On the Structure of the Global Attractor for Non-Autonomous Dynamical Systems with Weak Convergence

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX ON THE STRUCTURE OF THE GLOBAL ATTRACTOR FOR NON-AUTONOMOUS DYNAMICAL SYSTEMS WITH WEAK CONVERGENCE Tom´ as Caraballo Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160 41080-Sevilla (Spain) David Cheban State University of Moldova Department of Mathematics and Informatics A. Mateevich Street 60 MD–2009 Chi¸sin˘au, Moldova Abstract. The aim of this paper is to describe the structure of global attractors for non-autonomous dynamical systems with recurrent coefficients (with both continuous and discrete time). We consider a special class of this type of systems (the so–called weak convergent systems). It is shown that, for weak convergent systems, the answer to Seifert’s question (Does an almost periodic dissipative equation possess an almost periodic solution?) is affirmative, although, in general, even for scalar equations, the response is negative. We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our paper to the study of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of different classes of differential equations. 1. Introduction. Denote by Rnthe n–dimensional real Euclidian space with the norm |·|, and by C(R×Rn,Rn) the space of all continuous functions f:R×Rn→Rn equipped with the compact-open topology. Consider the differential equation x′=f(t, x),(1) where f∈C(R×Rn,Rn). Assume that the right-hand side of (1) satisfies hypotheses ensuring the existence, uniqueness and extendability of solutions of (1), i.e., for all (t0, x0)∈R×Rnthere exists a unique solution x(t;t0, x0) of equation (1) with initial data t0, x0, and defined for all t≥t0. Recall (see, for example, [18, 32]) that equation (1) is said to be uniformly dissipative (or uniformly ultimately bounded) if there exists a number r0>0 such that, 2000 Mathematics Subject Classification. primary:34C11,34C27,34D05,34D23,34D45,34K14, 37B20,37B55,37C55, 7C60, 37C65,37C70,37C75. Key words and phrases. Non-autonomous dynamical systems; skew-product systems; cocycles; global attractor; dissipative systems; convergent systems; quasi-periodic, almost periodic, almost automorphic, recurrent solutions; asymptotically almost periodic solutions. 1 2 TOM´ AS CARABALLO AND DAVID CHEBAN for every r > 0, there is L(r)>0 such that, if |x0| ≤ r, then |x(t;t0, x0)| ≤ r0, for all t≥t0+L(r). Equation (1) (respectively, the function f) is called regular, if for every x0∈Rn and g∈H(f) := {fτ:τ∈R}(where by bar we denote the closure in the space C(R×Rn,Rn) and fτ(t, x) := f(t+τ, x) for all (t, x)∈R×Rn), the equation x′=g(t, x) (2) possesses a unique solution ϕ(t, x, g) passing through the point x0at the initial moment t= 0, and defined on R+:= {t∈R|t≥0}. Theorem 1.1. [9, Ch.II] Suppose that f∈C(R×Rn,Rn)and H(f)is a compact subset of C(R×Rn,Rn). Then, the following statements are equivalent: (i) equation (1) is uniformly dissipative; (ii) there exists a positive number R0such that lim sup t→+∞|ϕ(t, x, g)| ≤ R0,(3) for all (x, g)∈Rn×H(f). At light of Theorem 1.1, it is said that equation (1) is dissipative (in fact, the family of equations (2) is collectively dissipative, but we use this shorter terminology) if (3) holds. Seifert’s Problem (see [19] for more details): Suppose that equation (1) is dissipative and the function fis almost periodic (with respect to the time variable). Does equation (1) possess an almost periodic solution? Fink and Fredericson [19] and Zhikov [33] established that, in general, even when equation (1) is scalar, the answer to Seifert’s question is negative. Related to this result, there are the following interesting questions: a) To extract some classes of dissipative differential equations for which the response to Seifert’s problem is positive; b) To indicate the additional (it is desirable “optimal”) conditions which, jointly with dissipativity, guarantee the