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

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Pullback attractors under discretization 347

(Θ

(h, ω), ψ

(n, x, (h, ω)) based on the numerical scheme, where we include the su-

perscript δ on π

and ψ

for emphasis. Then J

converges to A as δ → 0 uniformly

in the sense that

lim

δ→0

sup

,ω

)∈J



, x



, J



= 0. (10.13)

Suppose that (10.13) is not true. Then there exist sequences δ

→0 and



, x

, ω



⊂ J

and an ε

> 0 such that



, ω



, A



≥ ε

Let B be a absorbing compact set in R

that is independent of δ. Since J

⊂

× B × Ω and H

×B × Ω is compact, we can select a subsequence (we use the

same index for convenience) such that ω

→ ω

and x

→ x

as n → ∞. Hence



, ω



, J



≥ ε

. (10.14)

On the other hand, since B is compact and the absorption is uniform in ω ∈ Ω,

there exists a t > 0 such that

β (π(t, B, Ω), J) <

In addition, by invariance, π

(j, J

) = J

for all j ∈ N. Now choose for j the

integer N(t, h

), where N(t, h) is deﬁned in Lemma 10.4. Then we can ﬁnd a



, ˆω

, ˆx



∈ H

×B × Ω such that



N(t, h

, ˆx



, ˆω



, x



, ω

;

indeed we can deﬁne



, ˆω



by Θ

−N(t,h

)

, ω

). By a similar compactness

argument, there is subsequence of this subsequence (again we use the original index)

such that



ˆx

, ˆp



→



ˆx

, ˆω



∈ B × Ω. By Lemma 10.4 we then have

= ψ



N(t, h



, ˆx

, ˆω



→ ϕ



t, ˆx

, ˆω



= x

while

N(t,h

)



, ˆω





, ω



→



0, ω





0, σ

ˆω



Combining these results we have



N(t, h

, ˆx

, ˆω



→



0, π



t, ˆx

, ˆω





0, x

, ω



and hence β



, ω



, J



, which contradicts (10.14). Hence the original

assertion (10.13) must be true. This completes the proof of the main theorem,

Theorem 10.2. 

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 10

348 Global Attractors of Non-autonomous Dissipative Dynamical Systems

10.5 Singleton set-valued pullback attractor case

Let u

(t) and u

(t) be two solutions of the diﬀerential equation (10.3) with the

same initial parameter ω but diﬀerent initial values in the positively absorbing ball

B[0; R

] and write

∆(t) = u

(t) − u

(t), ∆

f,ω

(t) = f(σ

ω, u

(t)) −f(σ

ω, u

(t)).

Let L

be the local Lipschitz constant of f in B[0; R

], which by assumption is

uniform in ω ∈ Ω, so

|f(ω, u

(t)) − f(ω, u

(t))| ≤ L

(t) − u

(t)|

or |∆

f,ω

(t)| ≤ L

|∆(t)| in B[0; R

]. We assume that L

≤ α

/2. Then similarly to

earlier (but now we do not use the inner product inequality on the f)

|∆(t)|

= 2



∆(t), ∆(t)



= 2 (A(σ

ω)∆(t), ∆(t)) + 2 (∆

f,ω

(t), ∆(t))

≤ −2α

|∆(t)|

+ 2|∆

f,ω

(t)||∆(t)|

≤ −2α

|∆(t)|

+ 2L

|∆(t)|

≤ −2 (α

−L

) |∆(t)|

≤ −α

|∆(t)|

and so

|∆(t)| ≤ |∆(0)|e

−(α

/2)t

which means the solution operator u 7→ φ(t, ω, u) is a contraction mapping on the

ball B[0; R

] for each t > 0 and ω ∈ Ω.

Now consider the numerical scheme. Let

= u

+ hF (h, ω, u

), y

= u

+ hF (h, ω, u

)

where h ∈ [δ/2, δ] and u

, u

∈ B[0; R

]. Write

∆x = u

−u

, ∆y = y

−y

, ∆

(h) = F (h, ω, u

) − F (h, ω, u

so |∆F | ≤ L

|∆x| in B[0; R

], where L

is the local Lipschitz constant of F in u

on B[0; R

], uniformly in (h, ω) ∈ [0, 1] × Ω. We assume the Lipschitz consistency

condition N4 here, so

|∆

(h) − A∆x − ∆

f,ω

(h)| ≤ ¯µ

(h)|∆x|

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 10

Pullback attractors under discretization 349

where we omit the parameter ω for convenience. Thus we have

|∆y|

= (∆y, ∆y) = (∆x + h∆

(h), ∆x + h∆

(h))

