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

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Pullback attractors under discretization 337

(see Temam

[

314

]

, Chapter I.2.3). Assuming that p and r are diﬀerentiable functions

of R into itself, we can rewrite this system after the transformation u

:= u

−r(t)−

p(t) in the form (10.3) with the matrix

A(ω) =







−p(0) p(0) 0

−p(0) −1 0

0 0 −1







and the mapping

f(u, ω) =







−u

− ˙σ(0) − ˙r(0) − b(0)(r(0) + σ(0)) − (b(0) − 1)u







where u = (u

, u

) and ω := ω(·) = (b(·), r(·), σ(·)) ∈ C(R, R) × C

(R, R) ×

(R, R).

Suppose that b is almost periodic in C(R, R) and that σ, r are almost periodic

in C

(R, R) with

0 < σ

min

:= inf

t∈R

σ(t), 1 < inf

t∈R

b(t).

Then deﬁne σ

for each t ∈ R by σ

p(·) := ω(· + t). Finally, deﬁne the parameter

set Ω to be the closed hull

[

292

]

P :=

[

t∈R

(b(·), r(·), p(·)),

which is a compact subset of C(R, R) × C

(R, R) × C

(R, R).

We need to check that the assumptions D1–D4 are satisﬁed by the diﬀerential

equation with this matrix A and function f . The ﬁrst assumption D1 follows

straighforwardly and the second D2 holds with α(ω) ≡ α

:= min(σ

min

, 1) > 0

with the assumptions on p and r ensuing the continuity of A. These assumptions

on p and r also ensure the continuity of f needed in D3 and the estimate is given

(f(u, ω), u) = (−˙p(0) − ˙r(0))u

−b(0)(r(0) + p(0))u

−(b(0) −1)u

≤

( ˙p(0) + ˙r(0))

2(b(0) − 1)

(0)(r(0) + p(0))

2(b(0) − 1)

=: c(ω)

with a(ω) ≡ 0. Finally, it is obvious from these deﬁnitions of the constants that

D4 is also satisﬁed, in particular with c

:= sup

ω∈Ω

c(ω) < ∞.

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

338 Global Attractors of Non-autonomous Dissipative Dynamical Systems

The Euler scheme for this diﬀerential equation also satisﬁes assumptions N1–

N4. Since F (h, u, ω) ≡ A(ω)u + f(u, ω) here, assumptions N3 and N4 hold triv-

ially, while assumption N1 follows from D1–D3 above. Assumption N2 also follows

from D1–D3 with the proof being almost the same as in the ﬁrst part of the proof

in the Appendix. Note that if b is also assumed to be continuously diﬀerentiable,

then we have the usual second order local discretization error here. One can show

that assumptions N1–N4 are also satisﬁed by higher order schemes such as Runge–

Kutta schemes, but the details are not as straightforward or clean as for the Euler

scheme.

10.3 Cocycle property

The solution ϕ(t, u

, ω) of the diﬀerential equation (10.3) satisﬁes assumptions N1–

N4 the initial condition

ϕ(0, u

, ω) = u

for all (u

, ω) ∈ R

×Ω

and the cocycle property

ϕ(s + t, u

, ω) = ϕ(s, ϕ(t, u

, ω), σ

p) for all s, t ∈ R

, (u

, ω) ∈ R

×Ω

with respect to the autonomous dynamical system generated by the group {σ

}

t∈R

on Ω. (Existence of such solutions for all t ∈ R

is assured by that of an absorbing

set to be established in the proof of Theorem 10.2 below). By our assumptions the

mapping (t, u, ω) 7→ ϕ(t, u, σ

ω) is continuous. Moreover, the mapping ϕ := (ϕ, σ)

deﬁned on R

×R

×Ω

ϕ(t, u, ω) := (ϕ(t, u, ω), σ

ω) for all (t, u

, ω) ∈ R

×R

×Ω

generates an autonomous semi-dynamical system, that is a skew product ﬂow, on

the state space X := R

×Ω.