existence of at least one almost periodic solution of equation (1). Below we include a short survey on results concerning the questions a) and b). a) For the following classes of dissipative equations of type (1), the response to Seifert’s question is affirmative: linear equations (see [9, Ch.II]), quasi-linear equations (weak non-linear perturbations of linear equations) (see [7, 9]); holomorphic equations (see [5, 6, 8, 9]). b) Zubov (see [36]) established that equation (1) admits a unique almost periodic solution if it is convergent, i.e., it admits a unique solution which is bounded on Rand also uniformly globally asymptotically stable. This result was generalized for equations (1) with recurrent coefficients by Cheban [9, Ch.II] and with pseudo recurrent coefficients by Caraballo and Cheban [4]. Levitan and Zhikov (see, for example, [23]) proved that, for low-dimensional equations (namely, for n≤3), equation (1) admits at least one almost periodic solution if (1) is uniformly positively stable, i.e., for all ε > 0 there exists δ= δ(ε)>0 such that |x1−x2|< δ implies |ϕ(t, x1, g)−ϕ(t, x2, g)|< ε, for all t≥0 and g∈H(f). The main result for ODEs (Theorem 4.2 and its generalizations) that we prove in this paper is the following: we show that if equation (1) is weak convergent (i.e., ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 3 there exists a positive number Lsuch that lim t→+∞|ϕ(t, x1, g)−ϕ(t, x2, g)|= 0 for all |xi| ≤ L(i= 1,2) and g∈H(f)), and fis pseudo recurrent with respect to the time variable (in particular, fis recurrent, almost automorphic, Bohr almost periodic or quasi periodic), then, equation (1) admits a unique pseudo recurrent (respectively, recurrent, almost automorphic, Bohr almost periodic, quasi periodic) solution. If this solution is Lyapunov stable, then the Levinson center (the compact global attractor) is a minimal almost periodic set. If it is not Lyapunov stable, then the Levinson center contains a minimal almost periodic set, but it is not minimal (this means, in particular, that equation (1) admits a family (more than one) of solutions which are bounded on R). We present our results in the framework of general non-autonomous dynamical systems (cocycles) and we apply our abstract theory to several classes of differential equations. The paper is organized as follows. In Section 2, we collect some notions (global attractor, minimal set, point/compact dissipativity, non-autonomous dynamical systems with convergence, quasi periodicity, Levitan/Bohr almost periodicity, almost automorphy, recurrence, pseudo recurrence, Poisson stability, etc) and facts from the theory of dynamical systems which will be necessary in this paper. Section 3 is devoted to the study of a special class of non-autonomous dynamical systems (NDS): the so-called NDS with weak convergence. We give a generalization of the notion of convergent NDS. On the one hand, this type of NDS is very close to NDS with convergence (because they conserve some properties of convergent systems) and larger than that of convergent systems. On the other hand, we analyze the class of compact dissipative NDS with nontrivial Levinson center. The main results of our paper are proved in this section, namely Theorem 3.5 and Theorem 3.8 (see also Corollary 3.9 and Corollary 3.10) which provide sufficient conditions for the existence of a unique minimal set in the Levinson center which is homeomorphic to the base dynamical system (driving system). This means, in particular, that if the base dynamical system is a compact minimal set consisting of recurrent (respectively, almost automorphic, Bohr almost periodic, quasi periodic, periodic, stationary) points, then, under the conditions of Theorem 3.5, the Levinson center of a non-autonomous dynamical system contains a unique minimal set which consists of recurrent (respectively, almost automorphic, Bohr almost periodic, quasi periodic, periodic, stationary) points. In Section 4, we exhibit some applications of our abstract results to different classes of differential equations. Namely, almost periodic and asymptotically almost periodic solutions (Subsection 4.1), uniformly compatible (by the character of recurrence with the right hand side) solutions of strict dissipative equations (Subsection 4.2). 