Sistemas Dinâmicos Não Lineares Décima Sexta Aula · Pullback, uniform and skew-product...

Sistemas dinamicos nao-autonomos - continuacaoPullback, uniform and skew-product attractors

Sistemas Dinamicos Nao LinearesDecima Sexta Aula

Alexandre Nolasco de Carvalho

16 de Outubro de 2019

Alexandre N. Carvalho - USP/Sao Carlos Segundo Semestre de 2012


Given a nonautonomous differential equation as before, we need todeal with four different, but closely related, evolution operators:

(a) the semigroup Θ on Σ(f ), which is associated to thedynamics of the time-dependent nonlinearities appearing inthe equation;

(b) the nonautonomous dynamical system (K,Θ)(Rn ,Σ(f )) or thecocycle K;

(c) the skew-product semiflow Π defined on the product space X;

and lastly, for each global solution η : R → Σ(f ) of Θ, theevolution process Sη given by

(d) Sη(t, s)x = K(t − s, η(s))x for all t > s and x ∈ Rn.



To each of these objects, we have an associated notion ofattractor, and each one of them will play its part in the study ofthe asymptotic dynamics of our ordinary differential equation.

They are the following

(a) the global attractor S for Θ in Σ(f );

(b) the uniform attractor A for K in Rn;

(c) the global attractor A for Π in X;

(d) for each global solution η : R → S of Θ, the pullback

attractor {Aη(t)}t∈R of the evolution process Sη.



Next we present abstract definitions of all these objects, as well asconditions ensuring their existence in each case.

Moreover, we present the relationship between pullback anduniform attractors, which leads to a detailed description of theuniform attractor and provides further understanding of itsdynamical structures (associated to limiting evolution processes).

That will allow us to talk about upper and lower semicontinuityand topological structural stability, at least in the case of a suitablenonautonomous perturbation of a semigroup.



Motivated by the discussion carried out in the previous section, weestablish an abstract framework to work on.

In order to study the asymptotics of a nonautonomous problem weneed to consider the following families of operators: the drivingsemigroup and the associated cocycle, the skew-product semiflow,and the evolution processes obtained from the cocycle consideringglobal solutions of the driving semigroup.



Definicao

Let (X ,dX ) and (Σ, dΣ)be metric spaces. Consider Θ={Θ(t): t>0}⊂C (Σ) a semigroup in Σ, the driving semigroup. A cocycle

relative to Θ is a family {K(t, σ)∈C (X ): t>0, σ∈Σ} that satisfies

(i) K(0, σ)x = x, for all x ∈ X and σ ∈ Σ;

(ii) K(t + s, σ) = K(t,Θ(s)σ)K(s, σ), for all t, s > 0 and σ ∈ Σ;

(iii) R+ × Σ× X ∋ (t, σ, x) 7→ K(t, σ)x ∈ X is continuous.



Property (ii) is called cocycle property. Define in X = X × Σ,with the product metric, the skew-product semiflow orskew-product semigroup Π={Π(t): t>0}⊂C (X) defined by

Π(t)(x , σ) = (K(t, σ)x ,Θ(t)σ) for t > 0 and (x , σ) ∈ X.



The semigroup property of Θ and the cocycle property of K gives

Π(t + s)(x , σ) = (K(t + s, σ)x ,Θ(t + s)σ)

= (K(t,Θ(s)σ)K(s, σ)x ,Θ(t)Θ(s)σ)

= Π(t)Π(s)(x , σ),

and since the other properties of a semigroup are clearly satisfied,Π is in fact a semigroup in X.



When Σ is a unitary set, say Σ = {σ0}, we have Θ(t)σ0 = σ0 forall t > 0, and defining

T (t)x = K(t, σ0)x , for all t > 0 and x ∈ X , (1)

the skew-product semigroup Π becomes Π(t)(x , σ0)=(T (t)x , σ0)for t > 0 and T = {T (t) : t > 0} defines a semigroup in X .

This is the situation that arises when we consider an autonomousdifferential equation. For a nonautonomous differential equation, Σwill not be a unitary set.



Let us now derive conditions for the existence of an attractor forthe skew-product semigroup Π in X.

Recall that a semigroup has a global attractor if and only if it isasymptotically compact and bounded dissipative.

Define πX : X → X as the usual projection in the first coordinate,that is, πX (x , σ) = x for each (x , σ) ∈ X.



Thus we obtain

πX(

Π(t)(x , σ))

= K(t, σ)x , for all t > 0 and (x , σ) ∈ X, (2)

and defining πΣ : X → Σ as the usual projection in the secondcoordinate, that is, πΣ(x , σ) = σ for each (x , σ) ∈ X, we obtain

πΣ(

Π(t)(x , σ))

= Θ(t)σ, for all t > 0 and (x , σ) ∈ X. (3)



The bounded dissipativity property of Π is equivalent to say thatthere are bounded subsets B0 in X and S0 in Σ such that, forbounded subsets B ⊂ X and S ⊂ Σ there exists t0 = t0(B ,S) > 0such that Π(t)(B × S) ⊂ B0 × S0 for all t > t0.

