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

Подождите немного. Документ загружается.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

Pullback attractors of C-analytic systems 327

Deﬁnition 9.9 The equation (9.39) is called dissipative if there exists a positive

number r such that lim sup

t→+∞

|ϕ(t, z, g)| < r for all z ∈ C

and g ∈ H(f ).

Deﬁnition 9.10 The function f ∈ CH(R × C

, C

) is called positively (neg-

atively) Poisson stable in t ∈ R uniformly w.r.t. z on compact subsets of C

[

292

]

[

300

]

if there exists t

→ +∞ (t

→ −∞, respectively) such that f(t+t

, z) →

f(t, z) as n → ∞ uniformly on every compact subsets of R × C

Theorem 9.9 Suppose that the following conditions hold:

(1) The set H(f) ⊂ CH(R × C

, C

) is compact.

(2) Every function g ∈ H(f ) is positively Poisson stable (in this case function f is

called

[

300

]

[

304

]

quasi recurrent).

(3) The equation (9.39) is dissipative.

Then every equation (9.40) admits a unique bounded on R and positively Poisson

stable solution. This solution is globally uniformly asymptotically stable.

Proof. We consider the dynamical system of translations (Bebutov’s system

[

292

]

[

300

]

) (CH(R × C

, C

), R, σ). Since H(f) is invariant and closed subset of

CH(R × C

, C

), then on set H(f ) it is induced a dynamical system (H(f), R, σ).

Let Ω := H(f ), then on space C

it is deﬁned a C−analytic cocycle hC

, ϕ, (Ω, R, σ)i,

generated by equation (9.39). According to the general properties of equation (9.39)

with holomorphic right hand side f, the cocycle ϕ will be C−analytic (see, for

example,

[

122

]

). To ﬁnish the proof of this theorem it is suﬃcient to apply Theorem

9.4. 

Deﬁnition 9.11 (Bohr’s almost periodic function) The function f ∈ CH(R ×

, C

) is called almost periodic (in the sense of Bohr) in t ∈ R uniformly w.r.t. z

on compact subsets of C

[

238, 292, 300

]

if for every ε > 0 and nonempty compact

subset K ⊂ C

the set

T(ε, f, K) := {τ ∈ R|max

z∈K

|f(t + τ, z) −f(t, z)| < ε}

is relatively dense on R, i.e. there is a number l = l(ε, f, K) > 0 such that

T(ε, f, K)

[a, a + l] 6= ∅

for all a ∈ R.

Deﬁnition 9.12 The equation (9.39) is called pullback dissipative if for every

g ∈ H(f ) there exists a positive number r

such that for all R > 0

lim sup

t→+∞

|ϕ(t, g

−t

, z)| < r

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

328 Global Attractors of Non-autonomous Dissipative Dynamical Systems

uniformly w.r.t. |z| ≤ R.

Theorem 9.10 Let f ∈ CH(R × C

, C

) be an almost periodic function in t ∈ R

uniformly w.r.t. z on compact subsets of C

and the equation (9.39) be pullback

dissipative, then every equation (9.40) admits a unique bounded solution ν

(t), which

is almost periodic and satisﬁes the following conditions:

(1) ν

(t) is uniformly asymptotically stable (locally).

(2) lim

t→+∞

sup

|z|≤R

|ϕ(t, ω

−t

)z − ν

(0)| = 0.

Proof. The proof of this theorem uses the same type of argument as the proof of

Theorem 9.9 and is based on the Theorem 9.3. 

9.7.2 Caratheodory diﬀerential equations

Consider now the equation (9.39) with right hand side f satisfying the conditions

of Caratheodory (see, for example,

[

292

]

) and holomorphic w.r.t. variable z ∈ C

The space of all the Carateodory functions we denote by CH(R × C

, C

). The

topology on this space is deﬁned by family of semi-norm (see

[

292

]

)

n,m

(f) :=

−n

max

|z|≤m

|f(t, z)|dt.

This space is metrizable and on CH(R×C

, C

) the dynamical system of translations

(CH(R × C

, C

), R, σ) can be deﬁned.