The situation is somewhat more complicated for the discrete time system gen-

erated by the numerical scheme with variable time steps. For this we will restrict

the choice of admissible step-size sequences. For each δ > 0, we deﬁne H

to be the

set of all two sided sequences {h

}

n∈Z

satisfying

δ ≤ h

≤ δ (10.5)

for each n ∈ Z (the particular factor 1/2 here is chosen just for convenience). The

set H

is compact metric space with the metric



(1)

, h

(2)



∞

n=−∞

−|n|



(1)

−h

(2)



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

Pullback attractors under discretization 339

We then consider the shift operator ˜σ : H

→ H

deﬁned by ˜σh = ˜σ{h

}

n∈Z

:= {h

n+1

}

n∈Z

. The operator ˜σ is a homeomorphism with respect to the above

metric on H

and its group of iterates {˜σ

}

n∈Z

forms a discrete time autonomous

dynamical system on the compact metric space



, ρ



. Finally, for a given

sequence {h

}

n∈Z

we set t

= 0 and deﬁne t

= t

(h) :=

n−1

j=0

and t

−n

(h) := −

j=1

−j

for n ≥ 1.

Now we introduce the parameter space Q

:= H

×Ω for a ﬁxed δ > 0 and we

use the following lemma to introduce a discrete time autonomous dynamical system

Θ = {Θ

}

n∈Z

on Q

Lemma 10.1 The mappings Θ

: Q

→ Q

, n ∈ Z, generated by iteration of

:= id

δ , Θ

(h, p) := (˜σ

h, σ

p) , Θ

−1

(h, p) :=



˜σ

−1

h, σ

−h

−1



form a group of continuous mappings on H

×P .

Proof. We ﬁrst show that Θ

−1

= Θ

−1

= Θ

= id

δ . Indeed we have

−1

(h, p) = Θ



˜σ

−1

h, σ

−1





˜σ

−1

h, σ

−1

−h

−1



= (h, p)

−1

(h, p) = Θ

−1

(˜σ

h, σ

p) = (˜σ

−1

˜σ

h, σ

−h

p) = (h, p).

The continuity of the Θ

mappings follows from the facts that (t, p) 7→ σ

p is

continuous, ˜σ is a homeomorphism and the composition and cartesian products of

continuous mappings are continuous. 

We deﬁne a mapping ψ : Z

×R

×Q

→ R

ψ(0, u

, q) := u

, ψ(n, u

, q) = ψ(n, u

, (h, p)) := u

n ≥ 1,

where u

is the nth iterate of the numerical scheme (10.4) with initial value u

∈

, initial parameter ω ∈ Ω and step-size sequence h ∈ H

. These mappings are

continuous on R

×Q

. They also satisfy a cocycle property with respect to Θ. 

Lemma 10.2 ψ is a discrete time cocycle over (Q

, Z, Θ) R

with ﬁber R

Proof. We write the numerical scheme (10.4) for a given (h, p) and u

n+1

= u

+ h



, σ

(h)

p, u



, n ∈ Z

where t

(h) = 0 and t

(h) =

n−1

j=0

for n ≥ 1. Hence, in terms of the ψ mapping

we have

ψ(n + 1, u

, (h, p)) = ψ(n, u

, (h, p)) + h

F (h

, σ

(h)

p, ψ(n, u

, (h, p)))

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

340 Global Attractors of Non-autonomous Dissipative Dynamical Systems

in general and

ψ(1, u

, (h, p)) = u

+ h



, σ

(h)

p, u



)

for n = 0 since by deﬁnition ψ(0, u

, (h, p)) = u

, which gives the identity property.

The cocycle property is established as follows. Let n ≥ 0. Then

ψ(1 + n, u

, q) = ψ(1 + n, u

, (h, p))

= ψ(n, u

, (h, p)) + h



, σ

(h)

p, ψ(n, u

, (h, p))



= ψ(n, u

, (h, p)) + (˜σ



(˜σ

, σ

(˜σ

p, ψ(n, u

, (h, p))



= ψ (1, p, Θ

(h), ψ(n, u

, (h, p))) = ψ (1, ψ(n, u

, q), Θ

q) ,

that is u

1+n

= = ψ (1, u

, Θ

q). Iterating this n times, we obtain

= ψ (1, u

, Θ

n−1

q) ◦··· ◦ψ (1, u

, Θ

q) .

Similarly with m ≥ 2 we have

ψ(m + n, u

, q) = u

m+n

= ψ (1, ·, Θ

m+n−1

q) ◦ ··· ◦ψ (1, ·, Θ

q) ◦

◦ψ (1, ·, Θ

n−1

q) ◦ ··· ◦ψ (1, u

, Θ

= ψ (1, ·, Θ

m+n−1

q) ◦ ··· ◦ψ (1, u

, Θ

= ψ (1, ·, Θ

m−1

q) ◦ ··· ◦ψ (1, u

, Θ

= ψ (m, u

, Θ

q) = ψ (m, ψ (n, q, u

) , Θ

q) ,

which is the desired cocycle property.