= (∆x, ∆x) + 2h(∆

(h), ∆x) + h

(∆

(h), ∆

(h))

= |∆x|

+ 2h(A∆x + ∆

f,ω

(h), ∆x) + h

|∆

(h)|

+2h(∆

(h) − A∆x −∆

f,ω

(h), ∆x)

≤ |∆x|

−2hα

|∆x|

+ 2h|∆

f,ω

(h)||∆x| + h

|∆x|

+2h|∆

(h) − A∆x − ∆

f,ω

(h)||∆x|

≤ |∆x|

(1 − 2hα

) + 2hL

|∆x|

+ h

|∆x|

+2h¯µ

(h)|∆x|

≤ |∆x|



1 − 2h(α

−L

) + h

+ 2h¯µ

(h)



≤ |∆x|



1 − hα

+ h

+ 2h¯µ

(h)



where we have used the assumption that L

≤ α

/2. By further restricting h from

above we can assure that

|∆y|

≤ |∆x|

(1 − hα

/2) ≤ |∆x|

γ(δα

/4)

for h ∈ [δ/2, δ] and δ suﬃciently small. This means that the numerical solution

satisﬁes the contractive condition



ψ(n, u

, (

h, ω)) −ψ(n, ¯x

, (

h, ω))



≤ |u

− ¯x

|γ

for all u

, ¯x

∈ B[0; R

], ω ∈ Ω and step-size sequence

h ∈ H

, where γ

γ(δα

/4).

From the Contraction Mapping Principal we conclude that the original and

numerical pullback attractors each consist of a single trajectory. The continuity

of their component elements with respect to the parameter follows from Theorem

10.2 and the fact that upper semi–continuity there reduces to continuity for the

singleton set-valued mappings.

Theorem 10.3 Let the Assumptions D1–D4 and N1–N4 hold and suppose

that 2L

≤ α

and that δ is suﬃciently small. Then the pullback attractors of

Theorem 10.2 consist of singleton component sets, that is I

= {a

∗

(ω)} and I

(h,ω)

= {a

∗

(h, ω)}, where the mappings ω 7→ a

∗

(ω) and (h, ω) 7→ a

∗

(h, ω) are continuous.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 10

350 Global Attractors of Non-autonomous Dissipative Dynamical Systems

These singleton valued pullback attractor–trajectories inherit the periodicity or

almost periodicity of the diﬀerential equation and of the diﬀerential equation and

step-size sequence, respectively. This is formulated in the following theorem, the

proof of which will be presented in the remainder of this section. The periodic case

is straightforward, while the almost periodic case is considerably more complicated

and requires the introduction of appropriate deﬁnitions and a number of auxiliary

results.

Recall that the set A ⊂ P is called minimal with respect to a dynamical system

(Ω, R, σ) if it is nonempty, closed and invariant and if no proper subset of A has

these properties.

Theorem 10.4 Suppose that the assumptions of Theorem 10.3 hold and that Ω

is minimal. Then the singleton valued pullback attractor–trajectory I

= {a

∗

(ω)} is

periodic (resp., almost periodic) if ω ∈ Ω is periodic (resp., almost periodic), whereas

the numerical singleton valued pullback attractor–trajectory I

(h,ω)

= {a

∗

(h, ω)} is

periodic (resp., almost periodic) if q = (h, ω) ∈ Q

is periodic (resp., almost peri-

odic).

Deﬁnition 10.1 A sequence h = {h

}

n∈Z

is m–periodic if h

n+m

= h

for all n

∈ Z or, equivalently, if ˜σ

h = h, where m is the smallest integer for which these

equalities hold.

Recall that we have deﬁned a time sequence {t

(h)}

n∈Z

by t

(h) = 0, t

(h) :=

n−1

j=0

and t

−n

(h) := −

j=1

−j

for n ≥ 1 corresponding to a given sequence

h = {h

}

n∈Z

Lemma 10.5 Let h ∈ H

be m–periodic and let ω ∈ Ω be τ–periodic with respect

to σ, that is with σ

ω = ω where τ ∈ R

. Then the point (h, ω) ∈ Q

= H

×Ω is

periodic with respect to Θ = (˜σ, σ) if and only if t

(h)/τ is rational.

