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

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Autonomous dynamical systems 47

1.11 Asymptotic stability

In this paragraph we study the problem of asymptotic stability of compact posi-

tively invariant sets of the dynamical systems with inﬁnite-dimensional (non locally

compact) phase space. The diﬀerent conditions that are equivalent to asymptotic

stability are given. The obtained results are the local version of the results proved

before for dissipative dynamical systems.

Lemma 1.34 Let M ⊆ X be a compact and asymptotically stable set of a

dynamical system (X, T, π). Then there exists δ

> 0 such that

lim

t→+∞

β(π

K, M) = 0 (1.59)

for every compact K ∈ C(B(M, δ

)).

Proof. Since M is asymptotically stable, then there exists δ

such that B(M, δ

) ⊂

(M). We will show that for every compact K ∈ C(B(M, δ

)) the equality (1.59)

holds. For this we note, that from the asymptotic stability of the set M follows the

existence for all ε > 0 and x ∈ B(M, δ

) of such positive numbers δ(ε, x) and l(ε, x)

that

ρ(yt, M) < ε (1.60)

for all t ≥ l(ε, x) and y ∈ B(x, δ). We will show that from (1.60) follows (1.59). If

we suppose that it is not true, then there are K

∈ C(B(M, δ

)), t

→ +∞ and

> 0 such that

β(π

, M) ≥ ε

. (1.61)

From the inequality (1.61) follows that there are x

∈ K

such that

ρ(x

, M) ≥ ε

. (1.62)

Since the set K

is compact, then we can suppose that the sequence {x

} is con-

vergent. We put x

:= lim

n→+∞

, then from the inequality (1.60) follows that for n

large enough the inequality

ρ(x

, M) < ε

, (1.63)

holds. But the inequality (1.63) contradicts (1.62). The obtained contradiction

proves our statement. 

Theorem 1.36 Let (X, T, π) be a dynamical system, M ⊆ X be a compact and

positively invariant subset. Then the following conditions are equivalent:

1. the set M is asymptotically stable;

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48 Global Attractors of Non-autonomous Dissipative Dynamical Systems

2. the set W

(M) is open and for each ε > 0 and x ∈ W

(M) there are δ =

δ(ε, x) > 0 and τ = τ(ε, x) > 0 such that

β(π

B(x, δ), M) < ε (1.64)

for all t ≥ τ;

3. the set W

(M) is open and M attracts all compact subsets from W

(M), i.e.

for every compact K ∈ C(W

(M)) the equality (1.59) holds.

4. the set W

(M) is open, every compact K ⊆ W

(M) is stable in the sense of

Lagrange in positive direction and ∅ 6= J

⊆ M for all x ∈ W

(M).

Proof. We will show that 1. implies 2. First of all, let us prove that the set

(M) is open. In fact, since M is a attractive set, then there exists δ > 0 such

that B(M, δ) ⊂ W

(M). Let now q ∈ W

(M) \ B(M, δ), then there is τ

> 0 such

that qτ

∈ B(M, δ). Since the set B(M, δ) is open, then there exists γ > 0 such

that

B(qτ

, γ) ⊂ B(M, δ). (1.65)

According to the continuity of the mapping π

: X → X there exists η > 0 such

that the inclusion

B(q, η) ⊂ B(qτ

, γ)

holds. By Lemma 1.34 B(q, η) ⊂ W

(M) and, consequently, the set W

(M) is

open.

Denote by ξ(ε) > 0 a number, chosen for ε > 0 out of the condition of orbital

stability of the set M. Since W

(M) = {x ∈ X : lim

t→+∞

ρ(xt, M) = 0 for t → +∞},

then for all ξ and x ∈ W

(M) there exists τ = τ(ε, x) > 0 such that ρ(xt, M) < ε

for all t ≥ τ. From the continuity of the mapping π

τ(ε,x)

: X → X follows that for

x ∈ W

(M) and ξ > 0 there is δ = δ(ε, x) > 0 such that

B(x, δ) ⊆ B(xτ, ξ). (1.66)

By the choice of the number ξ and from (1.66) we have β(π

B(x, δ), M) < ε for all

t ≥ τ.