Finally we deﬁne Ψ := (ψ, Θ) and observe that the mappings Ψ(n, ·.·) are con-

tinuous on R

×Q

for n ∈ Z

. 

Remark 10.4 The mappings ψ, Θ and Ψ are deﬁned in the same way for each

δ > 0, so we do not index them with δ.

From the cocycle and group properties we obtain

Lemma 10.3 Ψ = (ψ, Θ) is a discrete time autonomous semi–dynamical system

on the state space R

×Q

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Pullback attractors under discretization 341

10.4 Main result

Our main result is the to establish the existence of pullback attractors for the

non-autonomous dynamical systems (NDS) generated by the diﬀerential equation

and numerical scheme and to show that the components of the numerical pullback

attractor are upper semi–continuous in their parameter and converge upper semi–

continuously to the corresponding components of the diﬀerential equation’s pullback

attractor. Global attractors also exist for the corresponding skew product ﬂows and

converge upper semi–continuously in an appropriate sense.

Theorem 10.2 Let Assumptions D1–D4 and N1–N3 hold. Then the continuous

time NDS (ϕ, σ) generated by the diﬀerential equation (10.3) has a pullback attractor

I = {I

| ω ∈ Ω} and the discrete time NDS (Ψ, Θ) generated by the numerical

scheme (10.4) has a pullback attractor I

= {I

| q ∈ Q

}, provided the maximal

step-size δ is suﬃciently small, such that the set-valued mappings ω 7→ I

and (p, h)

7→ I

(p,h)

are upper semi–continuous with respect to the Hausdorﬀ semi-distance and

satisfy

lim

δ→0+

sup

h∈H



(ω,h)

, I



= 0 for each ω ∈ Ω.

Moreover, the corresponding skew product ﬂows have global attractors J and J

respectively, of the form

J =

[

ω∈Ω

×{ω}, J

[

q∈Q

×{q},

which satisfy

lim

δ→0+



×Ω

, J



= 0.

Proof. The proof of the convergence assertions in Theorem 10.2 will follow imme-

diately from an application of Theorem 10.1 after it has been shown that the ball

B[0; R

] in R

with centre 0 and radius R

:= 3

/α

is a forwards absorbing set

uniformly in all parameters for both the continuous time and discrete time cocycle

systems under consideration. The convergence assertions require additional work.

10.4.1 Existence of an absorbing set

Write x(t) for the solution ϕ(t, u

, ω) of (10.3), so

(t) = A(σ

ω)x(t) + f (σ

ω, x(t)).

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

342 Global Attractors of Non-autonomous Dissipative Dynamical Systems

The following estimate will also be used to construct an absorbing set.

|x(t)|

= 2



(t), x(t)



= 2 (A(σ

p)x(t) + f(σ

p, x(t)), x(t)) (10.6)

= 2 (A(σ

p)x(t)) + 2 (f(σ

p, x(t)), x(t))

≤ −2α(σ

p) |x(t)|

+ 2a(σ

p) |x(t)|

+ 2c(σ

≤ −2 (α(σ

p) − a(σ

p)) |x(t)|

+ 2c

≤ −2α

|x(t)|

+ 2c

and so

|x(t)|

≤ |u

−2α



1 − e

−2α



This implies that the ball B[0; R

] with radius R

= 3

/α

is a forwards absorb-

ing and positively invariant set for all solutions of the diﬀerential equation (10.3)

uniformly in ω ∈ Ω. Note for later purposes that the ball B[0; 2R

/3] is also posi-

tively invariant for the diﬀerential equation.

The proof that B[0; R

] is a uniform forwards absorbing set for the numerical

scheme (10.4) is more complicated. First we show that the inequality (10.6) implies

|x(t)| ≤ |u

−α



1 − e

−α



. (10.7)

as long as |x(t)| ≥

. To see this note that (10.6) can be rewritten as

|x(t)|

≤ −2α

|x(t)|

+ 2c

≤ −2α

|x(t)|

+ 2

|x(t)|

and hence as

|x(t)| ≤ −α

|x(t)| +

√

as long as |x(t)| ≥

We note also by continuity that there exists a T = T (c

, α

) > 0 such that |x(t)|

= |ϕ(t, u

, ω)| ≥

for all t ∈ [0, T ], u

with |u

| ≥

and ω ∈ Ω.

We ﬁx an R  R

and let K

be the constant in the local discretization

error estimate for the ball B[0; R]. Then from (10.7) with x(h) = ϕ(h, u

, ω) and

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

Pullback attractors under discretization 343

Assumption N2 we have

| ≤ |ϕ(h, u

, ω)|+ |ϕ(h, u

, ω) − u

≤ |u

−α



1 − e

−α



+ K

hµ

(h) (10.8)

for u

∈ A[R, 2R

/3] := B[0; R] \B[0; 2R

/3] and h ∈ (0, T ]. Now let δ

∈ (0, 1] ∩

(0, T ] be such that

hµ

(h) ≤

hµ

(h)

1 − e

−α

≤ R −

for h ∈ (0, δ

]. Then from (10.8) for u

∈ A[R, 2R

/3] and h ∈ (0, δ

] we have

| ≤ |u

−α



1 − e

−α



+ K

hµ

(h)

≤ Re

−α



1 − e

−α





R −





1 − e

−α



≤ R,

so u

∈ B[0; R]. In addition, for u

∈ B[0; 2R

/3] we have x(h) ∈ B[0; 2R

/3], so

| ≤ |ϕ(h, u

, ω)|+ |ϕ(h, u

, ω) − u

| ≤

+ K

hµ

(h) ≤

= R

for h ∈ (0, δ

], so u

∈ B[0; R] here too. Hence, the ball B[0; R] is positively invariant

for the numerical scheme for u

∈ B[0; R] and h ∈ (0, δ

]. We can thus apply the

inequality (10.8) iteratively as long as the u

∈ A[R, 2R

/3], that is we have

n+1

| ≤ |u

−α



1 − e

−α



+ K

hµ

(h) (10.9)

when u

, u

, . . ., u

∈ A[R, 2R

/3] and h ∈ (0, δ

Now further restrict the step-size so that

hµ

(h)

1 − e

−α

≤

−

for all h ∈ (0, δ

], where δ

∈ (0, δ

). Using a similar argument as above for the ball

B[0; R], we can show that the ball B[0; R

] is positively invariant for the numerical

scheme with step-sizes h ∈ (0, δ

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

344 Global Attractors of Non-autonomous Dissipative Dynamical Systems

To show that the ball B[0; R

] is absorbing, we further restrict the step-size so

that



1 + e

−α



≤ e

−

for all h ∈ (0, δ

], where δ

∈ (0, δ

]. Then, if u

∈ A[R, R

], from inequality (10.9)

we have

| ≤ |u

−α



1 − e

−α



+ K

hµ

(h)

≤ |u

−α



1 − e

−α





−





1 − e

−α



≤ |u

−α



1 − e

−α



≤



1 + e

−α



| ≤ e

−

In particular, when u

, u

, . . ., u

∈ B[0; R] \ B[0; R

] we can iterate the last

inequality to obtain

| ≤ e

−j

if we use variable step-sizes with δ/2 ≤ h

δ with δ ∈ (0, δ

]. Obviously, there

exists a ﬁnite integer J

) such that |u

| ≤ R

for all j ≥ J

), that is the ball

B[0; R

] is absorbing for the numerical scheme for all step-size sequences h ∈ H

with δ ∈ (0, δ

]. Note that this holds uniformly in ω ∈ Ω.

10.4.2 Upper semi–continuity of the pullback attractor component

sets

Let ψ(n, x, q) denote the numerical trajectory and Θ the shift operator on Q

. The

mappings q 7→ Θq and (x, q) 7→ ψ(n, x, q) for each integer positive n are continuous.

The absorbing set B = B[0; R

] is compact absorbing set and forwards in-

variant uniformly in q ∈ Q

. Hence, by the cocycle property, the compact sets

ψ(n, B, Θ

−n

q) are nested with increasing n and the pullback attractor has compo-

nent sets deﬁned by

n≥0

ψ(n, B, Θ

−n

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Pullback attractors under discretization 345

