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

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Non-autonomous Dissipative Dynamical Systems 107

(W, h, Y ) such that X ⊆ W and λx ∈ X for all λ ≥ 0 and x ∈ X (in this case we

will call the bundle (X, h, Y ) homogeneous).

2.10 Power-law asymptotic of homogeneous systems

Let (X, h, Y ) be a local, trivial normed ﬁber bundle with the ﬁber E. We will say

that a non-autonomous dynamical system h(X, T

, π), (Y, T

, σ), hi admits a power-

law asymptotic of order m > 1, if there exist positive numbers a

and b

(i = 1, 2)

such that

t + b

|x|

1−m

)

−

m−1

≤ |π

x| ≤ (a

t + b

|x|

1−m

)

−

m−1

(2.116)

for all t ≥ 0 and x ∈ X.

Theorem 2.32 In order that a non-autonomous dynamical system h(X, T

, π),

(Y, T

, σ), hi would admit a power-law asymptotic of order m > 1, it is necessary

and suﬃcient that there would exist a continuous function V : X → R

satisfying

the following conditions:

a) there exit positive numbers α and β such that

α|x| ≤ V (x) ≤ β|x| (2.117)

for all x ∈ X;

b) V is diﬀerentiable along the trajectories of the system (X, T

, π) and

dV (π



t=0

= −|x|

(2.118)

for all x ∈ X.

Proof. Necessity. Suppose that the equality (2.116) holds and

V (x) :=

+∞

|π

dτ. (2.119)

It is easy to see that under Condition (2.116) the equality (2.119) deﬁnes correctly

the mapping V : X → R

. In addition, under Condition (2.116) the integral (2.119)

is uniformly (in |x| ≤ r, for each r > 0) convergent and, hence, the function V is

continuous. Further, from (2.116) and (2.119) it follows that α|x| ≤ V (x) ≤ β|x|

for all x ∈ X, where α =

m−1

1−m

and β =

m−1

1−m

Suﬃciency. Let V : X → R

be a continuous function satisfying Conditions a)

and b) of the theorem. We will show that the non-autonomous dynamical system

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108 Global Attractors of Non-autonomous Dissipative Dynamical Systems

h(X, T

, π), (Y, T

, σ), hi admits a power-law asymptotic of order m. Indeed, let us

put ϕ(t) := V (π

x) and note that by (2.118)

dϕ(t)

= −|π

for all t ≥ 0 and x ∈ X. From Condition (2.117) it follows that |π

x| ≥

V (π

for all t ≥ 0 and x ∈ X and, consequently,

dϕ(t)

≤ −

(t).

Taking into account that ϕ(t) = V (π

x) > 0 for all t ≥ 0, if x 6= 0, we have

V (π

x) ≤



m − 1

t + α

−m+1

|x|

−m+1



−

m−1

On the other hand, |π

x| ≤

V (π

x) and, hence,

|π

x| ≤



t + b

|x|

1−m



−

m−1

for all t ≥ 0 and x ∈ X, where a

m−1

1−m

and b

= 1.

By analogy, there can be established a lower power-law asymptotic for |π

x|.

The theorem is completely proved. 

Theorem 2.33 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system. Then the following conditions are equivalent:

(1) there exit positive numbers N

and ν

(i = 1, 2) such that

−ν

|x| ≤ |π

x| ≤ N

−ν

|x|

for all t ≥ 0 and x ∈ X;

(2) (a) there exit positive numbers α and β such that the inequality (2.117) holds

for all x ∈ X;

(b) the function V is diﬀerentiable along the trajectories of the system

(X, T

, π) and

dV (π



t=0

= −|x|

for all x ∈ X.

Proof. This assertion can be proved using the same reasoning as in the proof of

Theorem 2.32. 

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Non-autonomous Dissipative Dynamical Systems 109

Lemma 2.18 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system and let exist positive numbers a, b and m > 1 such that

|π

x| ≤ (at + b|x|

1−m

)

−

m−1

(2.120)

for all x ∈ X and t ≥ 0. Then

lim

t→+∞

|π

x| = 0 (2.121)

uniformly in |x| ≤ r for each r > 0 and

|π

x| ≤ N|x| (2.122)

for all t ≥ 0 and x ∈ X, where N = b

−

1−m

Proof. Consider the function ω(r, t) = (at + br

1−m

)

−

m−1

deﬁned for all t ≥ 0 and

r ≥ 0. Note, that

∂ω(r, t)

∂t

= br

1−m

(at + br

1−m

)

1−m

≥ 0

for all t ≥ 0 and, consequently, for |x| ≤ r we have

|π

x| ≤ (at + b|x|

1−m

)

−

≤ (at + br

1−m

)

−

from which it follows (2.121).