Now we will show that from 2. follows 3. Let ε > 0 and ∅ 6= K ⊂ W

(M) be

compact. Then for every point x ∈ K there exist δ = δ(ε, x) > 0 and τ = τ (ε, x) > 0

such that the inequality (1.64) holds for all t ≥ τ(ε, x). Consider an open covering

{B(x, δ(ε, x))}

x∈K

of the set K. Since the set K is compact and the space X is

complete, we can extract from this covering a ﬁnite sub-covering {B(x

, δ(ε, x

)}

i=1

Let l(ε, K) := max{τ(ε, x

) |i ∈

1, n}, then from the inequality (1.64) follows that

β(π

K, M) < ε for all t ≥ l(ε, K).

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Autonomous dynamical systems 49

Let us prove now that 3. implies 4. Really, since the set M attracts every

compact K ⊂ W

(M), then by Theorem 1.5 the set Σ

(K) = ∪{π

K |t ≥ 0} is

relatively compact and, consequently, J

⊇ ω

6= ∅ for every point x ∈ W

(M).

We note that J

⊆ M for all x ∈ W

(M). In fact, if y ∈ J

, then there exist

→ x and t

→ +∞ such that x

→ y. Since the set K

} is compact

and the set M attracts K

, then y ∈ M, i.e. J

⊂ M.

Finally, we will prove that from 4. follows 1. For this aim we note that from

the openness of W

(M) and compactness of M follows the existence of δ > 0 such

that B(M, δ) ⊂ W

(M), i.e. the set M is attracting. Now show that the set M is

orbitally stable. Since the set M is positively invariant, then

M ⊆ D

(M) = Σ

(M) ∪ J

(M) ⊆ M ∪M = M

and, consequently, D

(M) = M . To ﬁnish the proof of the theorem is suﬃcient

to note that under the conditions of the theorem from the equality D

(M) = M

follows the orbital stability of the set M . If we suppose that it is not true, then

there are ε

> 0, ε

> δ

→ 0, x

∈ B(M, δ

) and t

→ +∞ such that

ρ(x

, M) ≥ ε

. (1.67)

As x

∈ B(M, δ

), δ

→ 0 and the set M is compact, then the set K :=

} ⊆

B(M, ε

) is compact and, consequently, the set Σ

(K) is relatively compact. Thus,

we may suppose that the sequence {x

} is convergent. Let y = lim

n→+∞

, then

y ∈ D

(M) = M. From the equality (1.67), it follows that y /∈ M. The obtained

contradiction ﬁnishes the proof of the theorem. 

Theorem 1.37 Let M ⊆ X be a nonempty compact positively invariant and

asymptotically stable subset of a dynamical system (X, T, π), then the following

aﬃrmations hold:

(1) ω(M) ⊆ M;

(2) the set ω(M) is invariant;

(3) ω(M) =

t≥0

(4) ω(M) is a maximal compact invariant set in W

(M).

Proof. The ﬁrst statement of the theorem is true because the set M is positively

invariant and closed.

Let us prove the second statement. Since the set M is stable in the sense of

Lagrange in positive direction and ω(M) ⊆ M, then by Lemma 1.3 the set ω(M)

is invariant.

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50 Global Attractors of Non-autonomous Dissipative Dynamical Systems

To prove the third statement of the theorem we note that

t≥0

M ⊆ ω(M ).

Since from the inclusion ω(M) ⊆ M and the invariance of the set ω(M) follows

that ω(M) ⊆

t≥0

M, then ω(M) =

t≥0

Finally we will show that the fourth statement is true. Let K ⊂ W

(M) be an

arbitrary compact invariant set. By Theorem 1.36 the set M attracts the compact

K and, consequently, K ⊆ ω(K) ⊆ ω(M). The theorem is proved. 

Theorem 1.38 Let M ⊆ X be a nonempty compact asymptotically stable set.

Then the following statements hold:

(1) the set ω(M) is orbitally stable;

(2) the set ω(M) attracts every compact from W

(M);

(3) J

(A) = ω(M), where A = ∪{α

|ϕ ∈ Φ

, x ∈ ω(M)}.