Proof. Suppose that t

(h)/τ = k/l for some k, l ∈ N. Then lt

(h) = kτ and

(h, ω) = (h, σ

(h)

ω) = (h, σ

kτ

ω) = (h, ω). On the other hand, suppose that

(h, ω) = (h, ω) for some k ∈ N. Then (˜σ

h, σ

(h)

ω) = (h, ω), which implies that

k = l

m and t

(h) = l

τ where l

, l

∈ N. Hence t

(h)/τ =

. 

The following result can be found in

[

32, 304

]

Theorem 10.5 Let (X, T, π) be a dynamical system on a compact metric space

(X, ρ). Then a point x ∈ X is almost periodic if and only if for every ε > 0 there

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Pullback attractors under discretization 351

exists a δ = δ(ε) > 0 such that

ρ (π(t + t

, x), π(t + t

, x)) < ε, for all t ∈ T,

whenever ρ (π(t

, x), π(t

, x)) < δ.

Deﬁnition 10.2 A sequence {c

} in R is said to be almost periodic if the function

ϕ : Z → R deﬁned by ϕ(n) := c

for n∈ Z is almost periodic.

Deﬁnition 10.3 A sequence {τ

} in R will be called regular if it has the form

= an + c

, for all n ∈ Z,

where a ∈ R is a constant and{c

} is an almost periodic sequence; see Samoilenko

and Troﬁmchuk

[

280, 281

]

Theorem 10.6 Suppose that the time sequence {t

(h)}

n∈Z

corresponding to h

∈ H

is regular. In addition, suppose that dynamical system (Ω, R, σ) is minimal

and almost periodic, that is Ω is minimal and every point ω ∈ Ω is almost periodic.

Then the point (h, ω) ∈ Q

is almost periodic for the dynamical system (Q

, Z, Θ).

Proof. Since the point ω ∈ Ω is almost periodic for the dynamical system (Ω, R, σ),

then by Theorem 10.5 the point ω ∈ Ω will be almost periodic relatively to the

discrete time dynamical system (Ω, Z, σ

(a)

), where σ

(a)

= {σ

}

n∈Z

and t

(h) =

an + c

is the regularity representation of {t

(h)}

n∈Z

. We apply Theorem 10.5 to

the dynamical system (Ω, R, σ). Given ε > 0, let δ(ε) ∈ (0, ε/3) be such that

ρ(σ

, σ

) <

(10.15)

for every t ∈ R and ω

, ω

∈ Ω with ρ(ω

, ω

) < δ. Then we use uniform continuity

on the compact space Ω: given the above δ(ε) > 0, let γ(ε) ∈ (0, δ(ε)) be such that

ρ(σ

ω, ω) < δ (10.16)

for every ω ∈ Ω and s ∈ R with |s| ≤ γ.

Now for this γ(ε) > 0 we denote by M

γ(ε)

the relatively dense subset of Z subset

for which

ρ(˜σ

n+m

h, ˜σ

h) < γ(ε), |c

n+m

−c

| < γ(ε), ρ(σ

a(n+m)

ω, σ

ω) < γ(ε) (10.17)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 10

352 Global Attractors of Non-autonomous Dissipative Dynamical Systems

for all m ∈ M

, n ∈ Z and ω ∈ Ω. From (10.15)–(10.17) we have

ρ(Θ

n+m

(h, ω), Θ

(h, ω)) = ρ(˜σ

n+m

h, ˜σ

h) + ρ(σ

n+m

(h)

ω, σ

(h)

ω)

= ρ(˜σ

n+m

h, ˜σ

h) + ρ(σ

a(n+m)+c

n+m

ω, σ

an+c

ω) < ρ(˜σ

n+m

h, ˜σ

+ρ(σ

a(n+m)

(σ

n+m

ω), σ

a(n+m)

(σ

ω)) + ρ(σ

a(n+m)

(σ

ω), σ

(σ

ω))

< γ(ε) +

+ γ(ε) <

= ε

for all n ∈ Z and m ∈ M

γ(ε)

. Hence the point (h, p) ∈ Q

is almost periodic for

the dynamical system (Q

, Z, Θ). 

