nonlinear dynamical systems fourteenth class · attractors for evolution processes existence of...

Attractors for evolution processesExistence of pullback attractors

Nonlinear Dynamical SystemsFourteenth Class

Alexandre Nolasco de Carvalho

October 10, 2017

Alexandre N. Carvalho - USP/Sao Carlos Second Semester of 2017


Semigroups: Pullback attractors and global attractors

Attractors for evolution processes

Let X be a metric space,T=R orZ andP={(t, s)∈T2 : t >s}. Anevolution process inX is a family of maps {S(t, s) : (t, s)∈P}⊂ C(X )with the following properties

1) S(t, t) = I , for all t ∈ T,2) S(t, s) = S(t, τ)S(τ, s), for all t > τ > s,

3) P × X 3 (t, s, x) 7→ S(t, s)x ∈ X is continuous.

If X is a normed vector space and S(t, s) is linear for each (t, s)∈Pwe say that {S(t, s) : t > s ∈ T} is a linear evolution process.




S(t, s) takes each state x of the system at the instant s andevolves it to the state S(t, s)x of the system at at a later instant t.

For a fixed σ ∈ T, S(σ + τ, τ) can be distinct for each τ ∈ T; thatis, besides the elapsed time σ, the initial time τ may also play animportant role in the evolution process.

The processes for which the elapsed time determines the evolution;that is, those for which S(t, s) = S(t − s, 0) for all t > s, arecalled autonomous evolution processes.




The family {T (t) : t > 0} given by T (t) := S(t, 0), t > 0, satisfies

1) T (0) = I ,

2) T (t + s) = T (t)T (s), for all t, s > 0,

3) [0,∞)× X 3 (t, x) 7→ T (t)x ∈ X is continuous.

In this case we say that {T (t) : t > 0} is a semigroup.

Reciprocally, given a semigroup {T (t) : t > 0} the family{S(t, s) : (t, s) ∈ P} defined by S(t, s) = T (t − s), t > s, is anevolution process. When X is a normed vector space T (t) is asemigroup and each T (t) is linear, we say that {T (t) : t > 0} is alinear semigroup.




For an autonomous evolution process {S(t, s) : (t, s) ∈ P}, thebehavior of solutions as t →∞, known as forwards dynamics, isthe same as the behavior of the solutions as s → −∞, known aspullback dynamics.

These two asymptotic behaviors are, in general, not related andmay lead to completely different qualitative properties.

One of our aims is to reveal some of these new dynamicalproperties that the pullback dynamics may contain.




The notion of attractor for an evolution process require that otheraspects be taken into account. To start, a subset A of X will not,in general, be fixed by an evolution process {S(t, s) : (t, s) ∈ P}and the notion of invariance needs to be weakened to a notion thatstill allow us to construct global solutions. Hence, we defineinvariance in the following way:

Definition (Invariance)

Let {B(t) : t ∈ R} be a family of subsets of X . We say that thisfamily is invariant under the action of {S(t, s) : (t, s) ∈ P} if

S(t, s)B(s) = B(t), for all (t, s) ∈ P.




Let X be a metric space and {S(t, s) : (t, s) ∈ P} be an evolutionprocess in X . We define:

I For each (t, s) ∈ P, the image of B by S(t, s),

S(t, s)B := {S(t, s)b : b ∈ B}.

I The orbit of B starting at the instant s ∈ T

γs(B) :=⋃t>s

S(t, s)B.

I The pullback orbit of B at time t ∈ T,

γ(B, t) :=⋃s6t

S(t, s)B.




Definition (Pullback Attraction/Absorption)

Given t ∈ T, a set B(t) ⊂ X pullback-attracts (absorbs) boundedsubsets of X at time t under the action of {S(t, s) : (t, s) ∈ P} if

distH(S(t, s)D,B(t))s→−∞−→ 0

(∃ T = T (t,D) 6 t tal que S(t, s)D ⊂ B(t), ∀s 6 T ).

for each bounded subset D of X . A family {B(t) : t ∈ T}pullback-attracts (absorbs) bounded subsets of X under the actionof {S(t, s) : (t, s) ∈ P} if B(t) pullback-attracts (absorbs)bounded subsets of X at time t, for each t ∈ T. If there is afamily{B(t) : t ∈ R} of bounded sets that pullback-attracts(absorbs) bounded sets we say that {S(t, s) : (t, s) ∈ P} ispullback-bounded dissipative.