Proof. Show that the set ω(M) is orbitally stable. If we suppose that it is not

true, then there exist ε

> 0 and δ

→ 0 such that for certain x

∈ B(ω(M), δ

)

and t

→ +∞

ρ(x

, ω(M)) ≥ ε

. (1.68)

Since x

∈ B(ω(M), δ

) and the set ω(M) is compact, then we can suppose that the

sequence {x

} is convergent. By Theorem 1.36 the set H

(K) :=

∪{π

K |t ≥ 0}

is compact. We note that along with the set M the set M

= M ∪ H

(K) also

is attracting for the family of compact subsets from W

(M) and, consequently,

ω(M

) = ω(M). In particular, ω(H

(K)) ⊆ ω(M

) = ω(M). By the compactness

of the set H

(K) we my suppose that the sequence {x

} is convergent. Let

y := lim

n→+∞

, then y ∈ ω(H

(K)) ⊆ ω(M). But from the inequality (1.68)

follows that y /∈ ω(M). The obtained contradiction proves the ﬁrst statement of

the theorem.

Let us prove now the second assertion. Let K

be an arbitrary compact subset

from W

(M). By Theorem 1.5 the set K

is stable in the sense of Lagrange in

positive direction and the set ω(K

) 6= ∅, is compact and β(π

, ω(K

)) → 0 as

t → +∞. According to Lemma 1.3 the set ω(K

) is invariant, and by Theorem

1.37 we have ω(K

) ⊆ ω(M) because ω(M) is a maximal compact invariant set in

(M).

We will show that J

(A) = ω(M). Since A ⊆ ω(M), then J

(A) ⊆ J

(ω(M)).

By the asymptotic stability of the set ω(M) we have J

(ω(M)) ⊆ ω(M) and,

consequently, J

(A) ⊆ ω(M). Now we will prove the reverse inclusion. Let x ∈

ω(M) and ϕ ∈ Φ

, then ∅ 6= α

⊂ A and, consequently, there are y ∈ α

and

→ +∞ such that ϕ(−t

) → y ∈ A and x = π

ϕ(−t

). Thus, x ∈ J

⊆ J

(A),

i.e. ω(M ) ⊆ J

(A). The theorem is completely proved. 

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Autonomous dynamical systems 51

Denote by {M

|λ ∈ Λ} a family of nonempty compact positively invariant sets

attracting all the compacts from W

(M) and I := ∩{M

|λ ∈ Λ}.

Theorem 1.39 Let M be a nonempty compact positively invariant and asymptot-

ically stable set. Then I = ω(M) (i.e. the set ω(M) is the least compact positively

invariant set attracting all the compact sets from W

(M)).

Proof. First of all we will prove that ω(M) ⊆ I and, consequently, I 6= ∅. In fact,

for all λ ∈ Λ we have ω(M) = ω(M

) ⊆ M

, i.e. ω(M) ⊆ I. To ﬁnish the proof of

the theorem it is suﬃcient to show that I ⊆ ω(M). Since the set ω(M) is nonempty,

compact and attracts all the compacts from W

(M), then ω(M) ∈ {M

|λ ∈ Λ}

and, consequently, I ⊆ ω(M). The theorem is proved. 

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52 Global Attractors of Non-autonomous Dissipative Dynamical Systems

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Chapter 2

Non-autonomous dissipative dynamical

systems

2.1 On the stability of Levinson’s center

Let us consider a non-autonomous dynamical system

h(X, T

, π), (Y, T

, σ), hi. (2.1)

Denote by 2

a family of all bounded closed subsets from X, equipped with the

Hausdorﬀ distance. Let K ⊆ X be a compact subset of X.

Deﬁnition 2.1 We will say that the set K satisﬁes the condition (C) if the

mapping F : Y → 2

deﬁned by the equality

F (y) := K

:= {x ∈ K | h(x) = y}, (2.2)

is continuous.

Note, that the condition (C) means that the mapping h : X → Y is open on K.

This condition plays an important role in our reasoning; therefore we give a simple

test, which guarantees that on a compact invariant set we have this property.