This implies that for any bounded subset S ⊂ Σ there existst0 = t0(S) > 0 such that Θ(t)S ⊂ S0 for all t > t0, and also thatfor bounded subsets B⊂X and S⊂Σ there exists t0= t0(B ,S)>0such that K(t, σ)B ⊂ B0, for all t > t0 and σ ∈ S .



The asymptotic compactness property of Π is equivalent to saythat given a sequence {tn} in R

+ such that tn → ∞ and abounded sequence {(xn, σn)} in X, {Π(tn)(xn, σn)} has aconvergent subsequence in X, which is equivalent to given asequence {tn} in R

+ such that tn → ∞, and bounded sequences{xn} in X and {σn} in Σ we have

◮ {Θ(tn)σn} has a convergent subsequence in Σ and

◮ {K(tn, σn)xn} has a convergent subsequence in X .

This reasoning proves the following result on the characterizationof the skew-product semigroups that posses a global attractor:



Teorema (1)

Let (X , dX ), (Σ, dΣ) be metric spaces and X = X × Σ with the

product metric. Consider Θ a semigroup in Σ and K a cocycle

relative to Θ. If Π is the associated skew-product semigroup then

it has a global attractor A in X if and only if

(i) Θ has a global attractor S in Σ;

(ii) K is bounded dissipative, that is, there is a bounded

subset B0 of X such that, for bounded subsets B ⊂ X and

S ⊂ Σ, there exists t0 = t0(B ,S) > 0 such that

K(t, σ)B ⊂ B0, for all t > t0 and σ ∈ S;

(iii) K is asymptotically compact, that is, given sequence

{tn} in R+ with tn

n→∞−→ ∞, and bounded sequences {xn} in

X and {σn} in Σ, the sequence {K(tn, σn)xn} has a

convergent subsequence in X .



If we now return to our ode example, it is easy to see that anyfunction in Σ(f ) must satisfy the same conditions as f . Now ifσ ∈ Σ(f ) and u0 ∈ R

n, the solution K(·, σ)u0 must satisfy

d

dt‖K(t, σ)u0‖

2 = 2〈K(t, σ)u0, σ(t,K(t, σ)u0)〉 6 −2δ

and consequently, integrating both sides from 0 to t, we obtain‖K(t, σ)u0‖

26‖u0‖2−2δt, for all t>0 such that ‖K(t, σ)u0‖>M.



Using this fact, we conclude that the solutions exists for all t > 0.Now if BM = {u0 ∈ R

n : ‖u0‖ 6 M}, then given a bounded subsetB of Rn we have

supσ∈Σ(f )

distH(K(t, σ)B ,BM) → 0, as t → ∞,

and since we have already seen that Θ has a global attractor inΣ = Σ(f ), all the conditions of Theorem 1 are satisfied.



As we have seen, η : R → Σ is a global solution for Θ, the familySη={Sη(t, s): t>s} defined by

Sη(t, s) = K(t − s, η(s)) for t > s,

is an evolution process in X .



Definicao

Let X be a metric space and S = {S(t, s) : t > s} be an evolution

process in X . A family {A(t)}t∈R is called a bounded pullback

attractor for S if ∪t∈RA(t) is bounded, A(t) is compact in X for

each t ∈ R, and {A(t)}t∈R is S-invariant and S-pullback attracts

bounded subsets of X at time t, for each t ∈ R.



The pullback attractor, with this definition, is unique. In fact if{A(t)}t∈R and {A(t)}t∈R are pullback attractors for S, sinceB = ∪t∈RA(t) is bounded and {A(t)}t∈R pullback attractsbounded subsets of X at time t, the S-invariance property ofpullback attractors implies that

distH(A(t),A(t)) = distH(S(t, s)A(s),A(t))

6 distH(S(t, s)B ,A(t))s→−∞−→ 0.

This implies that A(t) ⊂ A(t), for each t ∈ R. The other inclusionfollows in the same way, interchanging A(t) and A(t).



Within this context, the following result of characterization ofpullback attractors holds:

TeoremaLet X be a metric space and S be an evolution process in X . If Shas a pullback attractor {A(t)}t∈R then for all t ∈ R we have

A(t) = {ξ(t) : ξ is a bounded global solution of S}.



Proof: From the S-invariance of {A(t)}t∈R, proceeding as before,for each x ∈ A(t0), we can construct a global solution ξ such thatξ(t0) = x and ξ(t) ∈ A(t) for all t ∈ R.

On the other hand, if ξ is a bounded global solution of S then,since {A(t)}t∈R S-pullback attracts B = ξ(R) at time t, for eachn ∈ N

∗, there exists tn 6 t such that distH(S(t, s)B ,A(t)) 6 1nfor

all s 6 tn.

In particular

distH(S(t, tn)ξ(tn),A(t)) = distH(ξ(t),A(t)) 61

n, ∀ n ∈ N

∗.

It follows that ξ(t) ∈ A(t) and completes the proof.