Definition (Pullback Attractor)

We say that a family {A(t) : t ∈ R} of compact subsets of X is apullback attractor for {S(t, s) : (t, s) ∈ P} if it is invariant,pullback attracts bounded subsets of X and is the minimal familyof closed sets which pullback attracts bounded subsets of X .




Recall that, if {S(t) : t ∈ T+} is a semigroup with a globalattractor A, then A is compact, invariant and attracts boundedsubsets of X under the action of {S(t) : t ∈ T+}. Besides that,

A = {y ∈ X : there is a global solution

bounded x : T→ X with x(0) = y}.(1)




We will see later that, in the non-autonomous case and when∪t∈R−A(t) is bounded, the notion of pullback attractor is thenotion that preserves this property; that is, for all t ∈ T

A(t) = {ξ(t) : ξ : T→ X is a global

bounded solution of {S(t, s) : (t, s) ∈ P}}.(2)




RemarkThe minimality in Definition is additional relatively to the theory ofglobal attractors for semigroups and it is essential to ensureuniqueness of pullback attractors.

Its inclusion is related to the weakening of the invariance property,imposed by the non-autonomous nature of evolution processes,together with the possibility that the pullback attractors beunbounded at −∞; that is, it opens up the possibility that, for allt ∈ R, ∪s6tA(s) be unbounded.




Remark (Minimality/Uniqueness)

If {T (t) : t > 0} and {T (t − s) : (t, s) ∈ P} is the associatedevolution process, there may be a family {A(t) : t ∈ T} ofcompact invariant subsets that pullback attract bounded subsets ofX and it is not minimal.

In fact, if T (t − s) = e−(t−s)x0, x0 ∈ R, (t, s) ∈ P, thefamily{[−e−t , e−t ] : t ∈ R} is invariant, [−e−t , e−t ] compact andpullback attracts bounded subsets of R at time t for each t ∈ T.




ExerciseShow that the minimality requirement in Definition 3 can beeliminated if we ask that ∪s6tA(s) be bounded for some t ∈ R.

Definition (Global/Backwards-bounded solutions)

Recall that, a global solution for an evolution process{S(t, s) : (t, s) ∈ P} is a continuous function ξ : R→ X such thatS(t, s)ξ(s) = ξ(t) for all (t, s) ∈ P. We say that a solutionξ : T→ X is backwards-bounded if there exists τ ∈ T such thatthe set {ξ(t) : t 6 τ} is bounded.




ExerciseIf {S(t, s) : (t, s) ∈ P} has a pullback attractor {A(t) : t ∈ T}and ξ : T→ X is a backwards-bounded solution, then ξ(t) ∈ A(t)for all t ∈ T.

ExerciseShow that, if {A(t) : t ∈ T} is a pullback attractor for theevolution process {S(t, s) : (t, s) ∈ P} and

⋃s6t A(s) is bounded

for all t ∈ R, then A(t) is given by

A(t) = {ξ(t) : ξ : T→ X is a backwards-bounded

solution of {S(t, s) : (t, s) ∈ P}}, ∀t ∈ T.(3)





Next we relate the pullback attractors of autonomous evolutionprocesses and the global attractors of the correspondingsemigroups.

We show that the notion of pullback attractors extend to evolutionprocesses, in a natural way, the notion of global attractors forsemigroups.




TheoremIf {T (t) : t > 0} is a semigroup and S(t, s) = T (t − s), (t, s) ∈ Pthe associated autonomous evolution process, {T (t) : t > 0} has aglobal attractor A if and only if {S(t, s) : (t, s) ∈ P} has apullback attractor {A(t) : t ∈ T}. In both cases A(t) = A for allt ∈ T.




Proof: If {T (t) : t > 0} has a global attractor A, it is clear thatthe family{A(t) : t ∈ T} with A(t) := A for all t ∈ T pullbackattracts bounded subsets of X under the action of{T (t − s) : (t, s) ∈ P}. The minimality follows from the fact thatA is bounded and S(t, s)A = T (t − s)A = A for all (t, s) ∈ P.




On the other hand, assume that {T (t − s) : (t, s) ∈ P} has apullback attractor {A(t) : t ∈ T}. It is clear that {T (t) : t > 0}has a global attractor A and that A ⊂ A(t) for each t ∈ T.Besides that, the family {A(t) = A : t ∈ T} pullback attractsbounded subsets of X . It follows from the minimality of A(t) thatA(t) = A ⊃ A(t) and the proof is complete.



