Alexandre Nolasco de Carvalho - USPCharacterization of the semigroups with global attractors Lemma Let fT(t) : t 2T+gbe a semigroup in X and B ˆX such that!(B) is compact and attracts

Characterization of the semigroups with global attractors

Nonlinear Dynamical SystemsThird Class

Alexandre Nolasco de Carvalho

August 24, 2017

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


Recall that we have proved the following result

LemmaLet {T (t) : t ∈ T+} be a semigroup in X . If B ⊂ X , thenT (t)ω(B) ⊂ ω(B) for all t ∈ T+. If B is such that ω(B) iscompact and attracts B, then ω(B) is invariant.



LemmaLet {T (t) : t ∈ T+} be a semigroup in X and B ⊂ X such thatω(B) is compact and attracts B.

I If T = R and B is connected, then ω(B) is connected;

I If T = Z, B is connected and B ⊃ ω(B), then ω(B) isconnected.



Proof: Suppose that T=Z and ω(B) is not connected, then ω(B)is disjoint union of two compact sets (separated by a positive

distance 2ρ), but ω(B) attracts B, so distH(T (t)(B), ω(B))t→∞−→ 0,

but this implies (from the fact that T (t)B is connected) thatT (t)B must be contained in the ρ neighborhood of one of theconnected components of ω(B) for t suitably large. From Lemma 1we have that T (t)B contains ω(B) which leads to a contradiction.

The proof of the case T = R follows immediately from the factthat ω(B) = ∩t≥0γt(B) and from the fact that γt(B) is connectedfor each t ≥ 0 (since [0,∞)× X 3 (s, x) 7→ T (s)x ∈ X iscontinuous and takes the connected set [t,∞)× B to theconnected set γt(B)).



Proof: Suppose that T=Z and ω(B) is not connected, then ω(B)is disjoint union of two compact sets (separated by a positive

distance 2ρ), but ω(B) attracts B, so distH(T (t)(B), ω(B))t→∞−→ 0,

but this implies (from the fact that T (t)B is connected) thatT (t)B must be contained in the ρ neighborhood of one of theconnected components of ω(B) for t suitably large. From Lemma 1we have that T (t)B contains ω(B) which leads to a contradiction.

The proof of the case T = R follows immediately from the factthat ω(B) = ∩t≥0γt(B) and from the fact that γt(B) is connectedfor each t ≥ 0 (since [0,∞)× X 3 (s, x) 7→ T (s)x ∈ X iscontinuous and takes the connected set [t,∞)× B to theconnected set γt(B)).



LemmaIf B is a non-empty subset of X such that γ+t0 (B) is compact, forsome t0 ∈ T+, then ω(B) is non-empty, compact, invariant andω(B) attracts B.



Proof: For each t ∈ T+, t ≥ t0, γ+t (B) is non-empty and

compact. It follows from the fact that the family {γ+t (B) : t ≥ t0}has the finite intersection property that ω(B) =

⋂t≥t0 γ

+t (B) is

non-empty and compact.

Now let us show that ω(B) attracts B. Suppose not, then thereare ε0 > 0 and sequences{xn : n ∈ N} in B, {tn : n ∈ T+} in T+

with tnn→∞−→ ∞, such that d(T (tn)xn, ω(B)) > ε0 for all n ∈ N.



Proof: For each t ∈ T+, t ≥ t0, γ+t (B) is non-empty and

compact. It follows from the fact that the family {γ+t (B) : t ≥ t0}has the finite intersection property that ω(B) =

⋂t≥t0 γ

+t (B) is

non-empty and compact.

Now let us show that ω(B) attracts B. Suppose not, then thereare ε0 > 0 and sequences{xn : n ∈ N} in B, {tn : n ∈ T+} in T+

with tnn→∞−→ ∞, such that d(T (tn)xn, ω(B)) > ε0 for all n ∈ N.



Since γ+t0 (B) is compact and {T (tn)xn, n ≥ n1} ⊂ γ+t0 (B) for some

n1 ∈ N, there are subsequences tnjj→∞−→ ∞ and xnj ∈ B such that

T (tnj )xnj is convergent to some y ∈ X . It follows that y ∈ ω(B)and d(y , ω(B) ≥ ε0 leading to a contradiction. Hence ω(B)attracts B.

Now, from Lemma (1) it follows that ω(B) is invariant and theproof is complete.




The following concept plays an important role in thecharacterization of the semigroups which posses a global attractor.