The inequality (2.122) it follows from (2.120) and the equality

(at + b|x|

1−m

)

−

m−1

= |x|(at|x|

m−1

+ b)

−

m−1

The lemma is proved. 

Lemma 2.19 Let D be a family of functions m : R

→ R

satisfying the condi-

tions:

a. there exists M > 0 such that 0 < m(t) ≤ M for all t ≥ 0 and m ∈ D;

b. m(t) → 0 as t → +∞ uniformly in m ∈ D, i.e. for any ε > 0 there exists

L(ε) > 0 such that m(t) < ε for all t ≥ L(ε) and m ∈ D .

Then we have the following assertions:

(1) if m(t + τ) ≤ m(t)m(τ) for all t, τ ≥ 0 and m ∈ D, then there exit positive

numbers N and ν such that

m(t) ≤ Ne

−νt

(2.123)

for all t ≥ 0 and m ∈ D;

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110 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) if m(t + τ ) ≤ m(t)m(τm

α−1

(t)) (α > 1) for all t, τ ≥ 0 and m ∈ D, then there

exist positive numbers a and b such that

m(t) ≤ M(a + bt)

−

α−1

for all t ≥ 0 and m ∈ D.

Proof. We will prove the ﬁrst statement of the lemma. Let γ ∈ (0, 1) and τ > 0 be

such that m(τ) ≤ γ < 1 for all m ∈ D. Let us put ν := −τ

−1

ln γ and N := Me

ντ

then

m(nτ) ≤ γ

for all n = 1, 2, . . .. Let t ≥ 0 and n be a natural number, such that t ∈ [nτ, nτ + τ].

Since

m(t) = m(t − nτ + nτ ) ≤ m(t − nτ)m(nτ ) ≤ Mγ

then the following inequality

m(t) ≤ e

ν(t−nτ)

m(t − nτ )γ

−νt

holds too. Taking into account that e

ντ

= γ

−1

, by the choice of the number N we

have the required inequality (2.123).

To prove the second statement of the lemma we will choose τ > 0 so that the

inequality

m(t) ≤

(2.124)

would hold for all t ≥ τ and m ∈ D. Since 0 < m(t) ≤ M and m(t + τ) ≤

m(t)m(τm

α−1

(t)) for all t, τ ≥ 0 and m ∈ D, then

m(t) ≤ M (0 ≤ t ≤ qτ, q = 2

α−1

) (2.125)

and

m(t) ≤

(τ ≤ t < +∞)

for all m ∈ D. Set t

= 0, t

i+1

= t

+ τq

= q

) and note that

m(t

) ≤

(i ≥ 1) (2.126)

for all m ∈ D. Indeed, according to (2.124), we have m(t

) ≤ 2

−1

. Moreover,

m(t

i+1

) = m(t

+ τq

) ≤ m(t

)m(τq

α−1

)) (2.127)

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Non-autonomous Dissipative Dynamical Systems 111

for all m ∈ D. Suppose that (2.126) is valid for all i ≤ n; then from (2.127) it

follows that

m(t

n+1

) ≤

n+1

since τ q

α−1

) ≥ τ (for all m ∈ D) in view of the choice of q

and the inductive

assumption (2.126). Thus, from (2.125) and (2.126) it follows that

m(t) ≤

for all t ≥ t

and n ≥ 1. Note. that t

n+1

= τ(q

n+1

−1)(q − 1)

−1

and, hence,

−n

= 2



α−1

−1

n+1

+ 1



−

α−1

Now let t ∈ [t

, t

n+1

), then we have

−n

< 2



1 +

α−1

−1



−

α−1

. (2.128)

From (2.125) and (2.128) it follows that

m(t) ≤ M



1−α

1 − 2

1−α



−

α−1

for all t ≥ 0 and m ∈ D. By setting a = 2

1−α

and b = τ

−1

(1 − 2

1−α

), we obtain

the required assertion. The lemma is proved. 