Corollary 10.1 Let h ∈ H

be m–periodic and let ω ∈ Ω be almost periodic for

the dynamical system (Ω, R, σ). Then the point (h, ω) ∈ Q

is almost periodic for

the dynamical system (Q

, Z, Θ). In particular, if t

(h)/τ is irrational, then point

(h, p) is almost periodic, but not periodic.

As the ﬁnal step in our proof of Theorem 10.4, we need the following lemma,

which we prove directly here noting that the result also follows from Theorems 1

and 2 in

[

276

]

Lemma 10.6 Suppose that the assumptions of Theorem 10.3 hold and let

∗

(q)}

q∈Q

δ denote the singleton valued pullback attractor for the numerical scheme

(10.4). Then the function n 7→ a

∗

(Θ

q) for n ∈ Z is periodic (resp., almost pe-

riodic) if the point q is periodic (resp., almost periodic) for the dynamical system

, Z, Θ).

Proof. Let q ∈ Q

be m–periodic, that is Θ

q = q. Then a

∗

(Θ

n+m

q) = a

∗

(Θ

= a

∗

(Θ

q) for every n ∈ Z. Hence n 7→ a

∗

(Θ

q) is periodic.

The function a

∗

: Q

→ R

deﬁned by q 7→ a

∗

(q) for each q ∈ Q

is continuous,

hence uniformly continuous, on the compact space Q

. That is, for every ε > 0

there exists a δ(ε) > 0 such that |a

∗

) − a

∗

)| < ε whenever ρ

δ (q

, q

) < δ.

Now let the point q be almost periodic and for δ = δ(ε) > 0 denote by M

the

relatively dense subset of Z such that ρ(Θ

n+m

q, Θ

q) < δ for all m ∈ M

and n ∈

Z. From this and the uniform continuity we have

∗

(Θ

n+m

q) − a

∗

(Θ

q)| < ε

for all n ∈ Z and m ∈ M

δ(ε)

. Hence n 7→ a

∗

(Θ

q) is almost periodic. 

In conclusion, we can restate the assertions of Theorem 10.4 in more detail as

follows.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 10

Pullback attractors under discretization 353

Corollary 10.2 Suppose that the assumptions of Theorem 10.3 hold and that Ω

is minimal. In addition, let {a

∗

(h, ω)}

(h,ω)∈Q

δ denote the singleton valued pullback

attractor for the numerical scheme (10.4).

1. Let h ∈ H

be m–periodic and ω ∈ Ω be τ–periodic. Then n 7→ a

∗

(Θ

(h, ω)} is

periodic if t

(h)/τ is rational and almost periodic if t

(h)/τ is irrational.

2. Let the time sequence {t

(h)}

n∈Z

corresponding to h ∈ H

be regular and let the

point ω ∈ Ω be almost periodic. Then n 7→ a

∗

(Θ

(h, ω)} is almost periodic.

10.6 Appendix: Proof of Lemma 10.4

Consider an autonomous dynamical system on a compact metric space Ω described

by a group σ = {σ

}

t∈R

of mappings of Ω into itself such that the mapping (t, ω)

7→ σ

ω is continuous. Consider also an ordinary diﬀerential equation

˙u = g(u, ωt) (ω ∈ Ω)

on R

with a unique solution ϕ(t, u

, ω) satisfying the initial value problem

ϕ(t, u

, ω) = g(ϕ(t, u

, ω), σ

ω), x(0, u

, ω) = u

We assume that (u, ω) 7→ g(u, ω) is continuous on R

× Ω and locally Lipschitz in

u uniformly in ω, that is for each R > 0 there exists an L

such that

|g(x, ω) −g(y, ω)| ≤ L

|x − y|, ∀x, y ∈ B[0; R].

In particular, then for a sequence of times t

and time steps h

= t

n+1

− t

this

gives in integral equation form

ϕ(t

n+1

, u

, ω) = ϕ(t

, u

, ω) +

n+1

g(σ

ω, ϕ(τ, u

, ω)) dτ.

In future we just write x(t) for this solution. By the Mean Value Theorem there

exists τ

∈ [0, 1] such that

ϕ(t

n+1

, u

, ω) = ϕ(t

, u

, ω) + x(t

n+1

) = x(t

) +

g(σ

+τ

ω, ϕ(t

+ τ

, u

, ω)).