DefinitionA semigroup {T (t) : t ∈ T+} is said to be asymptotically compactif, for each non-empty, closed, bounded and positively invariantsubset B of X , there is a compact set JB ⊂ B that attracts Bunder the action of {T (t) : t ≥ 0}.



LemmaAssume that {T (t) : t ∈ T+} is asymptotically compact and thatB is a subset of X such that γ+t0 (B) is bounded, for some t0 ∈ T+.Then ω(B) is non-empty, compact, invariant and attracts B underthe action of {T (t) : t ≥ 0}.



Proof: Since T (t) is continuous and T (t)γ+t0 (B) ⊂ γ+t0 (B), it

follows that T (t)γ+t0 (B) ⊂ γ+t0 (B) for all t > 0. Since{T (t) : t ∈ T+} is asymptotically compact we have that there

exists a compact set J ⊂ γ+t0 (B) that attracts γ+t0 (B).

Hence, there are sequences εnn→∞−→ 0 and tn

n→∞−→ ∞ so that

T (t)(γ+t0 (B))⊂Oεn(J) for all t > tn. So, ∅ 6= ω(B) ⊂ J.

Since ω(B) is closed and J is compact, we have that ω(B) iscompact.



Let us show that ω(B) attracts B. If not, there are ε0 > 0 andsequences xn∈B and tn

n→∞−→ ∞ such that d(T (tn)xn, ω(B))>ε0.

From the compactness of J, from the fact that J attracts γ+t0 (B)(therefore B) and from a previous result, there are subsequences

xnj ∈B, tnjj→∞−→∞ and z∈J such that T (tnj )xnj

j→∞−→ z .

It follows that z ∈ ω(B) and dist(z , ω(B)) > ε0, which leads to acontradiction and proves that ω(B) attracts B. Consequently,ω(B) is non-empty, compact and attracts B and, from Lemma 1,the invariance follows.



Proposition

Let {T (t) : t ∈ T+} be a semigroup in a metric space X . Supposethat {T (tn)xn : n ∈ N} is relatively compact whenever{xn : n ∈ N} and {T (tn)xn : n ∈ N} are bounded in X andtn

n→∞−→ ∞. Then {T (t) : t ∈ T+} is asymptotically compact.

Conversely, if {T (t) : t ≥ 0} is semigroup which is eventuallybounded and asymptotically compact, then {T (tn)xn : n ∈ N} iseventually compact whenever {xn : n ∈ N} is a bounded sequencein X and tn

n→∞−→ ∞



Proof: Suppose that {T (tn)xn : n ∈ N} is relatively compactwhenever {xn : n ∈ N} and {T (tn)xn : n ∈ N} are bounded in Xand tn

n→∞−→ ∞.

Let B be a non-empty, closed and bounded subset of X such thatT (t)(B) ⊂ B, for all t > 0. It is not difficult to se that ω(B) ⊂ Bis non-empty, invariant, compact and attracts B showing that{T (t) : t ∈ T+} is asymptotically compact.



On the other hand, if {T (t) : t ≥ 0} is an eventually boundedsemigroup and {xn : n ∈ N} is a bounded sequence in X , there is a

t0 > 0 such that B = γ+t0 ({xn : n ∈ N}) is a bounded set.

Since B is positively invariant and {T (t) : t ≥ 0} is asymptoticallycompact, there is a compact set J in X that attracts B.

In particular {T (tn)xn : n ∈ N} converges to J as n tends toinfinity and therefore it is relatively compact.



Definition (Study)

A semigroup {T (t) : t ∈ T+} is said to be conditionally eventuallycompact if, given a bounded and positively invariant set B, thereexists tB ∈ T+ such that T (tB)B is compact. A semigroup{T (t) : t ∈ T+} is said to be eventually compact if, given abounded set B, there exists tB ∈ T+ such that T (tB)B iscompact.



Theorem (Exercise)

A semigroup which is conditionally eventually compact isasymptotically compact.

Proof: Let B ⊂ X be a non-empty, closed, bounded and such thatT (t)B ⊂ B for all t > 0. Then, since {T (t) : t ∈ T+} is

conditionally eventually compact we have that γ+t (B) is compactfor sufficiently large t. Hence, from a prevous lemma,

ω(B) =⋂

t∈T+ γ+t (B) is non-empty, compact and attracts B. This

shows that {T (t) : t ∈ T+} is asymptotically compact.



