Alexandre Nolasco de Carvalho - icmc.usp.br · Topological structural stability Nonlinear dynamical systems Seventh Class Alexandre Nolasco de Carvalho September 14, 2017 Alexandre

Topological structural stability

Nonlinear dynamical systemsSeventh Class

Alexandre Nolasco de Carvalho

September 14, 2017

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



We are now ready to prove that dynamically gradient semigroupsare stable under perturbations ensuring that the structure of theattractors of dynamically gradient semigroups is robust underperturbations.

To present the result on robustness of the structure of attractorsunder small (autonomous) perturbations, we start with recallingthe definition of dynamically gradient semigroups and establishingthe meaning of small perturbation:



Recall the definition of dynamically gradient semigroups.

Definition (Dynamically Gradient Semigroup)

Let X be a metric space, {T (t) : t > 0} be a semigroup in X witha global attractor A and a disjoint collection of isolated invariantsets Ξ = {Ξ1, · · · ,Ξn} in A . We say that {T (t) : t > 0} is adynamically gradient semigroup relatively to Ξ if the following twoconditions are satisfied:

(G1) Any global solution ξ : T→ X in A satisfies

limt→−∞

dist(ξ(t),Ξi ) = 0 and limt→∞

dist(ξ(t),Ξj) = 0,

for some 1 ≤ i , j ≤ n.

(G2) Ξ does not contain any chain recurrent isolated invariant set(or equivalently A does not contain any homoclinicstructure).



Definition (Continuity and collective asymptotic compactness)

We say that a family of semigroups {Tη(t) : t ≥ 0}η∈[0,1] is

a) continuos at η = 0 if for each M > 0 and compact subset Kof X we have

sup(t,x)∈[0,M]×K

d(Tη(t)x ,T0(t)x)η→0→ 0,

b) collectivelly asymptotically compact at η = 0 if givensequences {ηk}k∈N ⊂ (0, 1], {xk}k∈N bounded in X ,

{tk}k∈N ⊂ [0,∞) with ηkk→∞−→ 0 and tk

k→∞−→ ∞ we have that{Tηk (tk)xk} is relatively compact in X .



Now we will consider dynamically gradient semigroups underperturbations. To that end, the following result will play a key role.



LemmaLet {Tη(t) : t ≥ 0}η∈[0,1] be a family of semigroups which iscontinuous and collectively asymptotically compact at η = 0. Letηn

n→∞−→ 0, an < bn < cn such that bn − an, cn − bnn→∞−→ 0, and set

Jn = [an, cn]. Let ξn : Jn → X be a solution of {Tηn(t) : t ≥ 0}and assume that

Θ =⋃n

ξn(Jn) is bounded.

Then there exists a subsequence {nk}k∈N and a bounded globalsolution ξ0 : R→ X of {T0(t) : t ≥ 0} such that ξnk → ξ0uniformly on compact subintervals of R.



Proof: As a first step, find a subsequence n0,k such that

ξn0,k (bn) = Tηn0,k (bn − an)ξn0,k (an)k→∞−→ x0

for some x0 ∈ X as k →∞. Define ξ0 : [0,∞)→ X byξ0(t) = T0(t)x0 for t > 0.

Then

ξ0(t) = T0(t)x0 = limk→∞

Tηn0,k (t)ξn0,k (bn) = limk→∞

ξn0,k (t),

with the convergence being uniform in compact subsets of R; inparticular, ξ0([0,∞)) ⊂ Θ.



Suppose that we have constructed nested subsequences {nj ,k}, for0 ≤ j ≤ m − 1, (that is, {nj+1,k : k ∈ N} ⊆ {nj ,k : k ∈ N}), suchthat ξnj,k (·+ bn)→ ξ0(·) as k →∞ uniformly on compactsubintervals of [−j ,∞), where ξ0 : [−(m − 1),∞)→ X is asolution of {T0(t) : t > 0} that lies in Θ.

Now find a subsequence nm,k such that ξnm,k(−m + bn)→ xm for

some xm ∈ X , and extend the definition of ξ0(·) to [−m,∞) bysetting ξ0(s) = T0(s −m)xm for s ∈ [−m,−(m − 1)).



Then

T0(1)xm = limk→∞

Tηnm,k(1)ξnm,k

(−m + bn)

= limk→∞

ξnm,k(−(m − 1) + bn) = ξ0(−(m − 1)),

and so ξ0 : [−m,∞) is a solution of {T0(t) : t > 0}. Clearlyξnm,k

→ ξ0 uniformly on compact subintervals of [−m,∞), and

ξ0([−m,∞)) ⊂ Θ.

