⊂ B(M, δ) ⊂ W

(M)

holds and consequently the set W

(M) is open.

Let us prove the second statement of the lemma. Let ε > 0 and K be a compact

from W

(M). For the number ε > 0 we choose δ(ε) > 0 taking into account the

condition of the stability of M. As M attracts points from W

(M), for every point

x ∈ K there are γ(x, ε) > 0 and l(x, ε) > 0 such that

B(x, γ(x, ε)) ⊆ B(M, ε) (1.18)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

Autonomous dynamical systems 15

for all t ≥ l(x, ε). By the compactness of K from the open covering {B(x, γ(x, ε)) |

x ∈ K} ⊇ K we can extract sub-covering ∪{B(x

, γ(x, ε)) | i =

1, n} ⊇ K. Assume

L(M, ε) := max{l(x

, ε) | i =

1, n}, then π

M ⊆ B(K, ε) for all t ≥ L(M, ε), i.e. M

attracts K. The lemma is proved. 

Denote by {M

| λ ∈ Λ} the family of all nonempty compact positively invariant

and globally asymptotically stable sets from X and J

|λ ∈ Λ}.

Theorem 1.8 Let (X, T, π) be compact dissipative and J be its center of Levin-

son, then J = J

, i.e. the center of Levinson J of the compact dissipative system

(X, T, π) is the least compact positively invariant globally asymptotically stable set

in X.

Proof. According to Lemma 1.8 the inclusion J

⊆ J

takes place and according to

Theorem 1.7 J = J

⊆ J

. The reverse inclusion also holds. For that it is suﬃcient

to note that for certain λ

∈ Λ the equality M

= J holds and consequently

= ∩{M

|λ ∈ Λ} ⊆ J. The theorem is proved. 

Theorem 1.9 Let (X, T, π) be compact dissipative dynamical system and K be a

nonempty compact invariant set from X, then the following statements are equiva-

lent:

1. K is the center of Levinson of (X, T, π);

2. K is globally asymptotically stable;

3. K is maximal compact invariant set in X.

Proof. According to Theorem 1.6 from 1. follows 2. Now we will show that the

inverse statement holds. Denote by J the center of Levinson of (X, T, π), then

according to Theorem 1.8 J ⊆ K. On the other hand, the center of Levinson

according to Theorem 1.6 is the attractor of all compact subsets from X and by

invariance of K we have K ⊆ J. So, J = K.

From Theorem 1.6 follows that 1. implies 3. To complete the proof of the

theorem it is suﬃcient to show that the inverse implication also holds. Let J be the

center of Levinson of (X, T, π). Then from one hand in virtue of Theorem 1.6 the

set J is compact and invariant and by the maximality of K we have J ⊆ K. On

the other hand, according to Theorem 1.6 the center of Levinson is the attractor

of the family C(X) and by the invariance of K we have K ⊆ J and consequently

K = J. The theorem is proved. 

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16 Global Attractors of Non-autonomous Dissipative Dynamical Systems

1.4 Dissipative systems on the local compact spaces

As it was mentioned above from the local dissipativity of the system (X, T, π) follows

its compact dissipativity. On the other hand if the space X is locally compact then

is is easy to see that the converse also is true. So, in locally compact spaces compact

and local dissipativity are equivalent. In fact, as it will be shown bellow a stronger

statement holds, namely: in locally compact spaces point dissipativity implies local

dissipativity.

Deﬁnition 1.26 The dynamical system (X, T, π) we will call:

− locally completely continuous if for every point p ∈ X there exist δ = δ(p) > 0

and l = l(p) > 0 such that π

B(p, δ) is relatively compact;

− weakly dissipative if there exist a nonempty compact K ⊆ X such that for every

ε > 0 and x ∈ X there is τ = τ(ε, x) > 0 for which xτ ∈ B(K, ε). In this case

we will call K weak attractor.

Note that every dynamical system (X, T, π) deﬁned on the locally compact met-

ric space X is locally completely continuous.

Lemma 1.9 Let K ⊆ X be a nonempty compact, p

∈ X and δ

> 0 (i =

1, m).

If K ⊆ ∪{B(p

, δ

) | i =

1, m}, then there exists γ > 0 such that

B(K, γ) ⊆ ∪{B(p

, δ

) | i =

1, m}. (1.19)

Proof. Assume that the inclusion (1.19) does not hold for any γ > 0. Then there

are γ

↓ 0 and r

∈ B(K, γ

) such that r

6∈ ∪{B(p

, δ

) | i =

1, m}. As γ

↓ 0 then

for every point r

there exists the point q

∈ K such that

ρ(r

, q

) < γ

. (1.20)

By the compactness of K the sequence {q

} can be considered converging to some

point q ∈ K. Then according to (1.20) r

→ q, which contradicts the choice of the

sequence {r

}. 

Lemma 1.10 Let (X, T, π) be weakly dissipative and locally completely continuous

and K ⊂ X be the weak attractor of (X, T, π), then:

1) there exist a

> 0 and l

> 0 such that π

B(K, a

) is relatively compact for

every t ≥ l

;

2) there exist L

≥ l

such that for all t ≥ L

B(K, a

) ⊆ π

B(K, a