A reduction principle for attractors with applications to ... · 1) and (N 2). Pretopology....

Departamento deMatemáticas,

UAM-Iztapalapa

Luis [email protected]

abstract

Basic conceptsand notationsSets and quasifilters.Notational conventions

General systems

Neighbourhood Systems.Upper semicontinuity.

Topological conceptsbased on axioms (N1)and (N2). Pretopology.

Reductionprinciple forattractorsIterated attractionsinvolving quasifilters

A reduction principle forattractors

Stability, Stableattractors.Asymptoticstability.Stable and unstableattractors.

A neceessary condition

The threshold principle

A sufficient condition forstability depending onlyon an attraction property.

Reduction principle forstability

The case ofasymptoticcompactnessThe limit set of a set.Asymptotic compactness

Atractors

Asymptotic stability

An aplication tosystem coupledof partialdifferentialequations

A reduction principle for attractorswith applications to systems of partial

differential equations

Luis AguirreUniversidad Autónoma Metropolitana-Iztapalapa

MéxicoVIII Workshop on Partial Differential Equations

Río de Janeiro Brazil, August 25-28, 2009


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A very general reduction principle for attraction ispresented which permits reducing the problem for acomposed system to that of its subsystems, withemphasis on the stability of the attractor.

The attractor property of a closed, invariant set M isproved under the conditions that a closed invariantsubspace of the state space is attractive and weaklyattracted by M.

An analogous reduction theorem is presented forstable attractors (asymptotic stability).


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The theory is developed in the context of aneighbourhood space with axioms defining apretopology.

Uniqueness is not required; only uppersemi-continuity with respect to initial conditions isassumed. The time scale is an ordered Abeliansemigroup.

All concepts are defined as relations betweenquasifilters.

As an example for the applications to systems ininfinite-dimensional spaces a pair of coupleddiffusion equations with Dirichlet and Neumannboundary conditions, respectively, is analyzed.


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In general, sets are denoted by capital italics, theirelements by the corresponding small italics. Non-emptycollections of non-empty sets, called quasifilters, aredenoted by capital script letters, their elements by thecorresponding capital italics.

Quasifilters consisting of a single set A will be denoted by{A}.


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An expression involving a quasifilter A and a set is to beunderstood as the quasifilter obtained by substituting Aby each of its elements A, for instance:

A ∩ B = {A ∩ B|A ∈ A}.

An expression involving two quasifilters is to beunderstood as the quasifilter obtained by substitutingeach of the quasifilters independently by one of itselements.


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By quasifilter in a given set X is meant a quasifilter all ofwhose elements are subsets of X . If every point of Xbelongs to some set of a quasifilter in X , we say that thequasifilter is on X .


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A quasifilter A is said to be coarser than a quasifilter B, orB finer than A, in symbols

A −−< B,

iff every element of A contains an element of B as asubset:

(∀A ∈ A)(∃B ∈ B) A ⊃ B.


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A general evolutionary system, or simply general system,in the sense in which the term is used here, consists of atriplet (X ,T ,Φ), where X denotes an arbitrary non-emptyset (state space), T an ordered Abelian semigroup (timescale), its elements being denoted by the letters σ and τ ,and Φ a function (the dynamics) from the product setX × T onto a quasifilter in X :

Φ : X × T → 2X\{∅}.


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The elements x of the state space X will be referred to aspoints. We define xτ = Φ(x , τ), and analogously for a setA ⊂ X ,

Aτ = {xτ | x ∈ A}.

Moreover, we will use the following notation:

Aτ =⋃σ

{Aσ | σ ≥ τ}

(and similarly for a single point x). In particular, the set

A0 =⋃σ

{Aσ|σ ∈ T}

is called the orbit corresponding to A, where 0 is theneutral element of the semigroup T .

If A0 = A, A is called invariant.


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We call Aτ the τ -section and Aτ the τ -tail of the orbitcorresponding to A and define the segment [0, τ ] as theset

A[0, τ ] = {Aσ | σ ≤ τ}.

For the latter, we will use the abbreviations xτ , Aτ . It willbe assumed that the following semi-group axiom issatisfied:

(xσ)τ = x(σ + τ)


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For notational reasons it is necessary to associate to thetime scale T a quasifilter defined as

{{τ}| τ ∈ T}. (1.1)

In this special case, we will use for the quasifilter sodefined the same letter T . In all expressions in which Tappears it is to be understood as the quasifilter definedby (1.1). In this sense, and taking into account theconvention concerning expressions involving a quasifilterand a set, for instance,

AT = {Aτ | τ ∈ T}.


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We associate to every non-empty set A ⊂ X a quasifilterin X , called neighbourhood system of A, denoting it byNA, and assuming that the following axioms are satisfied.

(N1) N ∈ NA implies N ⊃ A.(N2) B ⊃ A implies NA ⊃ NB.

Throughout the rest of the conference, it will be assumedthat the space X satisfies these two axioms. Furtheraxioms will be added whenever they are required.


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In the case of a set consisting of a single point x , we willwrite Nx instead of N{x}, calling Nx the neighbourhoodsystem of the point x .

Definition 1The system is upper semicontinuous, uniformly withrespect to A ⊂ X, iff

∀ τ ∈ T : N (Aτ) −−< (NA)τ . (1.2)


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Open and closed sets are defined as usual:I A is open iff ∀a ∈ A : {A} −−< Na;I B is closed iff it is the complement of an open set;I compactness is also defined as usual.


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From the definition of openness it follows immediately thatunions of open sets are open. However, the openness offinite intersections of open sets does not follow from theaxioms (N1) and (N2), as the following example shows:define X = {x , y , z}, Nx = {N ′,X}, Ny = {N ′′,X},Nz = {N ′,N ′′,X}, where N ′ = {x , z}, N ′′ = {y , z}.Axioms (N1) and (N2) are obviously satisfied. The sets N ′

and N ′′ are open, but their intersection, {z}, is not open.


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We call a quasifilter on the set X , which is closed underunions, a pretopology on X , and the set X equipped withthis quasifilter, a pretopological space.Interior and closure of a set A are defined as usual andthe closure is denoted by A.


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We also assume that every set consisting of a simglepoint is closed. This is equivalent to no point belonging toall neighbourhoods of some other point. A simpleexample of a space satisfying (N1) and (N2) and not theabove is the following:

X = {x , y}, Nx ={{x},X

}, Ny = {X}.

For this reason we assume throughout the paper that thefollowing neighbourhood axiom is satisfied.

(N3) No point is contained in all neighbourhoods of someother point.

This is known as axiom T1.


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M is a non-empty closed, invariant proper subset of X .

Definition 2.1. M attracts A weakly, in symbols

A w−→M,

iff∀A : NM −−< AT .


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Definition 2.2. M attracts V, in symbols,

V −→ M,

iff∀V : NM −−< VT .


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Definition 2.3. A attracts V, in symbols,

V −→ A,

iff(∀V )(∃A) NA −−< VT .

The subsequent theorem requires the followingneighbourhood axiom.


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(N4). If A ⊂ X ,

(∀N1 ∈ NA)(∀N2 ∈ NA)(∃N ∈ NA) N1 ∈ NN ∧ N2 ∈ NN.

Theorem 2.4. Let A,V be quasifilters in X. Suppose thefollowing conditions are satisfied:

1. M attracts A weakly;2. A attracts V;3. the upper semicontinuity condition (1.2) holds on

every A;4. neighbourhood axiom (N4) is satisfied.

Then M attracts V.


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In [ ], a similar proof was given which assumed uniformstability of M. The above proof shows that this conditionis redundant.As a matter of routine, Theorem 2.4 can be extended tothe case of more than two successive attractions.Theorem 2.4’. Under the hypotheses 3 and 4 of Theorem2.4 (2. holding on all A and Bk ),

V −→ A w−→B1w−→· · · w−→Bn

w−→M

implies V −→ M.


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We now consider the case where there exists an invariantset Y between M and X :

φ 6= M & Y & X .

In this case the system (X ,T ,Φ) induces a subsystem(Y ,T ,ΦY ) on Y , where ΦY denotes the restriction of Φ toY × T . The quasifilter A of the preceding section is nowassumed to be a quasifilter in Y and moreover, of theform

A = Y ∩ B (2.8)

(= {Y ∩ B|B ∈ B}), where B is a quasifilter on X (takingthe place of the quasifilter of bounded sets in the case ofa normed space).


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Theorem 2.4 must now be adapted to the new setting.More specifically, the hypothesis V → A has to beadjusted to the case where A is of the form (2.8). In Fact,it will be replaced by the two conditions V → Y and all V0remaining in some B, in other words, a kind ofboundedness condition with the sets B acting as bounds,which indeed they are when the space is normed. Thisleads to the necessity of introducing a second relationbetween quasifilters which is appropriate for formalizingboundedness type properties.


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Definition 2.5. The quasifilter A is dominated by thequasifilter B, in symbol,

A )< B,

iff(∀A)(∃B) A ⊂ B.

(between the relations −−< and )< there exists aduality: assign to every quasifilter A its co-quasifilterA∗ = {X\A| A ∈ A}, then A −−< B iff A∗ )< B∗.) Thecondition described above can then be written in the form

V0 )< B (2.9)

(V0 denoting, of course, the quasifilter {V0 |V ∈ V}).The relation (2.9) will also be expressed by saying that Vis B-bounded (and that each individual V is B-bounded).Each B, such that V ⊂ B, will be called a bound for V .


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Theorem 2.6.

V → M if

1. V is B-bounded, associating to every V one of itsbounds BV ;

2. ∀V : V → Y ∩ BV ;3. Y ∩ B w−→M;4. Axiom (N4) holds;5. the upper semi-continuity condition (1.2) holds for all

A = Y ∩ B, B ∈ B.

The proof is the same as of theorem 2.4, only with Areplaced by Y ∩ BV .


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Definition 2.7. M is a global attractor iff it attracts everypoint of X .

Corollary 2.8. If in Theorem 2.6 V is a quasifilter on X, Mis a global attractor .

If bounds are required to remain below a certain B, thequasifilter B must be restricted correspondingly.

We now turn to the question of how different kinds ofattraction are related to the quasifilters V which areattracted to M.


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1. If V consists of the individual points of aneighbourhood A of M, we say M attracts Apointwise. The set A is called the region of attractionof M.

2. If V consists of a neighbourhood of every point of aneighbourhood A of M, M attracts A locally uniformly.

3. If V contains among its elements a neighbourhood ofM, M is called a uniform attractor.

4. If V coincides with the quasifilter B, we call M aB-attractor.


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If X is a topological space, the only useful way ofspecifying a neighbourhood system for a set M is todefine NM as the (quasi)filter of all sets which containopen sets containing M. These then are the topologicalneighbourhoods of M.

If a metric is defined on X , the other natural way ofdefining a neighbourhood system is to define NM as the(quasi)filter consisting of all subsets of X which containthe ε-neighbourhood; for some ε of M.


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These then are the metric, or uniform neighbourhoods ofM. We denote the two neighbourhood systems by NtMand NmM, respectively. If M is compact, both areequivalent: NtM >−−−< NmM, i.e. NmM −−< Nt andNtM −−< NmM while, in general, only the first of the tworelations holds.(N. Bourbaki [ ] distinguishes between the two kinds ofneighbourhoods, calling the topological ones voisinage,the uniform ones entourage).


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The concept of attraction in use at present, e.g. in [ ],and which we have adopted in this paper, is a ratherbroad one in the sense that it includes unstableattractors, for instance those involving homoclinic orbits.(Actually, the original concept of attractor, first introducedin [ ], was of this kind.) Formerly, for instance in thework of R. Thom, the property of stability was included inthe concept of attractor.


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From the point of view of the usual applications, unstableattractors are of little use, because they allow orbitsemanating from a set M to go far astray before eventuallyreturning to M. For this reason it seems convenient tocomplete the present theory by the consideration ofstable attraction, originally known as asymptotic stability.This will inevitably include a reduction principle forstability. (The simplest example of an unstable attractor isgiven by the equation on the circle, θ = sin θ/2,0 ≤ θ < 2π. The point θ = 0 is a global unstableattractor.)


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Let M be a non-empty, proper closed, invariant subset ofX . The following property of M is crucial for thedevelopment of the present theory.

Condition C. ∀t : NM −−< (NM)τ (explicitly,(∀N ∈ NM)(∀τ)(∃N1 ∈ NM) Nτ

1 ⊂ N).


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In order to be able to state sufficient conditions for this tohold, we assume the following neighbourhood axioms.

(N5) Every neighbourhood of a point x contains an openneighbourhood of x .

(N6) If M is compact, every open set containing M is aneighbourhood of M.

Proposition 3.1. Under axioms (N5) and (N6), theinvariant set M satisfies condition C if it is compact.


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Condition C, of course, does not hold in general fornon-compact sets M (it fails, for instance, for the systemx = 0, y = x , with M = {(x , y)|x ∈ R, y = 0} and NMbeing defined as {{(x , y)|x ∈ R, |y | < ε}, ε > 0}).


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As a special case of definition (2.1) we have that A w−→Miff

NM −−< AT .

Definition 3.2. If V and W are subsets of the state spaceX , we call restriction of the orbit V0 to W the set

V0|W =⋃

v∈V

{vτ |vτ ⊂W}.

V0|W consists of the sections of orbits with initial points inV and contained in W .

The corresponding concept, for quasifiltersW instead ofsets W , is defined as the quasifilter

V0|W = {V0|W : W ∈ W}.


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Definition 3.3. The pair (M,A) satisfies the thresholdcondition (or Seibert’s condition [ ]) iff

NM −−< (NM)0|NA. (3.3)

In conventional notation, this is equivalent to

(∀N ∈ NM)(∃N ′ ∈ NM)(∃U ∈ NA) x ∈ N ′ and xτ ⊂ U

imply xτ ⊂ N.

U is called a threshold for (N,N ′).

Lemma 3.4. (Threshold Lemma). If M attracts A weakly,and if (1.2), Condition C and axiom (N4) hold, then (M,A)satisfies the threshold condition (3.3).


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Lemma 3.4 does not hold, in general, for non-compactsets M, because in the case, condition C is general notsatisfied.


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Definition 3.5. The set M is stable iff

NM −−< (NM)0. (3.13)

We recall that V is attracted to M, V → M, iff

NM −−< VT .

Theorem 3.6. If M attracts a neighbourhood V of M, andcondition C and axiom (N4) hold, then M is stable. In thiscase M is called asymptotically stable.


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Let Y and M be closed invariant sets as in subsection 2.2.

Definition 3.7. Y is locally stable near M iff

∃N1 ∈ NM : NY −−< (NY ∩ N1)0∣∣N1. (3.18)

We assume that there is given a quasifilter B as in 2.2,and that NM contains a neighbourhood N whichB-bounded;

∃N ∈ NM : N )< B.


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Lemma 3.8. If Y is locally stable near M and (M,Y )satisfies the threshold condition, and if condition C andthe axiom (N4) are satisfied, then M is stable.


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Definition 3.9. M is U-asymptotically stable iff it is aU-attractor and stable.

Theorem 3.10. The closed invariant set M isU-asymptotically stable if the following conditions aresatisfied.


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1. Y is locally stable near M.2. U is B-bounded; to every U a bound BU is assigned.3. For every U, Y ∩ BU attracts U.4. M attracts Y ∩ B weakly.5. The upper semicontinuity condition (1.2) holds for all

Y ∩ B.6. M satisfies condition C.7. The space X satisfies the axiom (N4).

The Theorem follows directly from Theorem 2.6, lemma3.4, and lemma 3.8, putting A = Y .


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Theorem 3.11. The closed invariant set M isasymptotically stable if

1. there exists a neighbourhood V of M which isB-bounded with bound BV (V0 ⊂ BV );

2. Y ∩ BV attracts V ;3. M attracts Y ∩ BV weakly;4. The hypotheses 5, 6 and 7 of Theorem 3.10 are

satisfied.This is an immediate consequence of Theorems 2.6 and3.6.


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Actually, the theorem states that M is V -asymptoticallystable (attracting V and stable) for the V that figures incondition 1.In the case of X being a metric space, with theneighbourhood systems defined in one of the waysspecified at the end of section 2, the axioms (N1) through(N6) are satisfied and so the results formulated in thecontext of a neigbourhood space can be applied.


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In this section, we assume a condition of asymptoticcompactness, which in most important applications tosystems in infinite-dimension, dimensional spaces suchas partial differential equations of hyperbolic or parabolictype, or functional-differential equations with delay, issatisfied. In the other hand, one loses some ordinarydifferential equations, is satisfied. On the other hand, oneloses some ordinary differential equations, the simplestone being x = y , y = 0.


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Definition 4.1. The limit set of the set A is defined as

LA =⋂{Aτ |τ ∈ T}.

(This is the exact equivalent of the definition of limit set ofa point, to which it reduces in the case where A consistsof a single point.)It is easy to see that LA is invariant.

Definition 4.2. The system is asymptotically compactwith respect to the set A iff

LA 6= ∅, LA is compact, A→ LA.


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We denote this condition by (AC)A. If the same threeconditions hold for every point of A we say the system isasymptotically compact on A and denote this condition by(AC)A. If the first of these conditions holds for everyelement of a quasifilter A, we write (AC)A.


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The next theorem requires the following neighbourhoodaxiom.

(N7) A closed set and a point outside of it possess disjointneighborhoods.

Theorem 4.3. If both (AC)A and (N7) hold, and A isattracted to a closed set M, then A is attracted to acompact subset MA of M which is the limit set of A.


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The most important case is the one where the quasifilterA is defined as consisting of the one-point sets {x},x ∈ X .Here ∀x : x → Lx (compact) and

⋃x Lx is a global

attractor.


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In the last mentioned case, the attractor may be unstable.Consider, for instance, the flow on the infinite cylinderz ∈ R, εθ ∈ [0,2π) given by the equations z = 0,θ = sin θ

2 . Here θ = 0 is an unstable, global attractor.If the set U is AC, according to Proposition 3.1, LUsatisfies the condition C, provided the axiom (N5) and(N6) are satisfied, in particular iff X is a metric space.Therefore Theorems 3.10 and 4.3 yield the followingresult.


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Theorem 4.4. Given a cuasifilter U , the set M =⋃

U LU isasimptotically stable if the following conditions aresatisfied.

1. U is asymptotically compact.;2. The set Y (containing M as a subset) is locally stable

near each LU.;3. Conditions 2. through 5. of Theorem 3.10 are

satisfied.;4. The space X satisfies the axioms (N4), (N5), (N6).


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Now we consider the following system of two nonlinearlycoupled partial differential equations with boundaryconditions:

ut = uxx , u(0, t) = u(1, t) = 0,vt = vyy + uv , v(0, t) = v(1, t) = 0,

(1)

with Dirichlet boundary conditions. The system iswell-posed for X = H0 ×H0,where

H0 = {χ ∈ L2(0,1) χ, χ′ absolutely continuous,

χ′′ ∈ L2(0,1) and χ(0) = χ(0) = 0}.

X is a linear space with norm


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‖ (ϕ,ψ) ‖X :=‖ ϕ ‖X0 + ‖ ψ ‖L2

‖ ϕ ‖X0 :=‖ ϕ ‖L2 + ‖ ϕ′′ ‖L2


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But, what happen with the system

ut = uxx , ux (0, t) = ux (1, t) = 0,vt = vyy + uv , vy (0, t) = vy (1, t) = 0,

(2)

with Neumann boundary conditions?Systems like this is my homework, now!

A reduction principle for attractors with applications to ... · 1) and (N 2). Pretopology....

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