Global and Pullback Attractors of Set-Valued Skew Product Flows

Global and pullback attractors of set-valued skew product flows T. Caraballo Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160 41080 Sevilla, Spain email: [email protected] P.E. Kloeden FB Mathematik, Johann Wolfgang Goethe Universit¨at, D-60054 Frankfurt am Main, Germany email: klo[email protected] and P. Mar´ın–Rubio Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160 41080 Sevilla, Spain email: [email protected] February 26, 2003 Abstract We investigate the asymptotic behaviour of a general set-valued skew product flow (SVSPF), that is, a set-valued cocycle mapping (coming from a nonautonomous differential equation or inclusion) driven by another, autonomous, system. Absorptivity conditions which ensure the existence of several types of attractors for such set-valued systems are established. The topological properties of and relations between these attractors, in forward and pullback senses and their strong and weak versions are analyzed. Several illustrative examples are also provided. Key words: set-valued skew product flow, pullback attraction, weak invariance 1991 MSC: 34D45 37B25 37B75 58C06 1 Introduction Set-valued analysis and attainability set functions are used to handle problems arising from differential equations without uniqueness, differential inclusions, or problems arising in control theory, viability theory, finances and economics among 1 others, and have been widely studied by several authors in the last decades (cf. [3, 20, 19, 24, 13, 14] among others). The investigation of the asymptotic behaviour of these phenomena, global, strong or weak stability and attraction properties, needs the concept of pullback attractors when non-autonomous equations are considered, in particular stochastic and random ones which are intrinsically non-autonomous (cf. [11, 10, 21, 15] and [16, 5] for the weak pullback case). However, the non-autonomous case can also be viewed within the framework of skew-product flows, which allows us to transform the problem into an autonomous one (cf. [22, 6, 7, 9, 17, 8]) and to apply the classical theory for autonomous systems in a different, extended, phase space. On the other hand, if such an attractor exists, then we can recover the dynamics in the original phase space Rdand its asymptotic properties by using suitable projections, which is important as this original phase space is often the one that is of interest or meaningful in modelling. The paper is organized as follows: In Section 2, we recall the concepts of skewproduct flows coming from a cocycle set-valued mapping with a related driving system. Conditions for existence of strong and weak attractors for a autonomous set-valued semidynamical system in an abstract metric space and its properties are analyzed in Section 3. Then, in Section 4 we apply these results to the case of a skew-product flow as introduced above, distinguishing between strong and weak cases. Here, the sectorial components in the original phase space Rdof the attractor in the extended phase space (under several suitable conditions) will be analyzed, namely, for strong and weak asymptotic concepts as well as for their connections with the underlying non-autonomous semi-flow. The formalism used here allows us to obtain several different conclusions from precedent works ([4, 5]). We also illustrate our theory with some examples and, for the sake of clarity, give most of the proofs at the end of the paper. 2 Set-valued skew product flows Hereafter, we will use the following notation: P(X) and K(X) for the set of nonempty and nonempty compact subsets of a given space X, respectively; H∗for the Hausdorff semi-distance, H∗(A, B) = supa∈Adist(a, B), and Hfor the Hausdorff distance, H(A, B) = max(H∗(A, B), H∗(B, A)). To establish an appropriate framework for our analysis, we consider given an autonomous driving system, θ:R×P→P, where Pis a metric space, i.e. a group of homeomorphisms under composition on Pwith the properties i) θ0p=pfor all p∈P, ii) θt+sp=θtθspfor all s, t ∈R. iii) the mapping (t, p)7→ θtpis continuous. For instance, one can think of an ordinary differential system in P=Rlwith an autonomous globally Lipschitz and dissipative vectorfield g, i.e. p0=g(p). 2 A set-valued skew product flow (SVSPF for short) consists of an autonomous driving system θon a metric space Pand a set-valued cocycle mapping (attainability set mapping) Φ : R+×P×Rd→ K(Rd) satisfying the following properties: 1. Compactness Φ(t, p, x) is a nonempty compact subset of Rdfor all t≥0, p∈P, x∈Rd; 2. Initial condition Φ(0, p, x) = {x} for all p∈Pand x∈Rd; 3. Cocycle property Φ(t+s, p, x) = Φ (t, θsp, Φ(s, p, x)) for all t≥0, p∈P,x∈Rd; 4. Continuity in time lim s→tH(Φ(s, p, x),Φ(t, p, x)) = 0 for all s,t≥0 and all p∈Pand x∈Rd; 5. Upper semi continuity in parameter and initial conditions lim q→p,y→xH∗(Φ(t, q, y),Φ(t, p, x)) = 0 uniformly in t∈[T0, T1] for any 0 ≤T0< T1<∞for all p∈Pand x∈Rd. Remark 1. Assumptions 4 and 5 imply that Φis globally upper semi continuous (u.s.c.), i.e. if (tn, pn, xn)→(t, p, x)as n→ ∞, then H∗(Φ(tn, pn, xn),Φ(t, p, x)) → 0. Indeed, H∗(Φ(tn, pn, xn),Φ(t, p, x)) ≤H∗(Φ(tn, pn, xn),Φ(tn, p, x)) +H∗(Φ(tn, p, x),Φ(t, p, x)) →0as n→ ∞, since the first term in the right hand side goes to zero by the u.s.c. in the second and third variables uniformly in time (property 5), and the second term goes to zero by the continuity (hence u.s.c.) of Φon its first variable (property 4). A trajectory of a set-valued cocycle Φ is a single-valued mapping φp: [0, T]→ Rdwhich, for the indicated p∈P, satisfies φp(t)∈Φ(t−s, θsp, φp(s)) for all 0 ≤s≤t≤T. (1) A trajectory φis called an entire trajectory if it is defined on all of Rand satisfies (1) for all s≤t. (If necessary a particular p∈P, we will use the notation p-trajectory for the above definition). Now let us denote Tp,x([0, T]) = {φp,trajectory φp(0) = x}. Then, we can establish the following result in a similar way as in [3, 20, 13]: Theorem 2. The following properties holds: 3 1. Tp,x([0, T]) 6=∅(there exist trajectories for all p, x,and T > 0) 2. Tp,x([0, T]) ⊆C([0, T]; Rd)(continuity) 3. Tp,x([0, T]) is a compact subset of C([0, T]; Rd) 4. Tpn,xn([0, T]) → Tp,x([0, T ]) (in H∗on C([0, T]; Rd)) as pn→p,xn→x. Remark 3. It is worth noticing that Theorem 2 also holds true if we consider a Banach space Xinstead of Rd(see the proof in Section 7). We now consider a general autonomous set-valued semidynamical system (SVSDS for short) as in Szeg¨o and Treccani [24], that is, a set-valued mapping Π : R+×Y→ Ywhere Yis a connected metric space satisfying suitable properties. To avoid unnecessary repetitions, such properties are the stated below for the special case in which the set Yis the extended phase space P×Rd. A particular case of an autonomous set-valued semidynamical system is be that generated by a set-valued skew product flow, namely with Π(t, p, x) = (θtp, Φ(t, p, x)) and satisfying the following properties: 1. Π(t, p, x) is nonempty and compact; 2. Π(0, p, x) = {(p, x)}; 3. The semigroup property: Π(t+s, p, x) = Π (t, Π(s, p, x)) ; 4. t7→ Π(t, p, x) is continuous in the Hausdorff metric, for all p, x, i.e. HP×Rd(Π(s, p, x),Π(t, p, x)) →0,as s→t; 5. (p, x)7→ Π(t, p, x) is upper semicontinuous in the H∗sense, i.e. H∗ P×Rd(Π(t, q, y),Π(t, p, x)) →0,as q→p, y →x, uniformly in compact intervals t∈[T1, T2]. A trajectory for an SVSDS is a single-valued mapping π: [0, T]→Ywith π(t)∈Π(t−s, π(s)) for all 0 ≤s≤t≤T. Analogously, a trajectory (or ptrajectory) for a SVSPF is a single-valued mapping πp: [0, T]→P×Rdwith πp(t)∈Π(t−s, πp(s)) for all 0 ≤s≤t≤Tand the first component of πp(0) equal to p. Proposition 4. πpis a trajectory for the SVSPF if and only if there exists a trajectory φpof Φsuch that πp(t) = (θtp, φp(t)) ∀t∈[0, T]. The result holds true for any trajectory defined in any interval of time, and also for entire trajectories. 4 Remark 5. An analogous result to Theorem 2 holds for an SVSDS Π(and therefore for an SVSPF), replacing C([0, T]; Rd)by C([0, T]; P×X)in all the statements, actually, in any time interval not necessarily in the positive half line. This is straightforward since the multi-valued mapping F:{(t, t0) : t≥t0} × P×X→ P(P×X)defined by F(t, t0, p, x) := Π(t−t0,(p, x)) satisfies the required conditions, since θand Φdo (cf. [3, 20, 13, 5]). 3 Attractors of SVSDS Now we consider a SVSDS Π : R+×Y→ P(Y) and recall the basic concepts on attractors, which we will be apply in the following section to our skew product formulation, denoting then the extended phase space space P×Rdby Y. For the sake of clarity, subscripts sand won attractors will denote strong and weak concepts. Definition 6. A strong global attractor for an SVSDS Πis a nonempty compact subset As⊂Ysatisfying 1. strong invariance: Π(t, As) = Asfor all t≥0 2. strong attraction: for every nonempty bounded subset Dof Y, distY(Π(t, D),As)→0,as t→ ∞ A weak global attractor for an SVSDS Πis a nonempty compact subset Aw⊂Y satisfying 1. weak invariance: ∀y∈ Awthere exists an entire trajectory π:R→Ywith π(0) = yand π(t)∈ Awfor all t∈R 2. weak attraction: for every nonempty bounded subset Dof Yand yn∈D, there exist trajectories πn:R+→Yand numbers τn→ ∞ with πn(0) = ynand distY(πn(τn),Aw)→0,as n→ ∞ For completeness, we recall some results ensuring the existence of such attractors. 3.1 Strong global attractor of SVSDS The most simple case of an autonomous semi-flow with a compact absorbing set Bs (i.e. for every bounded set Dthere exists tD≥0 such that Π(t, D)⊂ Bsfor all t≥tD) is well known. Assume that Bsis a nonempty compact absorbing set in Y. Without loss of generality, we can assume that Bsis Π–positively invariant (i.e. Π(t, Bs)⊂ Bs, for all t≥0). Define As=\ t≥0 Π(t, Bs). Then, a∈ Asiff a∈ Bsand there exist τn→ ∞, an∈Π(τn,Bs) with an→aas n→ ∞. 5 Proposition 7. The set Ashas the following properties: 1. It is nonempty and compact and attracts bounded sets. 2. It is Π–invariant, therefore it is a global strong attractor. Actually it is the maximal invariant compact set, and also the minimal closed set that attracts bounded sets. 3. If Π(t, x)is connected for all (t, x)∈R+×Y, then Asis also connected. Remark 8. The existence of an absorbing set for the construction of the attractor and its properties can be relaxed to that of an attracting set (see [18, Th.1]), which can be more appropriate in some other situations (e.g. hyperbolic systems). 3.2 Weak attractors of SVSDS We consider now an SVSDS Π(t, x) and establish existence of weak attractors. To this end, we introduce the concept of a weak absorbing set Bwwhich is a nonempty, compact set which in addition is •weakly positively invariant: for all b∈ Bwthere exists at least a trajectory π with π(0) = band π(t)∈ Bwfor all t≥0. •weakly absorbing: for all compact subset Dthere exists TD≥0 such that for any d∈Dthere exists a trajectory πwith π(0) = dand π(t)∈ Bwfor all t≥TD. Theorem 9. Assume there exists a weak absorbing set Bwfor an SVSDS Π. Then, there exists the maximal weak attractor Aww.r.t. Bw, which is defined as the set of points a∈ Bwsuch that there exist bn∈ Bw,τn→ ∞ and trajectories πn:R+→ Bw with πn(0) = bnand dist(πn(τn), a)→0as n→ ∞. Remark 10. As in [5, lemma 13], we observe that an entire trajectory π:R→Y satisfies that π(t)∈ Bwif and only if π(t)∈ Aw. Therefore, Awbecomes the set of points reached by entire trajectories contained in Bw. 4 Attractors of SVSPF Now specialize to Π = (θ, Φ) with our skew product structure, i.e. we consider now that Y=P×Rd. We again split our analysis into two cases concerning the strong and weak situations. 6 4.1 (Strong) Global Attractor of SVSPF Suppose that there exists a compact positive invariant absorbing set Bs⊂P×Rd for Π. Then, there exists a strong global attractor As=\ t≥0 Π(t, Bs). Let P∗= PrP(As) be the projection of Asonto the space Pand consider the decomposed notation As=[ p∈P∗ {p} × As(p). Proposition 11. Under the previous assumptions the following properties hold: 1. P∗is nonempty, compact and θtP∗=P∗. In fact, P∗is the global attractor of the (single-valued) autonomous driving system θon P. 2. As(p∗)is nonempty and compact for each p∗∈P∗. It also satisfies the invariance property As(θtp∗) = Φ(t, p∗, As(p∗)). 3. The mapping P∗3p7→ As(p)is upper semi continuous. Now consider the restriction Π∗of Π to P∗×Rd. Since θtP∗=P∗for all t∈R, it follows that Π∗is an SVSPF on P∗×Rd. Consider B∗ s=Bs∩(P∗×Rd). Then, B∗ sabsorbs sets in P∗×Rdunder Π∗(≡Π). Also B∗ sis nonempty, compact and Π∗-positively invariant, so Π∗has a maximal global attractor A∗ s=[ p∈P∗ {p} × A∗ s(p), in P∗×Rd. Then, we have the following result: Proposition 12. The strong global attractors of Πand Π∗coincide: A∗ s≡ As. 4.2 Weak Attractors of SVSPF Suppose that an SVSPF Π has a weak attractor relative to the compact weak absorbing set Bw⊂P×Rd, which is also given by Aw=[ p∈P∗ {p} × Aw(p), (again we denote P∗= PrPAw) as described above for general set-valued semi dynamical systems. Then, we have Proposition 13. The following properties hold: 1. P∗is the strong global attractor for the driving system. 2. Aw(p)is a nonempty compact set for each p∈P∗. 7 3. The map P∗3p7→ Aw(p)∈ K(Rd)is u.s.c. 4. Awis weakly invariant, i.e. if (p, a)∈ Aw, there exists an entire trajectory π= (θ, ϕp)such that ϕp(0) = aand π(t) = (θtp, ϕp(t)) ∈ Awfor all t∈R, i.e. ϕp(t)∈Aw(θtp). Moreover, Awis the largest weak invariant set in the absorbing set Bw. If we restrict Π to Π∗on P∗×Rd, we have that Π(t, (p, x)) = Π∗(t, (p, x)) for all t≥0 and (p, x)∈P∗×Rdsince P∗is θ-invariant. Define B∗ w=[ p∈P∗ {p} × Bw(p)⊂ Bw=[ p∈P {p} × Bw(p). Note that Aw(p)⊂Bw(p) for all p∈P∗. So, Bw(p) is nonempty and compact. We also have that B∗ wis weakly positive invariant since P∗is θ-invariant and Bwis weakly positive Π-invariant. Then, we obtain a maximal weak attractor for Π∗,A∗ w, with respect to B∗ wand we will use the notation A∗=[ p∈P∗ {p} × A∗ w(p). Remark 14. Note that a∈A∗ w(p)if and only if there exist sequences tn→ ∞, (pn, bn)∈ B∗ w, trajectories πn= (θ, ϕpn)with ϕpn(0) = bnand θtnpn→pand ϕpn(tn)→a. Observe that A∗ wconsists of entire Π∗-trajectories, and, since it is weak invariant, A∗ w(p) is nonempty and compact for all p∈P∗. Aw⊂ B∗ w⊂ Bwbut Awis the maximal Π-weak invariant family contained in Bw and A∗ wis Π∗-weak invariant (and so Π-weak invariant), therefore one has that A∗ w⊂ Aw. Indeed, as Π and Π∗coincide in B∗ w, entire Π-trajectories in Aware Π∗-trajectories and conversely, by the same argument of maximality, we conclude that Aw=A∗ w. Thus, we can restrict ourselves to the dynamics on B∗ wand we will study the relations between weak and strong skew-product attractors and their sections with respect to pullback weak and strong attractors in the following section. 5 Pullback structure of SVSPF attractors Once again we split our analysis into two cases: strong and weak attractors. 8 5.1 The strong case As before, Bsdenotes a Π-positive invariant compact absorbing set and B∗ s= Bs∩(P∗×Rd) is a Π∗-positive invariant compact absorbing set (we keep using the notation Bs=Sp∈P{p} × Bs(p), so Bs(p)6=∅for p∈P∗). This implies Φ-positive invariance for the sections Bs(p) in P∗: Φ(t, p, Bs(p)) ⊂Bs(θtp)∀t≥0,∀p∈P∗. Indeed, Π∗(t, B∗ s)⊂ B∗ simplies that Π(t, (p, Bs(p))) = (θtp, Φ(t, p, Bs(p))) ⊂ B∗ s=[ q∈P∗ (q, Bs(q)) what necessarily means our claim if p∈P∗. Lemma 15. Define ˆ As(p) = \ t≥0 Φ(t, θ−tp, Bs(θ−tp)) for p∈P∗. Then, ˆ As(p)is nonempty and compact for all p∈P∗. Remark 16. If Bs(p)and Φ(t, p, x)are connected for all t≥0,p∈P∗and x∈ Rd, then ˆ As(p)is also connected, since it is the intersection of a nested family of nonempty, compact, connected sets. Proposition 17. The following identities hold: ˆ As(p) = A∗ s(p) = As(p)∀p∈P∗.(2) Moreover, we also have the following result. Proposition 18. ˆ As(p)is the pullback attractor for Φon P∗×Rd. Remark 19. (i) Notice that Propositions 17 and 18 implies that the p-components of the strong global Π-attractor As, which attracts in the forward and pullback senses, are the strong pullback attractor for Φwhen the dynamics are restricteed to P∗×Rd. (ii) Here we started with a global attractor for the skew-product flow Πand have obtained a pullback attractor for Φ. The converse is not true in general, i.e. if {As(p), p ∈P∗}is a strong pullback attractor for Φ, then A=Sp∈P∗{p} × As(p)may not be a global attractor for Π(see [8] for a counterexample in the single-valued case). (iii) In general, pullback attractors for set-valued flows are only negatively invariant, and strict invariance needs additional assumptions (the easiest is a lower semi continuous property for the flow). Here, the strict invariance holds here since the driving system has a global attractor. 9 Now we prove that Tp,x([0, T ]) is compact in C([0, T]; Rd). Let be {φn} ⊂ Tp,x([0, T]). As Φ(T, p, x) is compact, there exists a subsequence {φn1(T)} ⊂ {φn(T)} converging to a point denoted φ(T). By the same reason, there exists another subsequence {φn2(T/2)} ⊂ {φn1(T/2)}converging to a point φ(T/2) and we iterate this procedure. By a diagonal argument, we obtain a subsequence relabeled again with index m, converging in all the dyadic numbers of [0, T]: φm(pT/2q)→φ(pT/2q). As {φn}are trajectories, then φn(t)∈Φ(t−s, θsp, φn(s)) for all 0 ≤s≤t≤T, in particular for dyadic numbers, whence φ(t)∈Φ(t−s, θsp, φ(s)) by the u.s.c. of Φ. To extend it to the whole interval to obtain a trajectory, we proceed as before. Let us prove that φn→φin C([0, T]; Rd). The pointwise convergence follows easily. Indeed, for any t, let us write φn(t)−φ(t) = φn(t)−φn(tD) + φn(tD)−φ(tD) + φ(tD)−φ(t), with tDa dyadic number close enough to tsuch that |φ(t0)−φ(t)| ≤ ε/3, and with H(Φ(t, p, x),Φ(t0, p, x)) ≤ε/3. Then we can choose n(tD) such that for all n≥n(tD), one has |φn(tD)−φ(tD)| ≤ ε/3. However, for the uniform convergence one needs to be more careful. We follow the proof in [20, Th.6.2]. By a contradiction argument, if it does not hold, there exist a constant ε > 0, sequences tn, with tn→t∈[0, T], and φnsuch that |φn(tn)−φ(t)|> ε. (9) Consider a dyadic number τ∈½(t, T] if t < T {T}if t=T. As {φn(tn)} ⊂ Φ([0, T], p, x), which is compact, there exists a convergent subsequence (we do not relabel it) φn(tn)→z. Then, there exists nτ∈Nsuch that for all n≥nτ, we have that tn< τ. Since φn(τ)∈Φ(τ−tn, θtnp, φn(tn)) and τis dyadic, we have φn(τ)→φ(τ) and so, the global u.s.c. of Φ implies that φ(τ)∈Φ(τ−t, θtp, z). Using now the continuity of φand the density of dyadic numbers we have: φ(t) = lim τ→tφ(τ)∈lim sup τ→t Φ(τ−t, θtp, z) = Φ(0, θtp, z) = {z}, which contradicts (9). Finally, we prove the upper semicontinuity result claimed in (4): H∗(Tpn,xn([0, T]),Tp,x([0, T ]) →0 if (pn, xn)→(p, x). According to the last section, it is equivalent to prove ε-u.s.c. We proceed again by a contradiction argument. Suppose there exist a positive constant ε > 0, a sequence of pairs (pn, xn) converging to (p, x) in P×Rd, and trajectories φn∈ Tpn,xn([0, T ]) such that φn6∈ BC([0,T ];Rd)(Tp,x([0, T ]), ε). We will prove that for a subsequence φn0, it is satisfied that φn0→φ∈ Tp,x([0, T]), which will give us the contradiction. 16 First, we have that φn(0) = xn→x. As φn(T)∈Φ(T, pn, xn), by the compact values and u.s.c. of Φ, there exists a subsequence (which we do not relabel) converging to an element φ(T) in Φ(T, p, x). The same argument can be applied to this subsequence at time T/2, and, taking a diagonal subsequence, for a countable set of numbers (the dyadic in [0, T]), defining a set of points φ(kT/2m). Of course, they satisfy the trajectory property, as far as φnare: φn(t)∈Φ(t−s, θspn, φn(s)), for dyadic 0 ≤s≤t≤T. By the u.s.c. of Φ we have φ(t)∈Φ(t−s, θsp, φ(s). The extension to the whole interval is done as above, preserving the trajectory property. To finish, the uniform convergence of φnto φis deduced as in the previous case. 7.2 Autonomous strong attractors of SVSDS. Proof of Proposition 7 Clearly, Asis nonempty since it is the intersection of a nested family of compact sets. Moreover, it is compact too. Since any bounded set is absorbed by Bs, it is enough to see that Bsis attracted by As. If not, there exist ε > 0 and a sequence xn∈Π(tn,Bs), with tn→ ∞, such that dist(xn,As)≥ε > 0. But xn∈ Bsfor all n≥n(Bs) by the absorbing property of Bs, and as it is compact, there is a subsequence converging (we do not relabel) to x∈ As, which is a contradiction. We check Π–invariance in two steps: First, we prove that Π(t, As)⊂ Asas in the single-valued case. Indeed, Π(t, As) = Π³t, \ r≥0 Π(r, Bs)´⊂\ r≥0 Π(t, Π(r, Bs)) =\ r≥0 Π(t+r, Bs) = \ r≥t Π(r, Bs) = \ r≥0 Π(r, Bs) = As where we have used the semigroup property of Π and the positive invariance of Bs. For the converse, As⊂Π(t, As), pick a∈ As. Then, there exist sequences τn→ ∞, and an∈Π(τn,Bs) with an→a. Consider t > 0 and n(t) such that for all n≥n(t), τn−t≥0. Then, for all n≥n(t): an∈Π(τn,Bs) = Π(t, Π(τn−t, Bs) and thus there exists a sequence a0 n∈Π(τn−t, Bs) with an∈Π(t, a0 n). Since Bsis compact and positively invariant, we deduce from a0 n∈Π(τn−t, Bs)⊂ Bs 17 the existence of a convergent subsequence a0 nj→a0∈ Bsif j→ ∞. Of course, τnj−t→ ∞ and therefore, a0∈ As. By the upper semicontinuity we have that H∗(Π(t, a0 nj),Π(t, a0)) →0j→ ∞, and from anj→aand anj∈Π(t, a0 nj) we obtain a∈Π(t, a0)⊂Π(t, As) as desired. Since a compact set Kthat is Π-invariant satisfies by the attraction property, dist(Π(t, K),As)→0,and we have that dist(K, As) = dist(Π(t, K),As), then dist(K, As) = 0 and so K⊂ As. On the other hand, for a closed set Battracting bounded sets, we have that dist(As, B) = dist(Π(t, As), B)→0 and therefore dist(As, B) = 0 and As⊂B. Indeed, for the last statement, it is enough to prove that Asattracts a connected bounded set ˜ Bcontaining As, which is trivial here by taking ˜ B=B(0,kBsk)⊃ Bs⊃ As. Suppose by contradiction that Asis not connected, then there exists two disjoint open sets Oi(i=1,2) with As⊂O1∪O2and As∩Oi6=∅. Denote by ωi=As∩Oi, which are closed, and therefore compact sets. Take ² > 0 such that B(ω1, ²)∩B(ω2, ²) = ∅. By the attraction property, there exists t˜ Bwith Π(t˜ B,˜ B)⊂B(As, ²). But Π(t˜ B,·) is u.s.c. and ˜ Bis connected, and so it is Π(t˜ B,˜ B). Therefore, it can only be contained in one of the sets B(ωi, ²), which contradicts the inclusions As⊂Π(t˜ B,As)⊂Π(t˜ B,˜ B). 7.3 Autonomous weak attractors of SVSDS. Proof of Theorem 9 Nonempty and compact: consider any sequences τn→ ∞ and bn∈ Bw. By the weak positive invariance of Bw, there exist trajectories πnwith πn(0) = bnand πn(t)∈ Bwfor all t≥0. In particular, an=πn(τn)∈ Bw, and by the compactness of Bwwe can extract a subsequence anjconverging to an element ain Bwas j→ ∞. Taking {τnj, bnj, anj}jas the original sequences, we have that a∈ Awwhich is therefore nonempty. To show that Awis compact, we only need to see that it is closed since is contained in the compact set Bw. Suppose ak∈ Awand ak→aas k→ ∞. Then, there exist sequences τk,n → ∞ as n→ ∞ and trajectories πk,n with πk,n(t)∈ Bw for all t≥0 and πk,n(τk,n)→akas n→ ∞. Pick nkso that |πk,nk(τk,nk)−ak| ≤ 1/k and tk+1,nk+1 ≥tk,nk+ 1 ∀k∈Z+. Then |πk,nk(τk,nk)−a| ≤ |πk,nk(τk,nk)−ak|+|ak−a| ≤ 1/k +|ak−a| → 0 as k→ ∞. Taking {πk,nk, τk,nk}kas the original sequences, we have again that a∈ Aw, which is closed as desired, and hence compact. Weak positive invariance: Take a∈ Aw, then there exist a sequence τn→ ∞ and trajectories πnwith π(t)∈ Bwfor all t≥0, such that πn(τn)→aas n→ ∞. If we denote vn(t) := πn(τn+t), it is obvious that vnis a trajectory and vn(t)∈ Bwfor 18 all t≥0 and vn(0) →a∈ Aw. Applying Barbashin’s theorem [20, 13] on an interval, say [0, T ], we obtain a convergent subsequence vnj(t)→v(t) uniformly for t∈[0, T]. Naturally, v: [0, T]→ Bwis a trajectory and v(0) = a. Moreover, v(t)∈ Awsince πnj(τnj)∈ Bw,πnj(τnj+t)→v(t) and τnj+t→ ∞. A Cantor diagonal argument shows again that we can obtain a trajectory defined on all of R+. Weak negative invariance: The same idea can be used backwards in time. For any T > 0, consider nTsuch that τn−T≥0 for all n≥nT, and write vn: [−T, 0] → Bw:s7→ vn(s) := πn(τn+s). Barbashin’s theorem can be applied successively on intervals [−T, 0], [−2T, −T], ... and by a diagonal argument we obtain the existence of a trajectory ¯v:R−→ Bw, which indeed takes values in Awas before, with ¯v(0) = a. The concatenation of v and ¯vgives us the invariance of Awas desired. Weak attraction: Let Dbe a bounded subset of Rd. Since Bwis weakly absorbing, there exists a time TD>0 such that for each dn∈Dthere exists a trajectory πnwith πn(0) = dnand πn(t)∈ Bwfor all t≥TD. By the weak positive invariance of Bw, we can consider trajectories ˜πn:R+→ Bw with ˜πn(0) = πn(TD). Since Bwis compact, for any sequence τn,k → ∞ as k→ ∞, there exist subsequences ˜πn(τn,k0)→anas k0→ ∞ for some an∈ Bw(for each n). By definition of A, we have that an∈ Aw. Define π∗ n(t) := ½πn(t) 0 ≤t≤TD, ˜πn(t−TD)t≥TD. Then, π∗ n(0) = dnand π∗ n(τn,k0+TD)→anas k0→ ∞. Pick k0 nsuch that τn,k0 n< τn+1,k0 n+1 and dist(π∗ n(τn,k0 n+TD),Aw)≤1/n. Therefore we have obtained for the trajectories π∗ nwhich start at dnthat dist(π∗ n(τn,k0 n+TD),Aw)→0. Thus, we have weak attraction. The maximality statement w.r.t. Bwcomes from its definition. 7.4 Attractors for SVSPF and their restrictions 7.4.1 Proof of proposition 11 The first assertion is obvious. For the second one, as P∗is the projection onto Pof the attractor As, for every p∗∈P∗,As(p∗) is nonempty. Compactness follows from that of Asand the continuity of the projection of P∗×Rdonto Rd. The Φ-invariance of As(p) follows trivially from the Π-invariance of As. We now prove the third claim. Since As(p) is compact, it is equivalent to prove ε-u.s.c. (cf. [1]). Suppose not, then there exists a constant ε > 0 and pn→ 19 p(elements of P∗) such that As(pn)6⊂ B(As(p), ε), i.e. there exists a sequence xn∈As(pn) with xn6∈ B(As(p), ε). By the sectorial definition of As(p), for each (pn, xn), there exist sequences tn m→ ∞ as m→ ∞ and yn m∈Φ(tn m, pn m, bn m), with (pn m, bn m)⊂ Bs, such that (θtn mpn m, yn m)→(pn, xn) as m→ ∞. Pick m(n) strictly increasing such that tn m(n)is also strictly increasing. From (θtn m(n)pn m(n), yn m(n)) we can extract a subsequence converging to a pair (p, y)∈ {p}×As(p), as they belong to Bs asymptotically. But this means that a subsequence of xnapproximates y∈As(p), which is a contradiction. 7.4.2 Proof of Proposition 12 Obviously A∗ s⊂ As, since A∗ sattracts a smaller class of sets, in fact, just those from P∗×Rdrather than P×Rd. Now Π∗(t, As)≡Π(t, As)≡ As,∀t≥0 since As⊂P∗×Rdand Π∗≡Π on P∗×Rd. Since A∗ sattracts nonempty bounded subsets of P∗×Rdincluding As, one has that H∗(As,A∗ s) = H∗(Π∗(t, As),A∗ s)→0,as t→ ∞, i.e. H∗(As,A∗ s)≡0, thus A ⊂ A∗and therefore As≡ A∗ s. Alternatively, one can also argue in the following way: Asis a Π∗–invariant set and the global attractor A∗ sof Π∗in P∗×Rdis the maximal compact Π∗–invariant subset of P∗×Rd, so As⊂ A∗ s. 7.4.3 Proof of proposition 13 The statement about P∗is clear even in this weak framework since θis single-valued and the weak Π-invariance easily implies θ-invariance for P∗, and the weak attraction property of Awimplies the strong attraction property for θin P∗. That Aw(p) is nonempty is trivial, as mentioned before, since P∗is the projection of Awonto P. It is also compact, the proof is the same in Theorem 9, since for all ak∈P×Rdwe have that PrP(ak) = pand so the limit a. Let us prove that P∗3p7→ Aw(p) is u.s.c. We want to see that if p0→p, then H∗(Aw(p0), Aw(p)) →0. If not, there exist a constant ε > 0 and a sequence pn→p with ε≤H∗(Aw(pn), Aw(p)) = dist(an, Aw(p)), where we have used that Aw(pn) is compact. Therefore, ε≤dist(an, a)∀a∈Aw(p).(10) As Aw(pn)⊂PrRdAw, which is compact, from {an}we extract a convergent subsequence (which we do not relabel), and so (pn, an)→(p, a). From the weak invariance of Aw, there exists at least one entire trajectory πnpassing through each (pn, an). Barbashin’s theorem (Th. 2, see Remark 5) provides a converging subsequence {πn1}n1on the interval [−1,1] to a trajectory. Applying it again to this subsequence we obtain another one denoted by {πn2}n2, which is uniformly converging on [−2,2]. A diagonal argument gives an entire trajectory πsuch that π(0) = (p, a). By Remark 10, we have that (p, a)∈ Aw, so a∈Aw(p), which contradicts (10). The last statement is obvious. 20 7.5 Pullback structure of SVSPF attractors 7.5.1 Proof of Lemma 15 Thanks to the Φ-positive invariance of Bs(p) for p∈P∗, and the cocycle property, we have that Φ(t+r, θ−t−rp, Bs(θ−t−rp)) = Φ(t, θ−tp, Φ(r, θ−t−rp, Bs(θ−t−rp))) ⊂Φ(t, θ−tp, Bs(θ−tp)). Thus, these sets are nested, and by the u.s.c. of Φ on its third variable and the compactness of Bs(p) for p∈P∗, they are compact. Thus, ˆ As(p) is nonempty and compact. 7.5.2 Proof of Proposition 17 We only need to check the first identity since the second one has already been proved. First, let us prove that ˆ As(p)⊂A∗ s(p): {p} × ˆ As(p) = {p} × \ t≥0 Φ(t, θ−tp, Bs(θ−tp)) =\ t≥0 {p} × Φ(t, θ−tp, Bs(θ−tp)) =\ t≥0 {θt(θ−tp)} × Φ(t, θ−tp, Bs(θ−tp)) =\ t≥0 Π(t, (θ−tp, Bs(θ−tp))) ⊂\ t≥0 Π(t, B∗ s)≡ A∗ s. Here we have used that P∗is θ-invariant. Thus {p} × ˆ As(p)⊂ A∗ swhat implies that {p} × ˆ As(p)⊂ {p} × A∗ s(p) and therefore ˆ As(p)⊂A∗ s(p) as desired. As for the converse, notice that Π∗(t, A∗ s) = A∗ sfor all t≥0. So, in particular, Φ(t, p, A∗ s(p)) = A∗ s(θtp) for all t≥0, p ∈P∗. Moreover, we know that ˆ As(p)⊂A∗ s(p)⊂Bs(p). Thus, setting θ−tpinstead of p, A∗ s(p) = Φ(t, p, A∗ s(θ−tp)) ⊂Φ(t, θ−tp, Bs(θ−tp)) for all t≥0 and p∈P∗. Therefore, we finally obtain that A∗ s(p)⊂\ t≥0 Φ(t, θ−tp, Bs(θ−tp)) = ˆ As(p). 21 7.5.3 Proof of Proposition 18 Suppose not, then there exist a positive constant ε, a bounded set Dand sequences tn→ ∞,yn∈Φ(tn, θ−tnp, dn) with p∈P∗and dn∈Dsuch that dist(yn,ˆ As(p)) > ε > 0. There exists TD(p) such that Φ(t, θ−tp, D)⊂Bs(p) for all t≥TD(p). So, as Bs(p) is compact, there exists a converging subsequence (denoted the same) yn→x∈Bs(p). We will see that in fact x∈ˆ As(p), which will be a contradiction. Consider any τ > 0 and take n(τ) big enough such that tn−τ > 0 and Φ(tn− τ, θ−tnp, dn)⊂Bs(θ−τp) which is possible since tn→ ∞ and the family Bs(p) is Φ-pullback absorbing. 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