Theorem 2.34 Let a non-autonomous system h(X, T

, π), (Y, T

, σ), hi be homo-

geneous, Y be compact, and the ﬁber bundle (X, h, Y ) be normed. Then the following

conditions are equivalent:

a. the trivial section of the ﬁber bundle (X, h, Y ) is uniformly asymptotically

stable;

b. there exit positive numbers N and ν such that

|π

x| ≤ Ne

−νt

|x| (2.129)

for all x ∈ X and t ≥ 0.

Proof. To prove the theorem it is suﬃcient to establish the implication a. ⇒ b.,

since the converse assertion is obvious.

Note, that from the uniform asymptotic stability of the trivial section Θ of

the ﬁber bundle (X, h, Y ) and the homogeneity of the non-autonomous system

h(X, T

, π), (Y, T

, σ), hi it follows the equality X

= X. Now, to complete the

proof of the theorem it is suﬃcient to refer to Theorem 2.31. 

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112 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Theorem 2.35 Let X be a Banach space. For an autonomous homogeneous (of

order k > 1) dynamical system (X, R

, π) the following conditions are equivalent:

1. the trivial motion of (X, R

, π) is uniformly asymptotically stable;

2. there exit positive numbers α and β such that

|π

x| ≤ (α|x|

1−k

+ βt)

−

k−1

for all t ≥ 0 and x ∈ X.

Proof. Let us show that under the conditions of the theorem assumption 1. implies

2. Let x ∈ X (x 6= 0). Then, in view of the homogeneity of order k > 1 of the

system (X, R

, π), we have

|π(t, x)| = |x||π(t|x|

k−1

|x|

)| ≤ |x| sup

|y|≤1

|π(t|x|

k−1

, y)| = |x|m(t|x|

k−1

)

for all t ≥ 0 and x ∈ X, where

m(t) = sup

|y|≤1

|π(t, y)|.

From the uniform asymptotic stability of the trivial motion of (X, R

, π) and by

virtue of the homogeneity of order k > 1 of the system (X, R

, π) we have X

X. From Theorem 2.29 it follows the boundedness of the function m, but from

the uniform asymptotic stability of the trivial motion of (X, R

, π) it follows that

m(t) → 0 as t → +∞. In addition, note that

m(t + τ ) = sup

|y|≤1

|π(t + τ, y)| = sup

|y|≤1

|π(τ, π(t, y))|

= m(t) sup

|y|≤1





τm

k−1

(t),

π(t, y)

m(t)





≤ m(t)m(τ m

k−1

(t))

for all t, τ ≥ 0. According to Lemma 2.19, there exit positive numbers M, a and b

such that

m(t) ≤ M(a + bt)

−

k−1

for all t ≥ 0. Let us put α = M

1−k

a and β = M

1−k

b, then

|π(t, x)| ≤ |x|m(t|x|

k−1

) ≤ |x|M(a + bt|x|

k−1

)

−

k−1

= (α|x|

1−k

+ βt)

−

k−1

for all x ∈ X and t ≥ 0. To complete the proof of the theorem it is suﬃcient to

refer to Lemma 2.18. 

Remark 2.12 All results of this section hold for dynamical systems with homo-

geneous phase spaces (see Remark 2.11 b.).

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Non-autonomous Dissipative Dynamical Systems 113

2.11 Linear systems

Let (X, h, Y ) be a local trivial ﬁber bundle with the ﬁber E and h(X, T

, π),

(Y, T

, σ), hi be a linear non-autonomous dynamical system, i.e. the mapping

: X

→ X

is linear for all y ∈ Y and t ∈ T

. From the results of Chap-

ter 2 (section 10)follow the following statements.

Theorem 2.36 The following statements are equivalent:

(1) the dynamical system h(X, T

, π), (Y, T

, σ), hi is pointwise dissipative;

(2) X

= X.

Theorem 2.37 Let Y be compact. Then the following statements are equivalent:

(1) the dynamical system h(X, T

, π), (Y, T

, σ), hi is compactly dissipative;

(2) X

= X and the trivial section θ of the ﬁber bundle (X, h, Y ) is uniformly

stable, that is for all ε > 0 there exits δ(ε) > 0 such that |x| < δ implies

|xt| < ε for all t ≥ 0;

(3) if the ﬁber bundle (X, h, Y ) is normed, then there exits a positive number N

such that

|xt| ≤ N|x|

for all x ∈ X, t ≥ 0 and X

= X .

