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

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Linear almost periodic dynamical systems 457

1. The trivial solution of (13.116) is uniformly exponentially stable, i.e. there exist

positive numbers N and ν such that kU(t, A)U(τ, A)

−1

k ≤ Ne

−ν(t−τ)

for all

t ≥ τ.

2. There exist positive numbers N and ν such that kU(t, B)U(τ, B)

−1

k ≤

−ν(t−τ)

for all t ≥ τ and B ∈ H(A) = {A

: s ∈ [0, τ)}.

3. lim

t→+∞

kU(t, A)k = 0.

4. There exists p ≥ 1 such that

+∞

|U(t, A)u|

dt < +∞ for all u ∈ E.

Proof. Applying Theorem 13.36 to the cocycle h[E], U, (Y, R, σ)i, generated by

equation (13.116) we obtain the equivalence of conditions 2., 3. and 4. According

to Lemma 3

[

]

the conditions 1. and 2. are equivalent. The theorem is proved.



Theorem 13.39 Let A ∈ C(R, Λ) be τ-periodic and U(τ, A) be asymptotically

compact, then the following conditions are equivalent:

(1) The trivial solution of equation (13.116) is uniformly exponentially stable.

(2) lim

t→+∞

|U(t, A)u| = 0 for every u ∈ E.

Proof. Applying Theorem 13.37 to non-autonomous system h(X, R

, π),

(Y, R, σ), hi generated by equation (13.116), we obtain the equivalence of condi-

tions 1. and 2. The theorem is proved. 

Linear partial diﬀerential equations. Let Λ be some complete metric

space of linear closed operators acting into a Banach space E (for example Λ =

+ B|B ∈ [E]}, where A

is a closed operator that acts on E). We assume that

the following conditions are fulﬁlled for equation (13.116) and its H-class (13.117):

(a) for any v ∈ E and B ∈ H(A) equation (13.117) has exactly one mild solution

deﬁned on R

and satisﬁes the condition ϕ(0, v, B) = v;

(b) the mapping ϕ : (t, v, B) → ϕ(t, v, B) is continuous in the topology of R

E × C(R; Λ);

Under the assumptions above, (13.116) generates a linear cocycle h[E], U, (Y, R, σ)i,

where U(t, B) := ϕ(t, ·, B).

Applying the results from subsections 13.9.1 and 13.9.2 to this cocycle, we will

obtain the analogous assertions for diﬀerent classes of partial diﬀerential equations.

We will consider examples of partial diﬀerential equations which satisfy the

above conditions a. and b.

Consider the diﬀerential equation

= (A

+ A(t))u , (13.118)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 13

458 Global Attractors of Non-autonomous Dissipative Dynamical Systems

where A

is a sectorial operator that does not depend on t ∈ R, and A ∈ C(R, [E]).

The results of

[

122

]

imply that equation (13.118) satisﬁes conditions a. and b.

Under the assumptions above, (13.118) generates a linear cocycle h[E], U,

(Y, R, σ)i, where Y := H(A) and U(t, B) := ϕ(t, ·, B). Applying the results from

subsections 13.9.1 and 13.9.2 to this system, we will obtain the following results.

Theorem 13.40 Let A

- be the sectorial operator and A ∈ C(R, Λ) be τ -periodic,

then the following conditions are equivalent:

(1) The trivial solution of equation (13.118) is uniformly exponentially stable, i.e.

there exist positive numbers N and ν such that kU(t, A

+A)U (τ, A

+A)

−1

k ≤

−ν(t−τ)

for all t ≥ τ.

(2) There exist positive numbers N and ν such that kU(t, A

+B)U(τ, A

+B)

−1

k ≤

−ν(t−τ)

for all t ≥ τ and B ∈ H(A).

(3) lim

t→+∞

kU(t, A

+ A)k = 0.

(4) There exists p ≥ 1 such that

+∞

|U(t, A

+ A)u|

dt < +∞ for all u ∈ E.