This means, in particular, that β(ψ(n, B, Θ

−n

q), I

) → 0 as n → ∞. Let ε > 0 and

pick n

= n

(ε, q) > 0 so that

ψ(n

, B, Θ

−n

q) ⊂ B



, ε



where B



, ε



is the ball of radius ε about I

Now the compact set-valued mappings q 7→ ψ(n, B, Θ

−n

q) are continuous in q

with respect to the Hausdorﬀ semi-distance for each ﬁxed n. Fix ε > 0 and pick n

= n

(ε, ¯q) from above. Then there exists δ(ε, ¯q) = δ(ε, n

(ε, ¯q)) > 0 such that

β (ψ(n

, B, Θ

−n

q), ψ(n

, B, Θ

−n

¯q)) < ε

for all q ∈ Q

with ρ(q, ¯q) < δ(ε, ¯q). In particular,

ψ(n

, B, Θ

−n

q) ⊂ B (ψ(n

, B, Θ

−n

¯q), ε)

for all q ∈ Q

with ρ

δ (q, ¯q) < δ(ε, ¯q). Hence we have

⊂ ψ(n

, B, Θ

−n

q) ⊂ B (ψ(n

, B, Θ

−n

¯q), ε)

⊂ B



¯q

, ε



, ε



= B



¯q

, 2ε



that is I

⊂ B



¯q

, 2ε



or equivalently β



, I

¯q



< 2ε for all q ∈ Q

with ρ(q, ¯q) <

δ(ε, ¯q). This means the mapping q 7→ I

is upper semi–continuous.

The proof for the mapping ω 7→ A

is essentially the same.

10.4.3 Upper semi–continuous convergence of the discretized

pullback attractors

We will now prove the upper semi–continuous convergence of the discretized pull-

back attractor component sets to their continuous time counterparts. For the proof

we need the following lemma on the convergence of the numerical trajectories to the

corresponding continuous time trajectory with convergence of the maximum step

size to zero. Its proof is given in the appendix.

Lemma 10.4 For ﬁxed t > 0 and h = {h

}

n∈Z

∈ H

for some δ > 0, let N(t, h)

be the positive integer such that

−1

+ h

−2

+ ··· + h

−N(t,h)

≤ t < h

−1

+ h

−2

+ ··· + h

−N(t,h)−1

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

346 Global Attractors of Non-autonomous Dissipative Dynamical Systems

and consider a sequence (of step-size sequences) with h

∈ H

, where δ

→ 0 as

m → ∞. Then



N(t, h

), u

, Θ

−N(t,h

)

, ω

)



→ ϕ(t, u

, σ

−t

ω) as m → ∞

for any sequence u

→ u

∈ R

and ω

→ ω ∈ Ω.

We suppose that the upper semi–continuous convergence assertion of Theorem

10.2 is not true. Then there exists an ε

> 0 and sub-sequences (for convenience

we use the original index) h

in H

and a

∈ I

(ω,h

)

for m ∈ N such that

dist



, I

(ω,0)



≥ ε

. (10.10)

Note that the component sets I

(ω,h)

and A

are contained in the ball B[0; R

] where

= 3

/α

for each ω ∈ Ω. Then a

∈ A

(p,h

)

⊂ B[0; R

] for each for m ∈

N and B[0; R

] is compact, so there exists a convergent subsequence (again we use

the original index) a

→ a

∗

∈ B[0; R

]. Thus we have

dist



∗

, I

(ω,0)



≥ ε

Now choose t > 0 suﬃciently large so that by pullback attraction

dist



ϕ (t, σ

−t

p, B[0; R

]) , I

(ω,0)



. (10.11)

By the invariance property of a pullback attractor there exist b

∈ A

−N(t,h

)

(ω,h

)

such that



N(t, h

), Θ

−N(t,h

)

(p, h

), b



= a

. (10.12)

Since the A

−N(t,h

)

(ω,h

)

⊂ B[0; R

] for each for m ∈ N, there exists a convergent

subsequence (once again we use the original index) b

→ b

∗

∈ B[0; R

]. By (10.12)

and Lemma 10.4 we have

ϕ (t, σ

−t

ω, b

∗

) = a

∗

which contradicts (10.10) with respect (10.11). This contradiction proves the upper

semi–continuous convergence of the numerical pullback attractor components.

10.4.4 Upper semi–continuous convergence of the discretized

global attractors

Let J ⊆ R

× Ω be the global attractor of the continuous time skew prod-

uct ﬂow dynamical system π(t, u, ω) = (ϕ(t, u, ω), σ

ω) and J

⊂ H

× R

× Ω

the global attractor of the discrete time semi–dynamical system π

(n, h, u, ω) =