Theorem 2.38 Let Y be compact. Then the following statements are equivalent:

(1) the dynamical system h(X, T

, π), (Y, T

, σ), hi is locally dissipative;

(2) X

= X and the trivial section Θ of the ﬁber bundle (X, h, Y ) is uniformly

attracting, i.e. there exits γ > 0 such that

lim

t→+∞

sup

|x|≤γ

|π(t, x)| = 0;

(3) if the ﬁber bundle (X, h, Y ) is normed, then there exit positive numbers N and

ν such that

|π(t, x)| ≤ Ne

−νt

|x| (2.130)

for all x ∈ X, t ≥ 0.

Remark 2.13 a. Using Banach-Steinhaus theorem, it is easy to check that for

linear autonomous or periodic system the notions of pointwise and compact dissi-

pativity are equivalent.

b. In the example 1.8 the linear autonomous dynamical system deﬁned on the

Hilbert space X = L

[0, 1] by the diﬀerential equation x

= Ax with bounded operator

A is compactly dissipative but not locally dissipative.

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114 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Theorem 2.39 Let h(X, T, π), (Y, T, σ), hi be a group (i.e. T = R or Z) linear

non-autonomous dynamical system, Y be compact, (X, h, Y ) be a normed ﬁber bun-

dle and let exist positive numbers M and a such that |π

x| ≤ Me

a|t|

|x| (x ∈ X,

t ∈ T). Then the following statements are equivalent:

1. h(X, T, π), (Y, T, σ), hi is locally dissipative, that is there exit N, ν> 0 such that

|xt| ≤ Ne

−νt

|x| for all t ≥ 0 and x ∈ X;

2. there exits a function V : X → R

possessing the following properties:

2.a. V is some norm on (X, h, Y );

2.b. there exit positive numbers α and β such that α|x| ≤ V (x) ≤ β|x| for all

x ∈ X;

2.c.

(x) = −|x| (x ∈ X), where

(x) := lim

t↓0

−1

[V (xt) − V (x)] for T = R

and

(x) := V (π(1, x)) − V (x) for T = Z.

Proof. Let h(X, T, π), (Y, T, σ), hi be locally dissipative. Deﬁne a function V :

X → R

by the formula

V (x) =

+∞

|xt|dt (x ∈ X), (2.131)

if T = R. Directly from the equality (2.131) it follows that V is some norm on

(X, h, Y ). We will establish some properties of the function V .

|x| ≤ V (x) ≤

|x| (x ∈ X).

In fact,

V (x) =

+∞

|xt|dt ≤ N

+∞

−νt

|x|dt =

|x|.

On the other hand, |x| = |(xt)(−t)| ≤ Me

|xt| (x ∈ X, t ≥ 0) and, hence,

V (x) =

+∞

|xt|dt ≥

+∞

−at

|x|dt =

|x|.

(x) = −|x| (x ∈ X).

Indeed,

V (xt) − V (x) =

+∞

|(xt)s|ds −

+∞

|xs|ds

+∞

|xs|ds

+∞

|xs|ds = −

|xs|ds

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Non-autonomous Dissipative Dynamical Systems 115

and, consequently,

lim

t↓0

−1

[V (xt) − V (x)] = lim

t↓0

−t

−1

|xs|ds = −|x| (x ∈ X).

If T = Z, then we put V (x) =

+∞

n=0

|xn| (x ∈ X). It is obvious that V deﬁnes

some norm on (X, h, Y ). It is easy to see that the function deﬁned above possesses

the required properties. Thus, we established that 1. implies 2..

Now we will establish that 2. implies 1. Let x ∈ X and |x| 6= 0. Consider the

function ϕ(t) := V (xt). Since α|x| ≤ V (x) ≤ β|x| and

(x) = −|x| (x ∈ X), then

we have

˙ϕ(t) =

(xt) = −|xt| ≤ −

V (xt) = −

ϕ(t)

and, hence,

ϕ(t) ≤ ϕ(0)e

−

(t ≥ 0). (2.132)

On the other hand,

V (xt) ≤ |xt| ≤

αV (xt)

(x ∈ X, t ≥ 0). (2.133)

From the inequalities (2.132) and (2.133) it follows that

|xt| ≤

−

|x| (x ∈ X, t ≥ 0).

