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Gradient semigroups Dynamically gradient semigroups Nonlinear dynamical systems Sixth Class Alexandre Nolasco de Carvalho September 12, 2017 Alexandre N. Carvalho - USP/S˜ ao Carlos Second Semester of 2017

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Nonlinear dynamical systemsSixth Class

Alexandre Nolasco de Carvalho

September 12, 2017

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

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

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

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

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

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

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

Homoclinic structures or heteroclinic cycles

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

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

Homoclinic structures or heteroclinic cycles

ε���ε0

Ξ1sT (t1)ξ1

ξ1q q

qT (τ1)ξ1

6

Ξ3

ε@@@ε0

sΞ2s��ε

ε0

��εε0

sΞ1

qq

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

Homoclinic structures or heteroclinic cycles

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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