Finally, if we let nk = nk,k , the sequence {ξnk (·)}k∈N and thecorresponding global solution ξ0 have all the properties stated inthe theorem.



Corollary

Let {Tη(t) : t > 0}η∈[0,1] be a family of semigroups which iscontinuous and collectively asymptotically compact at η = 0.

Assume that, for each η ∈ [0, 1], {Tη(t) : t > 0} has a globalattractor Aη and that

⋃η∈[0,1] Aη is bounded.

If ηkk→∞−→ 0, ξk is a global solution of {Tηk (t) : t > 0} in Aηk and

sk ∈ R, for each k ∈ N; there is a subsequence {ξkj}j∈N of {ξk}k∈Nsuch that {ξkj (·+ skj )}j∈N converges uniformly in bounded intervalsof R to a global solution ξ0 of {T0(t) : t > 0} with ξ(R) ⊂ A0.

In particular, the family {Aη : η ∈ [0, 1]} is upper semicontinuousat η = 0.



LemmaLet {Tη(t) : t > 0}η∈[0,1] be a family of semigroups which iscontinuous and collectively asymptotically compact at η = 0.Assume also that {Tη(t) : t > 0}η∈[0,1] is collectively bounded;that is, given a bounded set B ⊂ X , there exists ηB ∈ (0, 1] such

that⋃t≥0

⋃η∈[0,ηB ]

Tη(t)B is bounded.

Assume that, for each η ∈ [0, 1], {Tη(t) : t > 0} has a globalattractor Aη with a disjoint collection of isolated invariant setsΞη = {Ξ1,η, · · ·Ξn,η} and such that {T0(t) : t > 0} satisfies (G1).

Given 0 < 2δ < min{d(Ξi ,0,Ξj ,0) : 1 ≤ i < j ≤ n} and a boundedset B ⊂ X , there are positive numbers t0 = t0(δ,B) and η0 > 0such that {Tη(t)u0 : 0 ≤ t ≤ t0} ∩

⋃ni=1Oδ(Ξi ,0) 6= ∅ for all

u0 ∈ B and η ∈ [0, η0].

Moreover, if maxj=1,··· ,n dH(Ξj ,η,Ξj ,0)η→0−→ 0, we can replace Ξi ,0

in the previous conclusion by Ξi ,η.



LemmaLet {Tη(t) : t > 0}η∈[0,1] be a family of semigroups which iscontinuous and collectively asymptotically compact at η = 0.Assume also that {Tη(t) : t > 0}η∈[0,1] is collectively bounded;that is, given a bounded set B ⊂ X , there exists ηB ∈ (0, 1] such

that⋃t≥0

⋃η∈[0,ηB ]

Tη(t)B is bounded.

Assume that, for each η ∈ [0, 1], {Tη(t) : t > 0} has a globalattractor Aη with a disjoint collection of isolated invariant setsΞη = {Ξ1,η, · · ·Ξn,η} and such that {T0(t) : t > 0} satisfies (G1).

Given 0 < 2δ < min{d(Ξi ,0,Ξj ,0) : 1 ≤ i < j ≤ n} and a boundedset B ⊂ X , there are positive numbers t0 = t0(δ,B) and η0 > 0such that {Tη(t)u0 : 0 ≤ t ≤ t0} ∩

⋃ni=1Oδ(Ξi ,0) 6= ∅ for all

u0 ∈ B and η ∈ [0, η0].

Moreover, if maxj=1,··· ,n dH(Ξj ,η,Ξj ,0)η→0−→ 0, we can replace Ξi ,0

in the previous conclusion by Ξi ,η.



Proof: We argue by contradiction. Assume that there is asequence {uk}k∈N in B, a sequence {ηk}k∈N in [0, 1] such that

ηkk→∞−→ 0 and a sequence of positive numbers {tk}k∈N with

tkk→∞−→ ∞ such that {Tηk (t)uk : 0 ≤ t ≤ tk}∩

⋃ni=1Oδ(Ξi ,0) = ∅.

Extracting subsequences we have (from Lemma 3 with ak = 0,bk = tk

2 and ck = tk) that there is a subsequence {kj} and a globalbounded solution ξ : R→ X of {T0(t) : t > 0} such that

Tηkj (t +tkj2 )ukj

j→∞−→ ξ(t) uniformly in compact subsets of R.

Hence, ξ(t) /∈⋃n

i=1Oδ(Ξi ,0) for all t∈R and this contradicts (G1).



LemmaLet {Tη(t) : t > 0}η∈[0,1] be a family of semigroups which iscontinuous and collectively asymptotically compact at η = 0.Assume also that {Tη(t) : t > 0}η∈[0,1] is collectively bounded.

Assume that, for each η ∈ [0, 1], {Tη(t) : t > 0} has a globalattractor Aη with a disjoint collection of isolated invariant sets

Ξη = {Ξ1,η, · · ·Ξn,η}, that max16i6n dH(Ξi ,η,Ξi ,0)η→0−→ 0 and that

{T0(t) : t > 0} is dynamically gradient.

Given 0 < 2δ < min16i<j6n dist(Ξi ,0,Ξj ,0), there are η0 > 0 andδ′ > 0 (independet of η ∈ [0, η0]) such that, for η∈ [0, η0] andd(z0,Ξi ,η)<δ′, 16 i6n, if d(Tη(t1)z0,Ξi ,η) > δ, for some t1 > 0,then d(Tη(t)z0,Ξi ,η) > δ′ for all t > t1.



Proof: Assume that, for some 1 6 i 6 n, there exist a sequence

{zk}k∈N in X , {ηk}k∈N with ηkk→∞−→ 0, d(zk ,Ξi ,ηk ) < 1

k andsequences σk < τk in R+ such that d(Tηk (σk)zk ,Ξi ,ηk ) > δ andd(Tηk (τk)zk ,Ξi ,ηk ) < 1

k .

This will lead us to a contradiction with property (G2) of{T0(t) : t > 0}, in fact with the non-existence of homoclinicstructures.



Note that {σk}k∈N can be chosen such that Tηk (s)uk ∈Oδ(Ξi ,ηk )

for all s < σk and, from Lemma 3, σkk→∞−→ ∞.

Taking subsequences we obtain a global solution ξ0 : R→ A such

that ξ0(τ)τ→−∞−→ Ξi ,0.

From the fact that {T0(t) : t > 0} is dynamically gradient we havethat there exists j 6= i such that ξ0(τ)

τ→∞−→ Ξj ,0.



From the fact that ξ0(τ)τ→∞−→ Ξj ,0 and from the construction of

ξ0, there exists a sebsequence {ηkm}m∈N of {ηk}k∈N andsubsequences {σkm}, {tkm} in R+, σkm < tkm < τkm , such that

d(Tηkm (σkm)zkm ,Ξj ,ηkm) <

1

mand d(Tηkm (tkm)zk ,Ξj ,ηkm

) > δ.



Proceeding exactly as in the previous step we obtain a solution

ξ1 : R→ A such that ξ1(s)s→−∞−→ Ξj ,0.

From the fact that {T0(t) : t > 0} is dynamically gradient we havethat there is a Ξ`,0∈Ξ0, Ξ`,0 /∈{Ξi ,0,Ξj ,0} so that ξ1(s)

s→∞−→ Ξ`,0.

Continuing with this process we arrive at a contradiction in a finitenumber of steps.



We are now ready to prove that small perturbations of adinamically gradient semigroups are dinamically gradientsemigroups; that is, for a dinamically gradient semigroup,conditions (G1) and (G2) are stable under perturbations.



TheoremLet {Tη(t) : t > 0}η∈[0,1] be a family of semigroups which iscontinuous and collectively bounded and collectively asymptoticallycompact at η = 0. Assume that

(a) {Tη(t) : t > 0} has a global attractor Aη, for each η ∈ [0, 1]and that

⋃η∈[0,1] Aη is bounded;

(b) for each η ∈ [0, 1], Aη contains a disjoint collection of isolated

invariant sets Ξη={Ξ1,η, · · ·,Ξn,η} with dH(Ξi ,η,Ξi ,0)η→0−→ 0,

for each i = 1, · · · , n;

(c) there exist δ > 0 and η0 ∈ (0, 1] such that Ξi ,η is the maximalinvariant set in Oδ(Ξi ,η), 1 ≤ i ≤ n and 0 ≤ η ≤ η0;

(d) {T0(t) : t > 0} is a dynamically gradient relatively to thedisjoint collection of isolated invariant sets Ξ0.

Then there exists η1>0 such that, {Tη(t) : t>0} is a dynamicallygradient semigroup with respect to Ξη, for each η∈(0, η1).



Proof: Note firstly that Ξi ,η → Ξi ,0 and together with hypothesis(c) this implies that there exists δ > 0 such that, for suitably smallη, if a solution ξη satisfies d(ξη(t),Ξi ,0) ≤ δ for all t > t0 and forsome t0 > 0, then ξη(t)→ Ξi ,η as t →∞.



We argue by contradiction to prove that for all suitably small η,{Tη(t) : t > 0} satisfies (G1).

Assume that there is a sequence ηkk→∞−→ 0 and corresponding

global solutions ξk in Aηk such that

lim supt→∞

dist(ξk(t),∪ni=1Ξi ,ηk ) > δ′. (1)



From Lemma 3 with sk=0 for all k , there is a subsequence (whichwe again denote by {ξk}) and a global solution ξ0 : R→ X of{T0(t) : t > 0} such that ξk(t)→ ξ0(t) uniformly in compactsubsets of R.

From (G1) ξ0(t)→ Ξi ,0, for some 1 ≤ i ≤ n. It follows that, foreach ` ∈ N∗, there are t` > 0 and k` ∈ N such thatd(ξk(t`),Ξi ,0) < 1

` , for each k > k`.

From (1), there exists t ′` > t` such that d(ξk`(t),Ξi ,0) < δ′ for allt ∈ [t`, t

′`) and d(ξk`(t`),Ξi ,0) = δ′.



From the continuity of {T0(t) : t > 0} and the invariance of Ξi ,0

we have that t ′` − t``→∞−→ ∞.

From Lemma 3 with s = {t ′`}∞`=1, taking subsequences if necessary,there is a global solution ξ1 : R→ X of {T0(t) : t > 0} such thatξ1(t) = lim`→∞ ξk`(t + t ′`) uniformly in compact subsets of R.

Then d(ξ1(t),Ξi ,0)6δ′ for all t ≤ 0 and, consequently,ξ1(t)→Ξi ,0 as t → −∞. From (G1) and (G2), ξ1(t)→ Ξj ,0 ∈ Ξ0

as t →∞ with i 6= j .



From the fact that ξk` → ξ1(t) uniformly in compact subsets of Rwe have that, for each m ∈ N, there is a time tm > 0 and km ∈ Nsuch that d(ξk(tm),Ξj) <

1m for all k > km.

Again, from (1), there exists t ′m > tm such that d(ξkm(t),Ξj) < δ′

for all t ∈ [tm, t′m) and d(ξkm(t ′m),Ξj) = δ′.

Proceeding exactly as before we obtain a global solutionξ2 : R→ X of {T0(t) : t > 0} such that ξ2(t)→ Ξj as t → −∞and ξ2(t)→ Ξr ,0 ∈ Ξ0 as t →∞ with r /∈ {i , j}.

In a finite number of steps we construct a homoclinic structure andarrive at a contradiction with (G2).



This proves that there is a η1 > 0 such that, for all global solutionξη in Aη with η 6 η1, we have that

limt→∞

d(ξη(t),Ξi ,η) = 0.

The proof that there is a η2 > 0 such that, for all global solutionξη in Aη with η 6 η2, we have that

limt→−∞

d(ξη(t),Ξj ,η) = 0,

is similar. This completes the proof that, for all suitably small η,{Tη(t) : t > 0} satisfies (G1).



Let us prove that, for all suitably small η, {Tη(t) : t > 0} satisfies(G2). Again we argue by contradiction.

Assume that there is a collection Ξ1,0, · · · ,Ξp,0 in Ξ0, a sequenceηk → 0, global solutions ξk in Aηk , and times tk1 , · · · , tkp such that

d(ξk(0),Ξi ,0) <1

k, d(ξk(tki ),Ξi+1,0) <

1

k, 1 ≤ i ≤ p,

where Ξ1,0 = Ξp+1,0.

One can choose a fixed sequence Ξ1,0, . . . ,Ξp,0 since there are onlya finite number of possible sequences from Ξ0.

Proceeding as in the proof of (G1) we construct a homoclinicstructure for {T0(t) : t > 0} and arrive at a contradiction.



ExerciseAssume that {Tη(t) : t > 0}η∈[0,1] is a family of semigroups whichis continuous and collectively asymptotically compact at η = 0,{Tη(t) : t > 0} has a global attractor Aη, for each η ∈ [0, 1], and⋃η∈[0,1] Aη is bounded. Then, for each τ > 0, there is an ε > 0

and η0 > 0 such that

⋃0≤t≤τ

⋃η∈[0,η0]

Tη(t)

Oε ⋃η∈[0,η0]

Aη

is bounded.

Use this to improve the hypothesis of the previous results.


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Alexandre Nolasco de Carvalho - icmc.usp.br · Topological structural stability Nonlinear dynamical systems Seventh Class Alexandre Nolasco de Carvalho September 14, 2017 Alexandre