Using the standard arguments for ODEs (see,for example,

[

122

]

and

[

186

]

) one

can prove that every equation (9.40) admits a unique solution ϕ(t, z, g) with initial

condition ϕ(0, z, g) = z and supplementary the mapping ϕ(t, g) := ϕ(t, ·, g) : C

→

is holomorphic. Thus if the solutions ϕ(t, z, g) are deﬁned on R

, the mapping

ϕ : R

× C

× H(f) → C

deﬁnes a C−analytic cocycle on C

with the base

H(f), where H(f) :=

|τ ∈ R} and the bar denotes the closure in the space

CH(R × C

, C

). Hence we may apply the general results from sections 1.–6. to

cocycle ϕ, generated by equation (9.39) with Caratheodory’s right hand side, and

we will obtain some results for this type of equations. For example the following

assertion holds.

Theorem 9.11 Let f ∈ CH(R × C

, C

) be an almost periodic function in t ∈ R

(in the sense of Stepanov

[

238

]

) uniformly w.r.t. z on compact subsets of C

, i.e.

for every ε > 0 and compact subset K ⊂ C

the set

T(ε, f, K) := {τ ∈ R|

max

x∈K

|f(t + τ + s, z) −f(t + s, z)|ds < ε}

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

Pullback attractors of C-analytic systems 329

is relatively dense on R. Suppose that the equation (9.39) is dissipative, then every

equation (9.40) admits a unique almost periodic (in the sense of Bohr) solution

(t) which is globally uniformly asymptotically stable.

9.7.3 ODEs with impulses

Let {t

}

n∈Z

be a sequence of real numbers, inf{t

n+1

−t

| n ∈ Z} > 0, p : R → C

be a continuously diﬀerentiable function on every interval (t

, t

n+1

), continuous to

the right in every point t = t

, almost periodic in the sense of Stepanov and

(t) =

n∈Z

where s

:= p(t

+ 0) − p(t

− 0) (i.e. the function p is piecewise constant). More

information about the function described above can be found in the books

[

170

]

and

[

279

]

Consider the equation with impulses

= f(t, z) +

n∈Z

(9.41)

or equivalently

= f (t, z) + p

(t). (9.42)

At the same time we consider the family of equations

= g(t, z) + q

(t), (9.43)

where (g, q) ∈ H(f, p) :=

{(f

, p

)|τ ∈ R} and by the bar we denote the closure in

the product-space CH(R × C

, C

) × C(R, C

Denote by ϕ(t, z, g, q) the unique solution of equation (9.43) (see

[

170

]

and

[

279

]

)

satisfying the initial condition ϕ(0, z, g, q) = z. This solution is continuous on every

interval (t

, t

n+1

) and continuous to the right in every point t = t

(see

[

170

]

and

[

279

]

Deﬁnition 9.13 The equation (9.41) is called dissipative if there exists a number

r > 0 such that lim sup

t→+∞

|ϕ(t, z, g, q)| < r for every (g, q) ∈ H(f, p) and z ∈ C

Using the transformation w := z + q(t) we can transform the equation (9.43)

into the equation

= g(t, w + q(t)). (9.44)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

330 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Remark 9.4 Denote by ˜ϕ(t, z, g, q) the cocycle deﬁned by the family of equations

(9.44), then it is clear that the cocycle ϕ generated by (9.42) is dissipative if and

only if the cocycle ˜ϕ generated by (9.44) is dissipative.

Theorem 9.12 Let f ∈ CH(R ×C

, C

) be a Bohr’s almost periodic function in

t ∈ R uniformly with respect to z on every compacts subsets of C

, locally Lipshitz

in z uniformly w.r.t. t ∈ T and p ∈ C(R, C

) be a Stepanov almost periodic function

bounded on R. If the equation (9.42) is dissipative, then for every (g, q) ∈ H(f, p)

the equation (9.43) admits a unique Stepanov almost periodic solution and this so-

lution is globally uniformly asymptotically stable .

Proof. Let ϕ(t, z, g, q) be the cocycle generated by equation (9.42) and let

˜ϕ(t, w, g, q) be the cocycle generated by equation (9.44). Then we have the following

equality

ϕ(t, z, g, q) = q(t) + ˜ϕ(t, z − q(0), g, q). (9.45)

Under the conditions of Theorem 9.12 the cocycle ˜ϕ is dissipative, C−analytic and

the right hand side of equation (9.44) under the conditions of Theorem 9.12 is

Stepanov almost periodic in t ∈ R uniformly on every compact of C

w.r.t. z. To

ﬁnish the proof of this theorem we apply the Theorem 9.11 to the equation (9.44)

and take into consideration the relation (9.45). The theorem is proved. 