2. Nonautonomous Dynamical Systems with Convergence. Let us start by recalling some concepts and notations about the theory of non-autonomous dynamical systems which will be necessary for our analysis. 2.1. Compact Global Attractors of Dynamical Systems. Let (X, ρ) be a metric space, R(Z) be the group of real (integer) numbers, R+(Z+) be the semigroup of nonnegative real (integer) numbers, Sbe one of the two sets Ror Zand T⊆S(S+⊆T) be a sub-semigroup of the additive group S. 4 TOM´ AS CARABALLO AND DAVID CHEBAN Adynamical system is a triplet (X, T, π), where π:T×X→Xis a continuous mapping satisfying the following conditions: π(0, x) = x(∀x∈X); π(s, π(t, x)) = π(s+t, x) (∀t, s ∈Tand x∈X). If T=R(R+) or Z(Z+), the dynamical system (X, T, π) is called a group (semigroup). When T=R+or R, the dynamical system (X, T, π) is called a flow, but if T⊆Z, then (X, T, π) is called a cascade (discrete flow). The function π(·, x) : T→Xis called the motion passing through the point xat the initial moment t= 0, and the set Σx:= π(T, x) is called the trajectory of this motion. A nonempty set M⊆Xis called positively invariant (negatively invariant,invariant) with respect to the dynamical system (X, T, π) or, simply, positively invariant (negatively invariant, invariant), if π(t, M)⊆M(M⊆π(t, M), π(t, M) = M) for every t∈T. A closed positively invariant set is said to be minimal if it does not contain any own closed positively invariant subset. It is easy to see that every positively invariant minimal set is invariant. Let M⊆X. The set ω(M) := \ t≥0[ τ≥t π(τ, M) is called the ω-limit of M. The set Ws(Λ), defined by the equality Ws(Λ) := {x∈X|lim t→+∞ρ(π(t, x),Λ) = 0} is called the stable manifold of the set Λ ⊆X. For p∈X,M⊂Xand δ > 0, let us denote by B(M, δ) = {x∈X|ρ(x, M)< δ}. The set Mis called: –orbitally stable if for every ε > 0, there exists δ=δ(ε)>0 such that ρ(x, M)< δimplies ρ(π(t, x), M)< ε, for all t≥0; –attracting if there exists γ > 0 such that B(M, γ)⊂Ws(M); –asymptotically stable if it is orbitally stable and attracting; –globally asymptotically stable if it is asymptotically stable and Ws(M) = X. The dynamical system (X, T, π) is called: −point dissipative if there exists a nonempty compact subset K⊆Xsuch that, for every x∈X, lim t→+∞ρ(π(t, x), K) = 0; (4) −compact dissipative if the equality (4) takes place uniformly w.r.t. xon the compact subsets of X; −locally complete (compact) if for any point p∈X, there exist δp>0 and lp>0 such that the set π(lp, B(p, δp)) is relatively compact. Let (X, T, π) be compact dissipative, and Kbe a compact set attracting every compact subset of X. Let us set J:= ω(K) := \ t≥0[ τ≥t π(τ, K).(5) It can be shown (see [9, Ch.I]) that the set Jdefined by equality (5) does not depend on the choice of the attractor K, but is characterized only by the properties ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 5 of the dynamical system (X, T, π) itself. The set Jis called the Levinson center of the compact dissipative dynamical system (X, T, π). Some properties of this set can be found in [9, 20]. 