Lemma 2.1 A nonempty compact invariant set K ⊆ X possesses the property

(C), if at least one of the following two conditions is fulﬁlled:

(1) there exist y

∈ Y and τ > 0 (τ ∈ T

) such that y

τ = y

and Y =

{σ(t, y

) | 0 ≤ t < τ}, i.e. τ is the least positive period for the point y

;

(2) the set Y is compact minimal one and the non-autonomous dynamical system

(2.1) is distal on the set K, i.e., for all diﬀerent points x

, x

∈ K such that

h(x

) = h(x

), we have

inf

t∈T

ρ(x

t, x

t) > 0. (2.3)

Proof. The ﬁrst statement of the lemma is evident. The second statement follows

from the proposal 4 from

[

238, p.107

]

. 

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54 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Deﬁnition 2.2 The non-autonomous dynamical system (2.1) is said to be point-

wise (compactly, locally, boundedly) dissipative if the autonomous dynamical

system (X, T

, π) possesses this property.

Deﬁnition 2.3 Let the non-autonomous dynamical system (2.1) be compactly di-

issipative and the set J be the center of Levinson of the dynamical system (X, T

, π).

The set J is said to be the Levinson’s center of the non-autonomous dynamical

system (2.1).

Above we have established that the Levinson’s center J is orbitally stable with

respect to the autonomous dynamical system (X, T

, π). Considering the set J as

the Levinson’s center of the non-autonomous dynamical system (2.1), the orbital

stability with respect to the non-autonomous system (2.1) is more naturally deﬁned

as follows:

Deﬁnition 2.4 The set K ⊆ X is said to be orbitally stable with respect to

the non-autonomous system (2.1) if for all ε > 0 there exists δ(ε) > 0 such that

ρ(x, K

) < δ (x ∈ X, y = h(x)) implies ρ(xt, K

) < ε for all t ≥ 0. If in addition

a. there is γ > 0 such that ρ(xt, K

) → 0 as t → +∞ for all x ∈ K

with

the condition that ρ(x, K

) ≤ γ, then the set K is said to be asymptotically

orbitally stable;

b. if ρ(xt, K

) → 0 as t → +∞ for all x ∈ X

, then the set K ⊆ X is said to

be globally asymptotically stable with respect to the non-autonomous system

(2.1).

Deﬁnition 2.5 The set K ⊆ X is called

[

332

]

globally asymptotically stable in

the sense of Lyapunov-Barbashin if

lim

t→+∞

ρ(xt, K

h(x)t

) = 0 (2.4)

for all x ∈ X, moreover the equality (2.4) holds uniformly with respect to the

variable x on compact subsets of X.

In the work

[

332

]

there it was shown that the Levinson’s center of a non-

autonomous dynamical system, generally speaking, is not globally asymptotically

stable in the sense of Lyapunov-Barbashin. We will establish below that the Levin-

son’s center J of the non-autonomous system (2.1) is globally asymptotically stable

if the set J possesses the property (C).

The following assertion may be made.

Lemma 2.2 Let K ⊆ X. If the set Σ

(K) is relatively compact and the set

M (ω(K) ⊆ M, h(M) = Y ) possesses the property (C). Then for each x ∈ K the

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Non-autonomous Dissipative Dynamical Systems 55

equality

lim

t→+∞

sup

x∈K

ρ(xt, M

) = 0 (2.5)

holds uniformly with respect to y ∈ h(H

(K)), where M

:= M ∩ h

−1

(y).

Proof. If we suppose the contrary, then there are ε

> 0, {y

} ⊆ h(H

(K)), x

∈

and t

→ +∞ such that

ρ(x

, M

) ≥ ε

. (2.6)

Since the set Σ

(K) is relatively compact, then we may suppose that the sequences

} and {y

} are convergent. Let ¯x := lim

n→+∞

and ¯y := lim

n→+∞

. We

note that

h(¯x) = lim

n→+∞

h(x

) = lim

n→+∞

h(x

= ¯y (2.7)

and, consequently, ¯x ∈ X

¯y

. On the other hand, according to Lemma 1.2 ¯x ∈ ω(K).

Thus, ¯x ∈ ω

¯y

(K) ⊆ M

¯y

Since the set M satisﬁes the condition (C), then passing to limit in the inequality

(2.6) as n → +∞, we obtain

ρ(¯x, M

¯y

) ≥ ε

. (2.8)

The inequality (2.8) contradicts the inclusion ¯x ∈ M

¯y

. The lemma is proved. 