Properties of pullback ω−limits

Existence of pullback attractors

As in the autonomous case, the notion of ω−limit will play animportant role in the theory of pullback attractors for evolutionprocesses. In what follows {S(t, s) : (t, s) ∈ P} is an evolutionprocess in a metric space X .

Definition (pullback ω−limit)

Let B ⊂ X . The pullback ω-limit of B is defined by

ω(B, t) :=⋂σ6t

⋃s6σ

S(t, s)B.




Exercise (Characterization of the pullback ω−limite)

For each subset B of X , we have that

ω(B, t) ={y ∈ X : there are {sk}k∈N ≤ t, skk→∞−→ −∞ and

{xk}k∈N in B, such that y = limk→∞

S(t, sk)xk}.(4)

Remark (ω−limits for semigroups)

If {S(t) : t > 0} is a semigroup and S(t, s)=S(t−s), (t, s)∈P,ω(B, t) :=

⋂s>0

⋃r>s S(r)B is independent of t and coincide with

the definition of ω−limit ω(B) of B for semigroups.





Some of the results presented next have proofs similar to those ofthe corresponding autonomous results. We present some of themto establish this analogy.

LemmaIf B ⊂ X , then S(t, s)ω(B, s) ⊂ ω(B, t) for each (t, s) ∈ P. If(t, s) ∈ P, ω(B, s) is compact and pullback attracts B at time s,then S(t, s)ω(B, s) = ω(B, t). Additionally, if ω(B, s) pullbackattracts B at time s for each s 6 t, B is a connected set and⋃

s6t ω(B, s) ⊂ B, then ω(B, t) is connected.




Proof: If ω(B, t) = ∅, there is nothing to prove. If ω(B, s) 6= ∅,from the continuity of S(t, s) and from (4) it immediately followsthat S(t, s)ω(B, s) ⊂ ω(B, t).

It remains to show that if, ω(B, s) is compact and pullbackattracts B, then ω(B, t) ⊂ S(t, s)ω(B, s). For x ∈ ω(B, t), thereare sequences σk → −∞, σk 6 t and xk ∈ B such that

S(t, σk)xkk→∞−→ x . Since σk → −∞ we have that there exists

k0 ∈ N such that σk 6 s for all k > k0. Hence,S(t, s)S(s, σk)xk = S(t, σk)xk → x for k > k0. Since ω(B, s)pullback attracts B at time s, we have that

dist(S(s, σk)xk , ω(B, s))k→∞−→ 0.




From that and the compactness of ω(B, s), it is easy to see that{S(s, σk)xk}k∈N has a subsequence (that we again denote byS(s, σk)xk) for some y ∈ ω(B, s). It follows from the continuity ofS(t, s) that S(t, s)y = x . Hence ω(B, t) = S(t, s)ω(B, s).

Now we prove the assertion about connectedness of ω(B, t).Suppose that ω(B, t) is disconnected , then ω(B, t) is a disjointunion of two compact sets (consequently, separated by a positivedistance), but ω(B, t) pullback attracts B and this is acontradiction with the fact that S(t, s)B is connected and containsω(B, t).




LemmaSuppose that B is non-empty subset of X and that, for eacht ∈ T, there exists σt 6 t such that

⋃s6σt

S(t, s)B is compact.Then, for each t ∈ T, ω(B, t) is non-empty, compact, pullbackattracts B at time t and {ω(B, t) : t ∈ T} is invariant.




Proof: Since⋃

s6σ S(t, s)B is non-empty and compact for eachσ 6 σt , we have that ω(B, t) is non-empty and compact.

We show that ω(B, t) pullback attracts B at time t. If not, thereexists ε > 0 and sequences {xk}k∈N in B, {σk}k∈N in T with

σk 6 t, σkk→∞−→ −∞ with dist(S(t, σk)xk , ω(B, t)) > ε for all

k ∈ N.




Since⋃

s6σtS(t, s)B is compact and for some k0 ∈ N,

{S(t, σk)xk , k > k0} ⊂⋃s6σt

S(t, s)B,

it follows that {S(t, σk)xk : k ∈ N} has a convergent subsequencefor some y ∈ ω(B, t).

This contradiction shows that ω(B, t) attracts B at time t.

From Lemma 7, ω(B, t) is invariant and the proof is complete.


nonlinear dynamical systems fourteenth class · attractors for evolution processes existence of...

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