The theorem is proved. 

Deﬁnition 2.23 Let h(X, T, π), (Y, T, σ), hi be a linear non-autonomous

dynamical system. A non-autonomous dynamical system h(W,T,µ), (Z,T,λ),%i is

said to be linear non-homogeneous, generated by linear (homogeneous) dynamical

system h(X, T, π), (Y, T, σ), hi, if the following conditions hold:

1. there exits a homomorphism q of the dynamical system (Z, T, λ) onto (Y, T, σ);

2. the space W

:= (q ◦ρ)

−1

(y) is aﬃne for all y ∈ (q ◦%)(W ) ⊆ Y and the vectorial

space X

= h

−1

(y) is an associated space to W

(

[

287, p.175

]

). The mapping

: W

→ W

is aﬃne and π

: X

→ X

is its linear associated function

(

[

287, p.179

]

), i.e. X

= {w

−w

| w

, w

∈ W

} and µ

−µ

= π

−w

)

for all w

, w

∈ W

and t ∈ T .

Remark 2.14 The deﬁnition of linear non-homogeneous system, associated by

the given linear system, is given in the work

[

]

, but our deﬁnition is more general

and sometimes more ﬂexible.

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116 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Lemma 2.20 Let h(W, T, µ), (Z, T, λ), %i be a linear non-homogeneous non-

autonomous dynamical system and (Z, T, λ) be compactly dissipative. Then the

following statements are equivalent:

1. h(W, T, µ), (Z, T, λ), %i is compactly dissipative;

2. h(W, T, µ), (Z, T, λ), %i is convergent;

Proof. Let h(W, T, µ), (Z, T, λ), %i be compactly dissipative and J

be the Levinson

center of (W, T, µ). We will show that J

∩ W

contains at most one point for all

z ∈ J

. If we suppose that it is not true, then there exit z

∈ J

and w

, w

∈

∩ W

such that w

6= w

. It is easy to see that w = w

+ λ(w

− w

) =

(1 −λ)w

+ λw

∈ J

∩W

for all λ ∈ R, but this contradicts the compactness of

. Thus, 1. implies 2. The reverse implication is obvious. The lemma is proved.

Theorem 2.40 Let Γ(Z, W ) 6= ∅ and h(X, S

, π), (Y, S, σ), hi be a linear ho-

mogeneous dynamical system, h(W, S

, µ), (Z, S, λ), %i be a linear nonhomogeneous

dynamical system generated by h(X, S

, π), (Y, S, σ), hi and q be a homomorphism

from (Z, S, λ) onto (Y, S, σ). If the spaces Y and Z are compact and (X, h, Y ) is a

normed ﬁber bundle, then the dynamical system h(W, S

, µ), (Z, S, λ), ρi is locally

dissipative if and only if the dynamical system h(X, S

, π), (Y, S, σ), hi is locally dis-

sipative.

Proof. Let h(W, S

, µ), (Z, S, λ), ρi be locally dissipative and J

be Levinson center

of (W, S

, µ). According to Lemma 2.20, the set J

∩ W

contains a single point

for all z ∈ J

. Denote this point by w

. Since (W, S

, µ) is locally dissipative, then

there exits γ > 0 such that

lim

t→+∞

ρ(µ

B(J

, γ), J

) = 0. (2.134)

We will show that the equality (2.134) implies

lim

t→+∞

ρ(µ

w, µ

) = 0 (2.135)

for all z ∈ J

and w ∈ B(w

, γ). Moreover, the equality (2.135) holds uniformly

in z and w. If we suppose that it is not true, then there exit ε

> 0, t

→ +∞,

∈ J

and x

∈ B(w

, γ) (h(x

) = z

) such that

ρ(µ

, µ

) ≥ ε

. (2.136)

From the equality (2.135) and the compactness of Z it follows that we can sup-

pose that the sequences {µ

}, {z

} and {µ

} convergent. We put

z = lim

t→+∞

and w = lim

t→+∞

, then lim

t→+∞

= w

. Passing to limit

in (2.136) as n → +∞, we obtain ρ(

w, w

) ≥ ε

. This contradicts the relation

w ∈ W

∩ J

. The obtained contradiction proves the equality (2.136) .