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

Chapter 10

Pullback attractors under discretization

A deﬁning characteristic of an autonomous dynamical system is its dependence

on the time that has elapsed only and not on the absolute time itself. Conse-

quently, limiting objects, such as attractors, actually exist for all time as invari-

ant sets under the evolution of the autonomous system. Although such concepts

can also be used for general non-autonomous systems, where the absolute start-

ing time is as important as the time elapsed since starting, they are often too

restrictive and exclude many interesting types of dynamical behaviour. A simple

example is that of an asymptotically stable solution that is neither constant nor

periodic. What are the limiting or attracting points here? What are the corre-

sponding invariant sets, and, important for numerical considerations, how can one

assure convergence to a particular point in such an invariant set? The forwards

running convergence of an asymptotically stable solution is of little direct use in

constructing the limiting solution since this solution may itself be changing with

increasing time. An alternative is to use pullback convergence, that is to hold ﬁxed

an absolute time instant and to consider the limiting values at this time instant of

other solutions that start progressively early in absolute time. The limiting sets now

depend on absolute time and are invariant under the evolution of the system, that

is, are carried forward onto each other as time increases. This idea was introduced

several years ago in the context of random dynamical systems

[

126, 153, 154, 284

]

which are intrinsically non-autonomous, but had been used already in the 1960s by

M.Krasnoselskii

[

223

]

to establish the existence of solutions of deterministic systems

that are bounded over the entire time axis. It has also been applied recently

[

217,

219

]

to investigate variable time-step (hence non-autonomous) numerical approxi-

mations of global attractors of autonomous systems governed by dissipative ordinary

diﬀerential equations.

Another key idea in

[

126, 153, 154, 284

]

is to formulate the non-autonomous dy-

namics on R

in terms of a cocycle mapping ϕ that is driven by an underlying

autonomous system σ on some parameter set Ω. At its simplest, Ω is just the abso-

lute time set R and σ is the shift operator that essentially resets the starting time

331

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

332 Global Attractors of Non-autonomous Dissipative Dynamical Systems

to the current absolute time value. More useful is to consider for Ω a function space

of admissible vector ﬁelds as proposed by G. R. Sell

[

292

]

or as a probability sample

space as in

[

126, 153, 154, 284

]

, where the current parameter value takes the role of

absolute time and is adjusted by σ with the passage of time. The advantage here

is, in the ﬁrst case at least, that the parameter space can be topologized (often as

a compact space) and the product system (σ, ϕ) is an autonomous semi–dynamical

system known as skew product ﬂow on the new product state space Ω × R

. The

extensive theory of autonomous dynamical systems can then be applied to such

skew product ﬂows, in particular concepts such as invariant sets, limit sets and at-

tractors, but just how these manifest themselves in terms of the original dynamics

on the original state space R

and what relationship, if any, they have with pullback

convergence need to be clariﬁed.

In this chapter we investigate the eﬀect of time discretization on the pullback

attractor of a non-autonomous ordinary diﬀerential equation for which the vector

ﬁelds depend on a parameter that varies in time rather than depending directly on

time itself. The parameter space is assumed to be compact so the skew product

ﬂow formalism as well as cocycle formalism also applies and the vector ﬁelds have a

strong dissipative structure that implies the existence of a compact set that absorbs

all compact sets under the resulting non-autonomous dynamics. The numerical

scheme considered is a general 1–step scheme such as the Euler scheme with variable

time-steps. Our main result is to show that the numerical scheme interpreted as

a discrete time non-autonomous dynamical system, hence discrete time cocycle

mapping and skew product ﬂow on an extended parameter space, also possesses a

cocycle attractor and that its component subsets converge upper semi–continuously

to those of the cocycle attractor of the original system governed by the diﬀerential

equation. This is a non-autonomous analogue of a result of P. E. Kloeden and

J. Lorenz

[

216

]

on the discretization of an autonomous attractor; see also

[

179,

308

]

. We will also see that the corresponding skew product ﬂow systems have

global attractors with the cocycle attractor component sets as their cross-sectional

sets in the original state space R

. Finally, we investigate the periodicity and almost

periodicity of the discretized pullback attractor when the parameter dynamics in

the ordinary diﬀerential equation is periodic or almost periodic and the pullback

attractor consists of singleton valued component sets, i.e. the pullback attractor is

a single trajectory.