(5) σ(U(τ, A

+ A)) ⊂ D.

Theorem 13.41 Let A

- be the sectorial operator with compact resolvant and

A ∈ C(R, Λ) be τ - periodic, then the following conditions are equivalent:

1. The trivial solution of equation (13.118) is uniformly exponentially stable.

2. lim

t→+∞

|U(t, A

+ A)u| = 0 for every u ∈ E.

3. |λ| < 1 for every multiplicator λ of operator of monodromy U (τ, A

+ A).

Proof. Since the sectorial operator A

admits a compact resolvant, then in view of

Lemma 7.2.2

[

122

]

the operator U (τ, A

+ A) is compact and, consequently (see,for

example

[

328, p.391-396

]

), every 0 6= λ ∈ σ(U (τ, A

+ A)) is a multiplicator for

operator of monodromy U (τ, A

+ A). Applying Theorem 13.40 (see also Remark

13.8) to linear cocycle h[E], U, (Y, R, σ)i generated by equation (13.118), we obtain

the equivalence of conditions 1., 2. and 3. The theorem is proved. 

Linear functional-diﬀerential equations. Let A = A(C, R

) be the Banach

space of all linear continuous operators acting from C into R

, equipped by operator

norm. Consider the equation

= A(t)u

, (13.119)

where A ∈ C(R, A). We put H(A) :=

: τ ∈ R}, A

(t) := A(t + τ) and the bar

denotes the closure in the topology of uniform convergence on compacts of R.

Along with equation (13.119) we also consider the family of equations

= B(t)u

, (13.120)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 13

Linear almost periodic dynamical systems 459

where B ∈ H(A). Let ϕ

(v, B) be a solution of equation (13.120) with condition

(v, B) = v deﬁned on R

. We put Y := H(A) and denote by (Y, R, σ) the

dynamical system of shifts on H(A). Let C := C([−r, 0], R

), X := C × Y and

π := (ϕ, σ) the dynamical system on X, deﬁned by the equality π(τ, (v, B)) :=

(ϕ

(v, B), B

). The non-autonomous dynamical system h(X, R

, π), (Y, R, σ), hi

(h := pr

: X → Y ) is linear. The following assertion takes place.

Lemma 13.17

[

]

Let H(A) be compact in C(R, A), then the non-autonomous

dynamical system h(X,R

,π),(Y,R,σ),hi generated by equation (13.119) is com-

pletely continuous, i.e. for every bounded set A ⊂ X there exists a positive number

` such that π

A is relatively compact.

Theorem 13.42 Let A be τ-periodic. Then the following assertions are equiva-

lent:

(1) The trivial solution of equation (13.119) is uniformly exponentially stable.

(2) lim

t→+∞

|U(t, A)u| = 0 for every u ∈ E.

(3) |λ| < 1 for every multiplicator λ of operator of monodromy U (τ, A).

Proof. Let h(X, R

, π), (Y, R, σ), hi be the linear non-autonomous dynamical

system, generated by equation (13.119). According to Lemma 13.17 this system

is completely continuous and, consequently, there exists a number k ∈ N such that

(τ, y

) = U(kτ, y

) is relatively compact. By virtue of theory of Riesz-Schauder

(see for example

[

328, p.391-395

]

) every 0 6= λ ∈ σ(U(τ, A)) is a multiplicator of op-

erator of monodromy U (τ, A). To ﬁnish the proof it is suﬃcient to refer to Theorems

13.36, 13.37 and Remark 13.8. 

Consider the neutral functional diﬀerential equation

= A(t)u

, (13.121)

where A ∈ C(R, A) and D ∈ A is non-atomic at zero operator

[

175, p.67

]

. As well

as in the case of equation (13.119), the equation (13.121) generates a linear non-

autonomous dynamical system h(X, R

, π), (Y, R, σ), hi, where X = C × Y, Y ; =

H(A) and π := (ϕ, σ). The following statement holds.

Theorem 13.43 Let A ∈ C(R, A) be τ-periodic and D is stable, then the following

assertions are equivalent:

(1) The trivial solution of equation (13.121) is uniformly exponentially stable;

(2) lim

t→+∞

|U(t, A)u| = 0 for every u ∈ E;

(3) |λ| < 1 for every multiplier λ of operator of monodromy U(τ, A).

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460 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. Let h(X, R

, π), (Y, R, σ), hi be the linear non-autonomous dynamical

system, generated by equation (13.121). According to Lemma 13.10 this system

is asymptotically compact. According to results of

[

175, Chapter 12

]

every 0 6= λ ∈

σ(U(τ, y

)) is a multiplier of operator of monodromy U(τ, y

). To ﬁnish the proof

of Theorem 13.43 it is suﬃcient to refer to Theorems 13.36, 13.37 and Remark 13.8.

The theorem is proved. 

Remark 13.11 The equivalence of conditions 1. and 3. in Theorem 13.41

(Theorem 13.42, Theorem 13.43) was proved in

[

188, p.219

]

(resp. in

[

188, p.233

]

[

175, p.365

]

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 14

Chapter 14

Triangular maps

Chapter 14 is devoted to the study of quasi-linear triangular maps: chaos, almost

periodic and recurrent solutions, integral manifolds, chaotic sets etc. This problem

is formulated and solved in the framework of non-autonomous dynamical systems

with discrete time. We prove that such systems admit an invariant continuous

section (an invariant manifold). Then, we obtain the conditions of the existence of

a compact global attractor and characterize its structure. We give a criterion for the

existence of almost periodic and recurrent solutions of the quasi-linear triangular

maps. Finally, we prove that quasi-linear maps with chaotic base admits a chaotic

compact invariant set.

This chapter is organized as follows.

In Section 1 we establish the relation between triangular maps and non-

autonomous dynamical systems with discrete time.

In Section 2 we study linear non-autonomous dynamical systems with discrete

time and prove that they admit a unique continuous invariant section – invariant

manifold (Theorem 14.2).

Section 3 is devoted to the study of the existence of invariant sections of quasi-

linear non-autonomous dynamical systems with discrete time (Theorems 14.3 and

14.4).

In Section 4 we prove the existence of compact global attractors of quasi-linear

dynamical systems (Theorems 14.5 and 14.7) and give the description of the struc-

ture of these attractors (Theorems 14.9 and 14.10).

Section 5 is devoted to the study of almost periodic and recurrent solutions of

quasi-linear diﬀerence equations (Theorem 14.13 and 14.14).

In section 6 we introduce the notion of pseudo-recurrent solution and prove

that quasi-linear dynamical systems with pseudo-recurrent base admit a pseudo-

recurrent solution (Theorem 14.15).

Section 7 is devoted to the study of chaos in triangular maps and non-

autonomous dynamical systems with discrete time (Theorem 14.17).

461

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462 Global Attractors of Non-autonomous Dissipative Dynamical Systems

14.1 Triangular maps and non-autonomous dynamical systems

Let W and Ω be two complete metric spaces and denote by X := W ×Ω its cartesian

product. Recall (see, for example,

[

214, 222

]

) that a continuous map F : X → X is

called triangular if there are two continuous maps f : W × Ω → W and g : Ω → Ω

such that F = (f, g), i.e. F (x) = F (u, ω) = (f(u, ω), g(ω)) for all x =: (u, ω) ∈ X.

Consider a system of diﬀerence equations

(

n+1

= f(ω

, u

)

n+1

= g(ω

(14.1)

for all n ∈ Z

, where Z

is the set of all non-negative integer numbers.

Along with system (14.1) we consider the family of equations

n+1

= f(g

ω, u

) (ω ∈ Ω), (14.2)

which is equivalent to system (14.1). Let ϕ(n, u, ω) be a solution of equation (14.2)

passing through the point u ∈ W for n = 0. It is easy to verify that the map

ϕ : Z

×W × Ω → W ((n, u, ω) 7→ ϕ(n, u, ω) ) satisﬁes the following conditions:

(1) ϕ(0, u, ω) = u for all u ∈ W ;

(2) ϕ(n + m, u, ω) = ϕ(n, ϕ(m, u, ω), σ(m, ω)) for all n, m ∈ Z

, u ∈ W and ω ∈ Ω,

where σ(n, ω) := g

ω;

(3) the map ϕ : Z

×W × Ω → W is continuous.

Denote by (Ω, Z

, σ) the semi-group dynamical system generated by positive

powers of the map g : Ω → Ω, i.e. σ(n, ω) := g

ω for all n ∈ Z

and ω ∈ Ω.

Deﬁnition 14.1 Recall

[

102

]

that a triplet hW, ϕ, (Ω, Z

, σ)i (or brieﬂy ϕ) is

called a cocycle (or non-autonomous dynamical system) over the dynamical system

(Ω, Z

, σ) with ﬁber W .

Thus, the reasoning above shows that every triangular map generates a cocycle

and, obviously, vice versa. Taking into consideration this remark we can study

triangular maps in the framework of non-autonomous dynamical systems (cocycles)

with discrete time.

Deﬁnition 14.2 A map γ : Z → Ω (respectively α : Z → X, where X := W ×Ω)

is called an entire trajectory of the dynamical system (Ω, Z

, σ) (respectively, of the

skew-product dynamical system (X, Z

, π), where π := (ϕ, σ) and hW, ϕ, (Ω, Z

, σ)i

is a cocycle over (Ω, Z

, σ) with ﬁber W ) passing through the point ω ∈ Ω (respec-

tively, x := (u, ω) ∈ X), if γ(0) = ω (resp. α(0) = x) and γ(n + m) = σ(m, γ(n))

(resp. α(n + m) = π(m, α(n)) for all n ∈ Z and m ∈ Z

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 14

Triangular maps 463

Denote by Φ

the set of all the entire trajectories of the semi-group dynamical

system (Ω, Z

, σ) passing through the point ω ∈ Ω at the initial moment n = 0 and

Φ :=

{Φ

| ω ∈ Ω}.

Deﬁnition 14.3 A map ν : Z → W is called an entire trajectory of the cocycle

hW, ϕ, (Ω, Z

, σ)i passing through the point u ∈ W if there are ω ∈ Ω and γ

∈ Φ

such that the map α : Z → X (X := W ×Ω) deﬁned by by the equality α := (ν, γ

)

is an entire trajectory of the skew-product dynamical system (X, Z

, π), i.e. α ∈ Φ

and x := (u, ω).

14.2 Linear non-autonomous dynamical systems

Let Ω be a compact metric space and (Ω, Z

, σ) be a semi-group dynamical system

on Ω with discrete time.

Deﬁnition 14.4 Recall that a subset A ⊆ Ω is called invariant (positively invari-

ant, negatively invariant) if σ

A = A (σ

A ⊆ A, A ⊆ σ

A) for all n ∈ Z

, where

:= σ(n, ·) : Ω → Ω.

Below we will suppose that the set Ω is invariant, i.e. σ

Ω = Ω for all n ∈ Z

Let E be a ﬁnite-dimensional Banach space with the norm |·| and W be a complete

metric space. Denote by L(E) the space of all linear continuous operators on E

and by C(Ω, W ) the space of all the continuous functions f : Ω → W endowed with

the compact-open topology, i.e. the uniform convergence on compact subsets in Ω.

The results of this section will be used in the next sections.

Consider a linear equation

n+1

= A(σ

ω)u

(ω ∈ Ω, σ

ω := σ(n, ω)) (14.3)

and an in-homogeneous equation

n+1

= A(σ

ω)u

+ f(σ

ω), (14.4)

where A ∈ C(Ω, L(E)) and f ∈ C(Ω, E).

Deﬁnition 14.5 Recall that a linear bounded operator P : E → E is called a

projection, if P

= P, where P

:= P ◦P .

Deﬁnition 14.6 Let U(n, ω) be the operator of Cauchy (a solution operator)

of linear equation (14.3). Following

[

117

]

we will say that equation (14.3) has an

exponential dichotomy on Ω, if there exists a continuous projection valued function

P : Ω → L(E) satisfying:

(1) P (σ

ω)U(n, ω) = U(n, ω)P (ω);

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464 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) U

(n, ω) is inversible as an operator from ImQ(ω) to ImQ(σ

ω), where

(n, ω) := U (n, ω)Q(ω);

(3) there exist constants 0 < ν < 1 and N > 0 such that

(n, ω)k ≤ N q

and kU

(n, ω)

−1

k ≤ Nq

for all ω ∈ Ω and n ∈ Z

, where U

(n, ω) := U (n, ω)P (ω).

Let ω ∈ Ω and γ

∈ Φ

. Consider a diﬀerence equation

n+1

= A(γ

(n))u

+ f(γ

(n)), (14.5)

and the corresponding homogeneous linear equation

n+1

= A(γ

(n))u

(ω ∈ Ω). (14.6)

Let (X, ρ) be a metric space with distance ρ. Denote by C(Z, X) the space of

all the functions f : Z → X equipped with a pointwise topology. This topology can

be metrized. For example, by the equality

d(f

, f

) :=

+∞

, d

)

1 + d

, d

)

where d

, d

) := max{ρ(f

(k), f

(k)) | k ∈ [−n, n]}, there is deﬁned a distance

on C(Z, X) which generates the pointwise topology.

If x ∈ X and A, B ⊆ X, then denote by ρ(x, A) := inf{ρ(x, a) | a ∈ A} and

β(A, B) := sup{ρ(a, B) | a ∈ A} the semi-distance of Hausdorﬀ.

Theorem 14.1 Suppose that linear equation (14.3) has an exponential dichotomy

on Ω. Then for f ∈ C(Ω, E) the following statements hold:

(1) the set I

:= {u ∈ E | ∃γ

∈ Φ

such that equation (14.5) admits a bounded

solution ψ

deﬁned on Z with the initial condition ψ

(0) = u} is nonempty and

compact;

(2) ϕ(n, I

, ω) = I

σ(n,ω)

for all n ∈ Z

and ω ∈ Ω, where ϕ(n, u, ω) is a solu-

tion of equation (14.4) with the condition ϕ(0, u, ω) = u and ϕ(n, M, ω) :=

{ϕ(n, u, ω) | u ∈ M};

(3) the map ω → I

is upper-semicontinuous, i.e.

lim

ω→ω

β(I

, I

) = 0

for every ω

∈ Ω, where β is the semi-distance of Hausdorﬀ;

(4) the set I :=

| ω ∈ Ω} is compact.

Proof. Let ω ∈ Ω. Since Ω is compact and invariant, the set Φ

6= ∅. We ﬁx

∈ Φ

. Under the conditions of Theorem 14.1 equation (14.6) has an exponential

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 14

Triangular maps 465

dichotomy on Ω with the same constants N and q that in equation (14.3). Then

equation (14.5) admits the unique solution ν

: Z → E with the condition

kν

∞

≤ N

1 + q

1 − q

kfk

∞

≤ N

1 + q

1 − q

kfk, (14.7)

where kfk := sup{|f(ω)| | ω ∈ Ω} and kν

∞

:= sup{|ν

(n)| | n ∈ Z} (see, for

example,

[

188, 120

]

). Thus, the set I

is not empty. From the continuity of the

function ϕ : Z

×E ×Ω → E and inequality (14.7) follows that the set I

is closed,

bounded and

|u| ≤ N

1 + q

1 − q

kfk

for all u ∈ I

and ω ∈ Ω.

The second statement of the theorem follows from the equality S

(Φ

) = Φ

σ(h,ω)

(h ∈ Z), where S

is an h-translation of the trajectory γ

, i.e. S

(n) :=

(n + h) for all n ∈ Z.

We will prove now the third aﬃrmation. Let ω

∈ Ω, ω

→ ω

, u

∈ I

and

→ u. To prove our statement it is suﬃcient to show that u ∈ I

. Since u

∈ I

there is a trajectory γ

∈ Φ

such that γ

converges to γ

∈ Φ

in C(Z, Ω)

and the equation

n+1

= A(γ

(n))u

+ f(γ

(n)) (14.8)

has a solution ν

with the initial condition ν

(0) = u

and satisfying inequality

(14.7), i.e.

|ν

(n)| ≤ N

1 + q

1 − q

kfk

∞

≤ N

1 + q

1 − q

kfk (14.9)

for all n ∈ Z and k ∈ N. It is clear that the sequence {ν

(n)

} converges for every

n ∈ Z. By Tihonoﬀ’s theorem the sequences {ν

} ⊂ C(Z, E) is relatively compact.

From equality (14.8) and inequality (14.9) follows that every limit point of the

sequence {ν

} is a bounded on Z solution of the equation

n+1

= A(γ

(n))u

+ f(γ

(n)). (14.10)

Taking into account that equation (14.10) admits a unique bounded on Z solution,

we obtain the convergence of the sequence {ν

} in the space C(Z, E). We put

:= lim

k→+∞

. It is easy to see that ν

(0) = u and, consequently, u ∈ I

To prove the fourth aﬃrmation it is suﬃcient to remark that for every ω ∈ Ω

the set I

is compact, the map F : ω → I

(F (ω) := I

) is upper-semicontinuous

and, consequently, the set I :=

| ω ∈ Ω} = F (Ω) is compact. The theorem is

completely proved. 

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466 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Lemma 14.1 Let ω ∈ Ω and γ

∈ Φ

(l = 1, 2). Under the conditions of Theorem

14.1 we have

|ν

(n) − ν

(n)| ≤ Nq

−u

| (14.11)

for all n ∈ N, where ν

is a bounded on Z solution of the equation

n+1

= A(γ

(n))u

+ f(γ

(n)) (14.12)

(l = 1, 2).

Proof. Denote ν(n) := ν

(n) − ν

(n), then the sequence ν is a bounded on Z

solution of equation (14.6), because γ

(n) = γ

(n) = σ(n, ω) for all n ∈ Z

Since equation (14.3) (and, consequently, equation (14.6) too) has an exponential

dichotomy, we have |ν(n)| ≤ N q

|ν(0)| for all n ∈ Z

. Taking into consideration

that ν(0) = u

−u

, we obtain the required statement. The lemma is proved. 

Lemma 14.2 Under the conditions of Theorem 14.1 for every ω ∈ Ω the set I

contains a single point u

Proof. Suppose that the statement of the lemma is not true. Then there exists

∈ Ω such that I

contains at least two points u

, u

6= u

). Let ν

(i = 1, 2) be

a bounded on Z solution of equation (14.10) with the condition ν

(0) = u

(i = 1, 2).

According to inequality (14.11), we have

|ν

(−n) − ν

(−n)| ≥ Nq

−n

−u

| (14.13)

for all n ∈ N. On the other hand, since ν

(l = 1, 2) is a bounded on Z solution of

equation (14.12), we have

sup

n∈Z

|ν

(n) − ν

(n)| < +∞. (14.14)

Inequalities (14.13) and (14.14) are contradictory. The obtained contradiction

proves our aﬃrmation. The lemma is proved. 

Theorem 14.2 Under the condition of Theorem 14.1 there exists a unique con-

tinuous function ν : Ω → E satisfying the following conditions:

a. the equality

ν(σ(n, ω)) = ϕ(n, ν(ω), ω)

holds for all n ∈ Z

and ω ∈ Ω, where ϕ(n, u, ω) is the unique solution of

equation (14.4) with the initial condition ϕ(0, u, ω) = u.