DefinitionWe say that a semigroup {T (t) : t ∈ T+} is point dissipative(bounded dissipative/compact dissipative) if there exist a boundedsubset B ⊂ X that attracts points (bounded subsets/compactsubsets) of X .

RemarkIn the above definition we can exchange the words attracts by theword absorbs without changing the notions that are being defined.



LemmaLet {T (t) : t ∈ T+} be a point dissipative and asymptoticallycompact semigroup. If the orbit of each compact subset of X isbounded, then {T (t) : t ∈ T+} is compact dissipative.



Proof: Since {T (t) : t ∈ T+} is point dissipative, there is abounded non-empty set B that absorbs points of X .

Let U = {x ∈ B : γ+(x) ⊂ B}. Since B absorbs points, we havethat U is non-empty. Clearly γ+(U) = U, U is bounded andabsorbs points.

We also know that T (t)(γ+(U)) ⊂ γ+(U), t > 0, and that{T (t) : t ∈ T+} is asymptotically compact. Therefore, there is acompact set K , with K ⊂ γ+(U) = U, such that K attracts U andconsequently K attracts points of X .



Now we show that there is a neighborhood V of K such thatγ+t (V ) is bounded for some t ∈ T+.

If that is not the case, there are sequences xn ∈ X , xn → y ∈ Kand tn →∞ such that {T (tn)xn : n ∈ N} is not bounded.

Consider A = {xn : n ∈ N}, hence A is compact and γ+t (A) is notbounded for each t ∈ T+. That contradicts our assumption theassumption that the orbit of a compact subset of X is bounded.



Let V be a neighborhood of K and tV ∈ T+ be such that γ+tV (V )is bounded.

Since K attracts points of X and T (t) is continuous, for all x ∈ Xthere is a neighborhood Ox of x and tx > 0 such thatT (t)(Ox) ⊂ γ+tV (V ) for t > tx ; that is, γ+tV (V ) absorbs aneighborhood of x for each x ∈ X .

From this it follows easily that γ+tV (V ) absorbs compact subsets ofX and that {T (t) : t ∈ T+} is compact dissipative.



Proposition

Let X be a metric space and {T (t) : t ∈ T+} be a semigroup. If

K is a compact set that attracts itself, then ω(K ) =⋂t∈T+

T (t)K .



Prova: Clearly⋂t∈T+

T (t)K ⊂ ω(K ). Now, for the converse

inclusion, we use a previous proposition with K1 = K to ensurethat ω(K ) ⊂ K and γ+(K ) is relatively compact. So, ω(K ) isnon-empty, compact, invariant, attracts K and

ω(K ) = T (t)ω(K ) ⊂ T (t)K , for all t ∈ T+,

which proves the result.



The following theorem characterizes the semigroups which haveglobal attractors.

TheoremA semigroup {T (t) : t ∈ T+} is eventually bounded, pointdissipative and asymptotically compact if and only if it has a globalattractor A.



Proof: Since {T (t) : t ∈ T+} is asymptotically compact, pointdissipative and eventually bounded, from the previous lemma it iscompact dissipative.

Let C be a bounded subset which absorbs compact subsets of Xand B = {x ∈ C : γ+(x) ⊂ C}.

Clearly B absorbs compact subsets of X , T (t)B ⊂ B and, since{T (t) : t ∈ T+} is asymptotically compact, there is a compact setK ⊂ B that attracts B.



It follows that K attracts compact subsets of X .

The set A=ω(K ) is non-empty, compact, invariant and attractsK . If J⊂X is compact ω(J)⊂K and ω(J) = T (s)ω(J) ⊂ T (s)Kfor each s > 0.

From the previous proposition ω(J) ⊂ ∩s∈T+T (s)K = ω(K ) andω(K ) attracts J.



Let B be a bounded subset X , since {T (t) : t ∈ T+} is eventuallybounded and asymptotically compact, it follows from Lemma 5that ω(B) is non-empty, compact, invariant and attracts B.

Since ω(B) is compact and invariant we have that ω(B) ⊂ A andconsequently A attracts B.

Clearly, if {T (t) : t ∈ T+} has a global attractor, it is eventuallybounded, point dissipative and asymptotically compact.


Documents

Alexandre Nolasco de Carvalho - USPCharacterization of the semigroups with global attractors Lemma Let fT(t) : t 2T+gbe a semigroup in X and B ˆX such that!(B) is compact and attracts