The corresponding higher order scheme solution is

n+1

= u

+ h

F (h

, σ

ω, u

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 10

354 Global Attractors of Non-autonomous Dissipative Dynamical Systems

where the increment function F (h, u, ω) is continuous and satisﬁes the consistency

condition

F (0, u, ω) = g(u, ω), ∀ x, ω.

Let x(t) := ϕ(t, u

, ω), then

x(t

n+1

) − u

n+1

= x(t

) − u

[g(x(t

+ τ

), σ

+τ

ω) −F (h

, σ

ω, u

)]

so the global discretization error E

:= |x(t

) − u

| is estimated by

n+1

≤ E

+ h

|g(x(t

+ τ

), σ

+τ

ω) −F (h

, σ

ω, u

≤ E

+ h

|g(x(t

+ τ

), σ

+τ

ω) −g(x(t

), σ

+τ

ω)|

|g(x(t

), σ

+τ

ω) − g(x(t

), σ

ω)|

|g(x(t

), σ

ω) −g(u

, σ

ω)|

|g(u

, σ

ω) −F (h

, σ

ω, u

≤ E

+ h

|x(t

+ τ

) − x(t

)| + h

(∆

ω; R)

|x(t

) − u

|+ h

ω; R)

= (1 + h

+ h



+τ

g(x(s), σ

ω) ds



(∆

ω; R) + h

ω; R)

≤ (1 + h

+ h

(∆

ω; R) + h

ω; R)

where M

:= max

ω∈Ω,x∈B[0;R]

|g(u, ω)| and ω

(δ; R) is the modulus of continuity

of g(σ

ω, x) uniformly in ω ∈ Ω and u ∈ B[0; R] and ω

ω; R) is the modulus of

continuity of F (·, u, ω) uniformly in ω ∈ Ω and u ∈ B[0; R] that is

(δ; R) := sup

0≤t≤δ

sup

ω∈Ω, x∈B[0;R]

|g(σ

ω, x) − g(ω, x)|

and

(δ; R) := sup

0≤h≤δ

sup

ω∈Ω, x∈B[0;R]

|F (h, u, ω) −F (0, u, ω)|.

Here ω

(δ; R) → 0 and ω

(δ; R) → 0 as δ → 0.

Now we consider an interval [0, T ] and restrict to step-sizes h

∈ [δ/2, δ] for

some δ > 0. Note then that t

n+1

j=0

then satisﬁes nδ/2 ≤ t

≤ nδ with t

≤ T , which means nδ ≤ 2T for these choices of n. The above diﬀerence inequality

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 10

Pullback attractors under discretization 355

thus satisﬁes

n+1

≤ (1 + L

δ)E

+ δ (ω

(δ; R) + ω

(δ; R) + L

δ)

and hence with E

= 0 yields

≤ δ (ω

(δ; R) + ω

(δ; R) + L

δ)

(1 + L

δ)

−1

(1 + L

δ) − 1

≤ (ω

(δ; R) + ω

(δ; R) + L

δ)

nδ

≤ (ω

(δ; R) + ω

(δ; R) + L

δ)

that is

|x(t

) − u

| ≤ (ω

(δ; R) + ω

(δ; R) + L

δ)

Hence for t ∈ (t

, t

n+1

), we have

|x(t) − u

| ≤ |x(t) − x(t

)| + |x(t

) − u

| ≤



g(σ

ω, x(s)) ds



+ (ω

(δ; R) + ω

(δ; R) + L

δ)

≤

δM

+ (ω

(δ; R) + ω

(δ; R) + L

δ)

Let us now consider variable parameters and initial values. let ω

→ ω in Ω

and u

→ u

in R

. Let ϕ(t, u

, ω) and u

, ω), etc, denote the corresponding

solutions. By continuity in initial conditions and parameters uniformly on a compact

time interval [0, T ], we have

ϕ(t, u

, ω

) → ϕ(t, u

, ω),

as j → ∞ for t ∈ [0, T ].

Combining all of these partial results for t ∈ (t

, t

n+1

) we obtain

, ω

) − x(t, u

, ω)| ≤ |u

, ω

) − x(t, u

, ω

)| +

|x(t, u

, ω

) − x(t, u

, ω)| ≤ δM

+ ((ω

(δ; R) + ω

(δ; R) +

δ)

+ |x(t, u

, ω

) − x(t, u

, ω)|

which converges to zero as the maximum step-size δ converges to zero and j tends

to ∞.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 10

356 Global Attractors of Non-autonomous Dissipative Dynamical Systems