Theorem 2.1 Let h(X, T

, π), (Y, T

, σ), hi be a compact dissipative non-

autonomous dynamical system and J be its Levinson center satisfying the condition

(C). If Y = h(J), then:

(1) the set J is orbitally stable in the positive direction with respect to the non-

autonomous dynamical system (2.1).

(2) the set J is a compact attractor of this system, i.e., the following equality

lim

t→+∞

ρ(xt, J

h(x)t

) = 0, (2.9)

holds uniformly with respect to x on every compact subset from X.

Proof. We will show that the Levinson center J is orbitally stable in positive

direction with respect to the non-autonomous dynamical system (2.1). If it is not

true, then there exist ε

> 0, δ

↓ 0, x

∈ B(J

h(x

)

, δ

) and t

≥ 0 such that

ρ(x

, J

h(x

) ≥ ε

. (2.10)

Under the conditions of the theorem we may suppose that the sequences {x

} and

{h(x

)} are convergent. Let x

:= lim

n→+∞

. Since the non-autonomous dynamical

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56 Global Attractors of Non-autonomous Dissipative Dynamical Systems

system (2.1) is compactly dissipative, then the set H

({x

}) is compact and, conse-

quently, we may suppose that the sequences {x

} and {h(x

} are convergent.

Put ¯x := lim

n→+∞

and ¯y := lim

n→+∞

h(x

We will show now that the sequence {t

} ﬁguring in the equality (2.10) tends to

+∞. If we suppose that it is not true, then we may suppose that it is convergent.

Denote by t

:= lim

n→+∞

and, passing to limit in the inequality (2.10) and taking

into consideration that the set J satisﬁes the condition (C), we obtain

ρ(x

, J

h(x

) ≥ ε

. (2.11)

On the other hand, ρ(x

, J

h(x

)

) < δ

and, consequently, x

∈ J

h(x

)

. Since the

set J is invariant, then from the last inclusion follows that x

∈ J

h(x

. But it

contradicts the inequality (2.11). Thus, t

→ +∞ and, consequently, ¯x ∈ ω(K

) ⊆

J. From the equality (2.5) follows that ¯x ∈ X

¯y

, i.e. ¯x ∈ ω

¯y

(K) ⊆ J

¯y

. From the

inequality (2.6) it follows that ¯x ∈ X

¯y

, i.e., ¯x ∈ ω

¯y

(K) ⊆ J

¯y

Passing to limit in the inequality (2.10) as n → +∞ and taking into account

that ¯x ∈ J

¯y

and the fact that the set J possesses the property (C), we obtain ε

≤ 0.

But this contradicts the choice of the number ε

. Thus, the orbital stability of the

set J in the positive direction is proved.

Now we will prove the second statement of the theorem. Let K ⊆ X be an

arbitrary compact, then under the conditions of the theorem the set H

(K) is

compact and ω(K) ⊆ J. To ﬁnish the proof of the theorem it is suﬃcient to refer

to Lemma 2.2. 

Remark 2.1 We note, that the problem of the stability in the sense of Lyapunov-

Barbashin of the Levinson center of a non-autonomous dynamical system has been

studied (in one special case) in the work

[

332

]

. We also note that in the work

[

332

]

stability means the equality (2.9), but not orbital stability. Using our terminology we

can formulate the results of the work

[

332

]

in the following way. Let X be a locally

compact space, the non-autonomous dynamical system (2.1) be pointwise dissipative

and Y be a compact minimal set. If the Levinson center J of the dynamical system

(X, T, π) satisﬁes the condition (C), then the non-autonomous dynamical system

h(X, T

, π), (Y, T

, σ), hi is compactly dissipative and its Levinson center attracts

every compact subset from X with respect to the non-autonomous dynamical system

(2.1).

This statement follows from Theorems 1.10 and 2.1. However, from the same

theorems follows that the Levinson center J is orbitally stable in the positive direc-

tion.

Theorem 2.2 Let h(X, T

, π), (Y, T

, σ), hi be a compact dissipative non-

autonomous dynamical system, its Levinson center J satisfy the condition (C) and

h(J) = Y , then the following conditions are equivalent: