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Gradient semigroupsDynamically gradient semigroups

Nonlinear dynamical systemsSixth Class

Alexandre Nolasco de Carvalho

September 12, 2017

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

Gradient semigroupsDynamically gradient semigroups

Gradient semigroups

Definition (Gradient Semigroups)

A semigroup {T (t) : t ≥ 0} with an invariant set Ξ is gradientrelatively to Ξ if there is a continuous function V : X → R suchthat

(i) T+ 3 t 7→ V (T (t)x) ∈ R is decreasing for each x ∈ X ;

(ii) If x is such that V (T (t)x) = V (x) for all t ∈ T+, then x ∈ Ξ.

(iii) V is constant in each connected component of Ξ.

A function V : X → R with these properties is called a Lyapunovfunction for {T (t) : t > 0}.

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

Gradient semigroupsDynamically gradient semigroups

Lemma (α and ω limits in Ξ)

Let {T (t) : t > 0} be a gradient semigroup relative to an invariantset Ξ. Then, ω(x) ⊂ Ξ for each x ∈ X and, if there is a globalsolution φ : T→ X through x , then αφ(x) ⊂ Ξ.

Furthermore, if Ξ is a disjoint union of closed invariant setsΞ1, · · · ,Ξn and x ∈ X , then ω(x) ⊂ Ξi , for some 1 ≤ i ≤ n, and ifhere is a global solution φ : T→ X through x , then αφ(x) ⊂ Ξj forsome 1 ≤ j ≤ n.

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

Gradient semigroupsDynamically gradient semigroups

Theorem (Characterization of Attractors)

If {T (t) : t > 0} is an is eventually bounded and asymptoticallycompact semigroup which is gradient relatively to a boundedinvariant set Ξ, then it has a global attractor A = W u(Ξ), where

W u(Ξ) := {y ∈ X : there is a global solution φ : T→ X

with φ(0) = y such that φ(t)t→−∞−→ Ξ}

is the unstable set of Ξ.

If Ξ =⋃n

i=1 Ξi where Ξ = {Ξ1, · · · ,Ξn} is a disjoint collection ofclosed invariant sets, then A = ∪ni=1W

u(Ξi ).

Finally, if there is a bounded connected set B that contains A,then A is connected.

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

Gradient semigroupsDynamically gradient semigroups

Lemma (Near invariant sets)

Let {T (t) : t > 0} be a semigroup and Ξ be a compact invariantset for {T (t) : t > 0}. Given t > 0 and ε > 0, there exists δ > 0such that {T (s)y : 0 6 s 6 t, y ∈ Oδ(Ξ)} ⊂ Oε(Ξ).

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

Gradient semigroupsDynamically gradient semigroups

Lemma (Chain of Local Attractors)

Let {T (t) : t > 0} be a gradient semigroup relative to an invariantset Ξ. Suppose that {T (t) : t > 0} has a global attractor A, thatΞ =

⋃ni=1 Ξi with Ξ = {Ξ1, · · · ,Ξn} being a disjoint collection of

closed invariant sets, n ∈ N∗, and that the associated Lyapunovfunction is constant on each Ξi , 1 ≤ i ≤ n. LetV (Ξ) = {n1, · · · , np} with ni < ni+1, 1 6 i 6 p − 1.

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

Gradient semigroupsDynamically gradient semigroups

If 1 6 j 6 p − 1 and nj 6 r < nj+1, then Xr = {z ∈ X : V (z) 6 r}is positively invariant under the action of {T (t) : t > 0} and{Tr (t) : t > 0}, the restriction of {T (t) : t > 0} to Xr , has aglobal attractor A(j) given by

A(j) = ∪{W u(Ξ`) : V (Ξ`) 6 nj}.

In particular, V (z) 6 nj for z ∈ A(j), n1 = min{V (x) : x ∈ X} andA(1) = ∪{Ξ ∈ Ξ : V (Ξ) = n1} consists of all asymptotically stableinvariant sets; that is, for each Ξ ∈ Ξ with Ξ ⊂ A(1) there is anε > 0 such that T (t)x

t→∞−→ Ξ whenever x ∈ Oε(Ξ).

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

Dynamically gradient semigroups

Now we present the notion of dynamically gradient semigroups.These semigroups have the dynamical properties of a gradientsemigroup, but its definition do not require the existence of aLyapunov function.

Under natural assumptions we show that dynamically gradientsemigroups are gradient and that dynamically gradient semigroupsare stable under perturbations.

Let {T (t) : t ≥ 0} be a semigroup with a global attractor A whichcontains a disjoint collection of isolated invariant setsΞ = {Ξ1, · · ·Ξn}. We define:

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

DefinitionLet Ξ ∈ Ξ and 0 < ε < ε0 := 1

2 min1≤i<j≤n

dist(Ξi ,Ξj). An ε−chain

from Ξ to Ξ consists of

1. A subcollection {Ξ`1 , · · · ,Ξ`k} of Ξ with Ξ`1 = Ξ =: Ξ`k+1;

2. Points {ξ1, · · · , ξk} in X , with d(ξi ,Ξ`i ) < ε, for i = 1, · · · , k ;

3. Positive real numbers {t1, · · · , tk} and {τ1, · · · , τk} with0 < τi < ti , for all i = 1, · · · , k such that

d(T (ti )ξi ,Ξ`i+1) < ε, for all i = 1, · · · , k

andd(T (τi )ξi ,∪kj=1Oε0(Ξ`j )) > 0.

An isolated invariant set Ξ ∈ Ξ is chain recurrent if there existsδ ∈ (0, ε0) and ε−chains from Ξ to Ξ, for each ε ∈ (0, δ).

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

ε���ε0

Ξ1sT (t1)ξ1

ξ1q q

qT (τ1)ξ1

6

�

Ξ3

ε@@@ε0

sΞ2s��ε

ε0

��εε0

sΞ1

qT (τ3)ξ3

?

I

T (τ2)ξ2

3T (τ1)ξ1

q qT (t3)ξ3

ξ1

qqT (t1)ξ1

ξ2

q qT (t2)ξ2

ξ3

Figure: Examples of ε−chains

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

We are now ready to define dynamically gradient semigroups.

DefinitionLet 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.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

LemmaLet {T (t) : t > 0} be a a dynamically gradient semigroup relativelyto a disjoint collection of isolated invariant sets Ξ = {Ξ1, · · · ,Ξn},with a global attractor A and let 0<2δ0<min16i<j6n dist(Ξi ,Ξj).

Then given 0 < δ < δ0, there exist a δ′ > 0 such that, ifd(z0,Ξi ) < δ′, 1 6 i 6 n, and for some t1 > 0 we haved(T (t1)z0,Ξi ) > δ, then d(T (t)z0,Ξi ) > δ′ for all t > t1.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

Prova: Assume that, for some 1 6 i 6 n, there is a sequence{uk}k∈N in X with d(uk ,Ξi ) <

1k and sequences σk < tk in T+

such that d(T (σk)uk ,Ξi ) > δ and d(T (tk)uk ,Ξi ) <1k . That

contradicts (G2) and proves the result.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

LemmaLet {T (t) : t > 0} semigroup with a global attractor A, a disjointcollection of isolated invariant sets Ξ = {Ξ1, · · ·Ξn} such that{T (t) : t > 0} satisfies (G1).

Given 0 < 2δ < min{d(Ξi ,Ξj) : 1 ≤ i < j ≤ n} and a bounded setB ⊂ X , there exist positive numbers t0 = t0(δ,B) such that{T (t)u0 : 0 ≤ t ≤ t0} ∩

⋃ni=1Oδ(Ξi ) 6= ∅ for all u0 ∈ B.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

Proof: We argue by contradiction. Assume that there is asequence {uk}k∈N in B and a sequence {tk}k∈N in T+ with

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

⋃ni=1Oδ(Ξi ) = ∅.

Extracting subsequences we have that there is a bounded globalsolution ξ :T→X of {T (t): t> 0} such that T (t + tk

2 )uk → ξ(t)uniformly in compact subsets of T. Hence, ξ(t) /∈

⋃ni=1Oδ(Ξi ) for

all t ∈ T and this contradicts (G1).

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

Now we prove that, for a dynamically gradient semigroup, theω−limit set of a point is a contained in one of the isolatedinvariant sets. We note that condition (G1) is imposed only forsolutions in the global attractor.

LemmaSuppose that {T (t) : t > 0} is a dynamically gradient semigroupwith disjoint collection of isolated invariant sets Ξ = {Ξ1, · · · ,Ξn}and global attractor A. Given x ∈ X there is a Ξj ∈ Ξ such that

d(T (t)x ,Ξj)t→∞−→ 0.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

Proof: It follows from Lemma 8 that, given δ ∈ (0, δ0) there is aδ′ ∈ (0, δ) such that d(y ,Ξi ) < δ′ and for some ty ,δ > 0,d(T (ty ,δ)y ,Ξi ) > δ, then d(T (t)v ,Ξi ) > δ′ for all t > ty ,δ. Onthe other hand, since γ+(x) is bounded, it follows from Lemma 9that, given δ′ there exists tδ′ = tδ′(γ

+(x)) ∈ T such that, for eachy ∈ γ+(x),

{T (t)y : 0 6 t 6 tδ′} ∩ ∪ni=1Bδ′(Ξi ) 6= ∅.

From the fact that Ξ is finite, there exists Ξj ∈ Ξ and, for each,δ ∈ (0, δ0), a sδ ∈ T+ such that T (s)x ∈ Bδ(Ξj) for all s > sδ.This completes the proof.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

Homoclinic structures or heteroclinic cycles

Later, we will prove the topological structural stability ofdynamically gradient semigroups; that is, small autonomousperturbations of dynamically gradient semigroups are stilldynamically gradient semigroups. To that end, the followingconcept is crucial and is a viable replacement for condition (G2) inDefinition 7.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

DefinitionLet {T (t) : t > 0} be a semigroup with disjoint collection ofisolated invariant sets Ξ = {Ξ1, · · ·Ξn}, assume that it has aglobal attractor A and let 0 < 2δ < min16i<j6n d(Ξi ,Ξj).

A homoclinic structure in A is a collection {Ξ`1 , · · · ,Ξ`k} ⊂ Ξ anda collection of bounded global solutions {ξi : R→ X , 1 ≤ i ≤ k}such that, with Ξ`k+1

:= Ξ`1 ,

limt→−∞

ξi (t) = Ξ`i , limt→+∞

ξi (t) = Ξ`i+1, 1 ≤ i ≤ k ;

andmin

j=1,··· ,ksupt∈R

d(ξj(t),∪ni=1Oδ(Ξi )) > 0. (1)

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

sΞ`3

Iξ2(t)qsΞ`2

:ξ1(t)qs

Ξ`1

?ξ3(t)q

Figure: Example of homoclinic structure

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

RemarkCondition (1) has a technical nature, and it is used only for thecase when k = 1, since a solution ξ : R→ A such that ξ(t) ∈ Ξi

for all t ∈ R and some i ∈ {1, · · · , n} in our definition is not ahomoclinic structure.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

LemmaIf {T (t) : t > 0} is a semigroup with a disjoint collection ofisolated invariant sets Ξ = {Ξ1, · · · ,Ξn} which has a globalattractor A and satisfies (G1), then (G2) is satisfied if and only ifA does not have any homoclinic structure.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

Proof: Assuming that A has a homoclinic structure it is easy tosee that the isolated invariant sets in it are chain recurrent.

On the other hand, if Ξ ∈ Ξ is chain recurrent, there is a subset{Ξk1 , · · · ,Ξk`}, Ξ = Ξk1 = Ξk` , which we denote (after reorderingof Ξ) by {Ξ1, · · · ,Ξ`} and, for each positive integer k, pointsxk1 , · · · , xk` and positive numbers tk1 , · · · , tk` such that

d(xki ,Ξi ) <1

k, d(T (tki )xki ,Ξi+1) <

1

k, 1 ≤ i ≤ `.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

We may assume that⋃∞

k=1

⋃t∈[0,tki ] T (t)xki

⋂Ξj 6= ∅ if j is

different from i and i + 1 (adding more isolated invariant sets tothe 1

k chains if needed).

Choose 0 < 2δ < min{d(Ξi ,Ξj) : 1 ≤ i < j ≤ n} and chooseτki > 0 such that d(T (t)xki ,Ξi ) < δ, for all 0 ≤ t < τki andd(T (τki )xki ,Ξi ) ≥ δ.

It is clear from the continuity of {T (t) : t ≥ 0} and from theinvariance of Ξi that τki →∞ as k →∞ (see Lemma 4).

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

For t ∈ [−τki ,∞) let ξki (t) = T (τki + t)xki . Taking subsequenceswe define the global solutions ξi : T→ X by ξi (t) = limk→∞ ξ

ki (t).

Since each ξi (t) must converge to an equilibrium solution ast → +∞ and as t → −∞ and since ξi (t) ∈ Oδ(Ξi ) for all t ≤ 0we have that ξi (t)→ Ξi as t → −∞.

Since⋃∞

k=1

⋃t∈[0,tki ] T (t)xki

⋂Ξj 6= ∅, j different from i and

i + 1, we must have that (from (G2)) ξ(t)t→∞−→ Ξi+1.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

The collection of isolated invariant sets {Ξ1, · · · ,Ξ`} ⊂ Ξ and theset of global solutions {ξi : T→ X , 1 ≤ i ≤ `} are such thatΞ = Ξ1 = Ξ` and

limt→−∞

ξi (t) = Ξi , limt→+∞

ξi (t) = Ξi+1, 1 ≤ i ≤ `.

Hence A has a homoclinic structure.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

Corollary

If {T (t) : t > 0} is a dinamically gradient semigroup with adisjoint set of isolated invariant sets Ξ = {Ξ1, · · · ,Ξn} and A isits global attractor, then there are isolated invariant sets Ξα andΞω such that Ξα has trivial stable set in A; that is,W s(Ξα) ∩ A = Ξα where

W s(Ξα) := {y ∈ X : T (t)y → Ξα as t →∞},

and Ξω has trivial unstable set; that is, W u(Ξω) = Ξω where

W u(Ξω) := {y ∈ X : there is a global solution ξ :T→X

such that ξ(0)=y and ξ(t)t→−∞−→ Ξω}.

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

Gradient semigroupsDynamically gradient semigroups

Homoclinic structures or heteroclinic cycles

Proof: Let us prove the existence of at least one isolated invariantset Ξω with a trivial unstable set.

If for each Ξi , 1 ≤ i ≤ n, there is a global solution ξi : T→ X suchthat ξi (t)→ Ξi as t → −∞, then it is easy to see that there mustbe a homoclinic structure; in fact; choose ξ1 : T→ A such that

ξ(t)t→−∞−→ Ξ1 =: Ξ`1 . From (G1), ξ1(t)

t→∞−→ Ξ`2 ∈ Ξ with`2 6= `1.

Choose ξ2 : T→ X such that ξ(t)t→−∞−→ Ξ`2 and let `3 be such

that ξ2(t)t→∞−→ Ξ`3 . In a finite number of steps we arrive at a

contradiction with (G2).

The existence of Ξα is similar.

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