The chapter is organized as follows. Pullback attractors, cocycles and skew prod-

uct ﬂows are deﬁned in Section 1 and a theorem is stated, summarizing results from

the literature on the relationship between pullback attractors and global attractors

of skew product ﬂows. The class of non-autonomous diﬀerential equations and the

corresponding variable time-step 1–step schemes to be considered are introduced in

Section 2 and their cocycle formalism is then established in Section 3. The main

result, Theorem 10.2, is formulated and proved in Section 4. Section 5 is devoted

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

Pullback attractors under discretization 333

to periodic and almost periodic behaviour when the pullback attractor is a single

trajectory. Finally, the section 6 contains the proof of a lemma used earlier, which

compares the cocycle mappings of the original continuous time system and of the

discrete time numerical systems.

10.1 Non-autonomous dynamical systems and pullback attractors

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

group σ = {σ

}

t∈T

of mappings of Ω into itself, where the time set T is either Z

(discrete time) or R (continuous time).

Let W be a complete metric space and consider a continuous mapping ϕ :

×W ×Ω → W satisfying the properties

ϕ(0, ·, ω) = id

, ϕ(τ + t, u, ω) = ϕ(τ, ϕ(t, u, ω), σ

ω)

for all t, τ ∈ T

, ω ∈ Ω and u ∈ W . The mapping ϕ is called a (continuous) cocycle

on W with respect to σ. Then the mapping π : T

×W ×Ω → W × Ω deﬁned by

π(t, u, ω) := (ϕ(t, u, ω), σ

ω)

for all t ∈ T

, (u, ω) ∈ Ω ×W forms an autonomous semi–dynamical system on the

state space W ×Ω, i.e. the set of mappings {π(t, ·, ·)}

t∈T

of W ×Ω into itself is a

semi–group, which is called a continuous skew product ﬂow

[

292

]

The usual concept of a global attractor for the autonomous semi–dynamical

system π on the state space Ω × W can be used here. It is the maximal nonempty

compact subset A of Ω × W which is π–invariant, that is

π(t, A) = A for all t ∈ T

and attracts all compact subsets of Ω × W , that is

lim

t→∞

β (π(t, D), A) = 0 for all D ∈ C(W × Ω),

where C(W × Ω) is the space of all nonempty compact subsets of Ω × W and β is

the Hausdorﬀ semi-distance on C(W × Ω).

Another type of attractor, called a pullback attractor, consists of subsets of

the original state space W , which is advantageous for discretizations of the non-

autonomous system.

Theorem 10.1 Let ϕ be a continuous cocycle on W with respect to a group σ

of continuous mappings on Ω and let π = (ϕ, σ) be the corresponding skew product

ﬂow on W ×Ω. In addition, suppose that there is a nonempty compact subset B of

W and for every D ∈ C(W ) there exists a T (D) ∈ T

, which is independent f ω ∈

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

334 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Ω, such that

ϕ(t, D, ω) ⊂ B for all t > T (D). (10.1)

Then

1. there exists a unique pullback attractor I = {I

| ω ∈ Ω} of the cocycle ϕ on

W , where

τ∈T

[

t>τ

ϕ (t, B, σ

−t

ω). (10.2)

2. there exists a global compact attractor A of the autonomous semi–dynamical

system π on W × Ω, where

A =

τ∈T

[

t>τ

π (t, B × Ω).

J =

[

ω∈Ω

×{ω}.

See Crauel and Flandoli

[

126

]

and Schmalfuß

[

284

]

for the proof of Assertion 1. and

Cheban and Fakeeh

[

]

and Hale

[

179

]

for the proof of Assertion 2. Assertion 3.

has been proved by Cheban

[

]

Remark 10.1 Assertion 1. true remains under weaker conditions. For instance,

the sets D in the absorbing condition (10.1) could be parameter dependent, the

parameter space Ω need not be compact nor the mappings σ

continuous (which is

the situation in random dynamical systems, see Arnold

[

]

). Note that the validity of

Assertion 3. for non-uniform absorbing times, a situation which occurs in important

applications, remains open. See

[

]

for a systematic investigation of the relationship

between the pullback and forwards attractors of the cocycle system and the global

attractor of the associated skew product ﬂow.