Teorema (3)

Let X be a metric space and S an evolution process in X .

Assume that there exists a compact subset K of X such that K

pullback attracts bounded subsets of X at time t, for each t ∈ R.

Then, S has a bounded pullback attractor {A(t) = ω(K , t)}t∈Rand ∪t∈RA(t) is a compact subset of X .



It follows directly from Theorems 1 and 3 the following:

CorolarioLet (X , dX ), (Σ, dΣ) be metric spaces and consider X = X × Σwith the product metric. Consider also a semigroup Θ in Σ and a

cocycle K relative to Θ.

If the associated skew-product semigroup Π has a global attractor

A in X, then for any bounded global solution η : R → Σ of Θ, the

evolution process Sη = {Sη(t, s) = K(t − s, η(s)) : t > s} has a

pullback attractor {Aη(t)}t∈R and ∪t∈RAη(t) ⊂ πX (A).



We can now characterize the global attractor of the skew-productsemigroup in terms of the pullback attractors of evolution processesassociated to bounded global solutions of the driving semigroup.



TeoremaLet (X , dX ), (Σ, dΣ) be metric spaces and consider X = X × Σwith the product metric. Consider also a semigroup Θ in Σ and a

cocycle K relative to Θ.

If the associated skew-product semigroup Π has a global attractor

A, then the driving semigroup Θ has a global attractor S in Σ.

If η : R → S is a bounded global solution for Θ then the evolution

process Sη = {Sη(t, s) : t > s}, given by Sη(t, s) = K(t − s, η(s)),possesses a pullback attractor {Aη(t)}t∈R with the property that

Aη(t) = {x ∈ X : (x , η(t)) ∈ A}. Moreover,

A =⋃

η

⋃

t∈R

Aη(t)× {η(t)},

where the first union is for all bounded global solutions η of Θ.



Proof: Note that a global bounded solution for Π is a continuousfunction (x(·), η(·)) : R → A that satisfies

(x(t+s), η(t+s)) = Π(t)(x(s), η(s)) = (K(t, η(s))x(s),Θ(t)η(s)),

for all t>0, s∈R. Thus x(t+s)=K(t, η(s))x(s)=Sη(t+s, s)x(s)and Θ(t)η(s) = η(t + s), for all t > 0 and s ∈ R. It follows thatx : R → X is a bounded global solution for Sη and η is a boundedglobal solution for Θ. Hence, any point of A is of the form(x(t), η(t)) where x(t) ∈ Aη(t) and η : R → Σ is a bounded globalsolution for Θ and therefore

A ⊂⋃

η

⋃

t∈R

Aη(t)× {η(t)},

where the first union is for all bounded global solutions η of Θ.



On the other hand, if η : R → S is a bounded global solution forΘ and x0 ∈ Aη(t0), there exists global solution x : R → X of Sη

such that x(t0) = x0 and x(t) ∈ Aη(t) for all t ∈ R. Since∪t∈RAη(t) is bounded, we have x : R → X a bounded globalsolution of Sη, and consequently (x(·), η(·)) : R → X is a boundedglobal solution of Π, showing that (x(t), η(t)) ∈ A for all t ∈ R.Then, in particular, (x0, η(t0)) ∈ A and

⋃

η

⋃

t∈R

Aη(t)× {η(t)} ⊂ A,

where the first union is taken over all bounded global solutions ηfor Θ, which completes the proof.



An important object that appears in Theorem 4 is πX (A), whichhas the property of K-attraction of bounded subsets of X ,uniformly in bounded subsets of Σ.

This object turns out to be, in our context, what is known in theliterature by uniform attractor for the cocycle K, which we explainnext.



Definicao

Let (X , dX ), (Σ, dΣ) be metric spaces and X = X × Σ with the

product metric. Consider Θ a semigroup in Σ and K a cocycle

relative to Θ. The uniform attractor for the cocycle K is the

minimal closed subset A of X such that for each bounded subset

B of X and each bounded subset Γ of Σ we have

limt→∞

supσ∈Γ

distH(K(t, σ)B ,A) = 0,

where the minimality condition means that if C is any closed

subset of X such that

limt→∞

supσ∈Γ

distH(K(t, σ)B ,C ) = 0,

for each bounded subset B of X and each bounded subset Γ of Σ,then we have A ⊂ C.



From Theorem 1 the following result holds:

TeoremaLet (X , dX ), (Σ, dΣ) be metric spaces and X = X × Σ with the

product metric. Consider Θ a semigroup in Σ and K be a cocycle

relative to Θ. If Θ has a global attractor S in Σ and K is

bounded dissipative and asymptotically compact, then the cocycle

K has a compact uniform attractor A, and if A is the global

attractor for the associated skew-product semigroup Π then

A = πX (A). Furthermore

A =⋃

η

Aη(0),

where the union is taken over all bounded global solutions η for Θ.



This result is a straightforward application of Theorem 1. Justnote that if η is a bounded global solution for Θ then so isηr (·) = η(·+ r), for each r ∈ R, and we can conclude thatAηr

(0) = Aη(r).


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