10.2 Non-autonomous quasi-linear diﬀerential equation

We consider a non-autonomous quasi-linear diﬀerential equation

˙u = A(ω)u + f(u, ωt) (10.3)

on R

where ω ∈ Ω, on which there exists a group of mappings σ

: Ω → Ω for all t

∈ R, where ωt := ωt. A solution x(t) = ϕ(t, u

, ω) of (10.3) with initial value x(0)

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

Pullback attractors under discretization 335

= u

satisﬁes the equation

dϕ

(t, u

, ω) = A(σ

ω)ϕ(t, u

, ω) + f(σ

ω, ϕ(t, u

, ω)).

Our assumptions are

D1. Ω is a compact metric space and (t, ω) 7→ σ

ω is continuous.

D2. ω 7→ A(ω) is continuous and satisﬁes

(A(ω)u, u) ≤ −α(ω) |u|

for all (u, ω) ∈ R

×Ω.

D3. (u, ω) 7→ f(u, ω) is continuous, is locally Lipschitz in u uniformly in ω ∈ Ω

and satisﬁes

(f(u, ω), u) ≤ a(ω) |u|

+ c(ω)

for all (u, ω) ∈ R

×Ω.

D4. α(ω) > 0, α(ω) − a(ω) ≥ α

> 0 and c(ω) ≤ c

< ∞ for all ω ∈ Ω.

Remark 10.2 . The mapping f in D3 denotes the remaining part of the dif-

ferential equation. It may also contain linear terms that has not been included in

linear part with the matrix A(ω). Though seemingly superﬂuous, it is sometimes

convenient to distinguish the matrix operator A(ω) in this way (especially in inﬁnite

dimensional generalizations, which are not considered here).

Remark 10.3 By the above continuity and the compactness of Ω we have the

ﬁnite uniform upper bounds

:= sup

ω∈Ω

kA(ω)k, F

:= sup

ω∈Ω,|u|≤R

|F (u, ω)|.

We also consider a variable time-step one-step explicit numerical scheme cor-

responding to the diﬀerential equation (10.3), such as the Euler scheme, which we

write as

n+1

= u

+ h

F (h

, σ

ω, u

) (10.4)

where t

= 0 and t

n−1

j=0

, t

−n

= −

j=1

−j

for n ≥ 1 for {h

}

n∈Z

a given

two sided sequence of positive terms and F : [0, 1] ×Ω ×R

→ R

is the increment

function. We make the following assumptions:

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

336 Global Attractors of Non-autonomous Dissipative Dynamical Systems

N1. F is continuous in all of its variables and locally Lipschitz in u uniformly in

(h, p) ∈ [0, 1] ×Ω;

N2. The numerical scheme (10.4) satisﬁes a local discretization error estimate of

the form

|ϕ(h

, u

, ω) −u

| ≤ h

), |u

| ≤ R,

for each R > 0, where µ

(h) > 0 for h > 0 and µ

(h) → 0 as h → 0+.

N3. F satisﬁes the consistency condition

F (0, u, ω) = A(ω)u + f(u, ω)

for all (u, ω) ∈ R

×Ω;

N4. F satisﬁes the Lipschitz consistency condition

|(F (h, u, ω) − A(ω)u −f(u, ω)) −

(F (h, v, ω) −A(ω)v − f(v, ω))| ≤ ¯µ

(h)|u − v|

for all |u|, |v| ≤ R uniformly in ω ∈ Ω, where ¯µ

(h) > 0 for h > 0 and ¯µ

(h) → 0

as h → 0+.

For example, F (h, u, ω) ≡ A(ω)u + f(u, ω) for the Euler scheme applied to the

diﬀerential equation (10.3). Note that for one-step order schemes such as the Euler

and Runge–Kutta schemes, µ(h) is typically of the form K

for some integer p

≥ 1. In our case here this would require the diﬀerentiability of F in ω as well as u

and of σ

in t, which is too restrictive for certain applications.

Our main result (see Theorem 10.2 below) is that non-autonomous dynamical

systems generated by the diﬀerential equation (10.3) and the numerical scheme

(10.4) both have pullback and global attractors, and that the numerical attractors

converge upper semi–continuously to the corresponding attractors of the diﬀerential

equation as the step size goes to zero. We will apply Theorem 10.1 to establish the

existence of such attractors, but ﬁrst we need to show in what sense the numerical

scheme (10.4) generates a discrete time cocycle mapping and skew product ﬂow.

An example

We consider the 3–dimensional Lorenz system with time dependent coeﬃcients

˙u

= −p(t)u

+ p(t)u

˙u

= r(t)u

−u

˙u

= −b(t)u

+ u