2.2. Non-Autonomous Dynamical Systems with Convergence. Given two dynamical systems (X, T1, π) and (Y, T2, σ), a triplet h(X, T1, π),(Y, T2, σ), hi, where his a homomorphism from (X, T1, π) onto (Y, T2, σ), is called a non-autonomous dynamical system (see [2, 9]). Recall (see, for example, [9, 12]) that the nonautonomous dynamical system h(X,T1, π),(Y, T2, σ), hiis said to be convergent if the following conditions are satisfied: (i) the dynamical systems (X, T1, π) and (Y, T2, σ) are compact dissipative; (ii) the set JXTXycontains no more than one point for all y∈JY, where Xy:= h−1(y) := {x∈X|h(x) = y}and JX(respectively, JY) is the Levinson center of the dynamical system (X, T1, π) (respectively, (Y, T2, σ)). Some sufficient conditions and criteria ensuring the convergence of a dynamical system can be found in [9, Ch.II]. Thus, a non-autonomous dynamical system h(X, T1, π),(Y, T2, σ), hiis convergent, if the systems (X, T1, π) and (Y, T2, σ) are compact dissipative with Levinson centers JXand JYrespectively, and JXhas “trivial” sections, i.e., JXTXyconsists of a single point for all y∈JY. In this case, the Levinson center JXof the dynamical system (X, T1, π) is a copy (a homeomorphic image) of the Levinson center JY of the dynamical system (Y, T2, σ). Thus, the dynamics on JXis the same as on JY. Remark 2.1. 1. We note that convergent systems are, in some sense, the simplest dissipative dynamical systems. If Yis compact and invariant, T2=S,h(X, T1, π), (Y, T2, σ), hiis a convergent non-autonomous dynamical system, and JXis the Levinson center of (X, T1, π), then (JX,T2, π) and (Y, T2, σ) are homeomorphic. Although the Levinson center of a convergent system can be completely described, it may be sufficiently complicated. 2. The concept of convergent system of differential equations is well developed (see, for example, B. P. Demidovich [16, 17, 18], Pavlov et al. [24], V. A. Pliss [25, 26], V. I. Zubov [35], and many others). The non-autonomous system of differential equations x′=f(t, x) (6) is called convergent, if it admits a unique solution defined and bounded on R, which is uniformly globally asymptotically stable. It is possible to show that the non-autonomous dynamical system generated by the convergent equation (6) is convergent. However, the concept of convergent non-autonomous dynamical system is much more general (see [9, 12] and the bibliography therein). Given ε > 0,a number τ∈Tis called an ε−shift (respectively, an ε−almost period) of x, if ρ(π(τ, x), x)< ε (respectively, ρ(π(τ+t, x), π(t, x)) < ε, for all t∈T). A point x∈Xis called almost recurrent (respectively, Bohr almost periodic), if for any ε > 0, there exists a positive number lsuch that, in any segment of length l, there is an ε−shift (respectively, an ε−almost period) of the point x∈X. If the point x∈Xis almost recurrent, and the set H(x) := {π(t, x)|t∈T}is compact, then xis called recurrent, where the bar denotes the closure in X. 6 TOM´ AS CARABALLO AND DAVID CHEBAN Denote by Nx:= {{tn} ⊂ T: such that {π(tn, x)} → xand {tn} → ∞} and Mx:= {{tn} ⊂ T: such that {π(tn, x)}is convergent and {tn} → ∞}. A point x∈Xis called Poisson stable in the positive direction if there exists a sequence {tn} ∈ Nxsuch that tn→+∞as n→ ∞. Let (X, T, π) be a two-sided dynamical system (i.e., T=S). A point x∈Xis called Poisson stable in the negative direction if there exists a sequence {tn} ∈ Nx such that tn→ −∞ as n→ ∞. The point x∈Xis called Poisson stable if it is Poisson stable in both directions. The dynamical system (X, T, π) is said to be (i) transitive if there exists a point x0∈Xsuch that H(x0) = X; (ii) pseudo recurrent if Xis compact, transitive, and every point x∈Xis Poisson stable. A point x∈Xis called pseudo recurrent (see [28, 30]) if the dynamical system (H(x),T, π) is pseudo recurrent. Remark 2.2. Every recurrent point is pseudo recurrent, but there exist pseudo recurrent points which are not recurrent (see [28, 30]). An m-dimensional torus is denoted by Tm:= Rm/2πZm.Let (Tm,T, σ) be an irrational winding of Tm, i.e., σ(t, ν) := (ν1t, ν2t,...,νmt) for all t∈Sand ν∈ Tm. A point x∈Xis called quasi-periodic, with frequency ν:= (ν1, ν2,...,νm)∈ Tm, if there exists a continuous function Φ : Tm→Xsuch that π(t, x) := Φ(σ(t, ω)) for all t∈T,where (Tm,T, σ) is an irrational winding of the torus Tmand ω∈ Tm. A point x∈Xof the dynamical system (X, T, π) is called Levitan almost periodic (see [2, 23]) if there exists a dynamical system (Y, T, σ), and a Bohr almost periodic point y∈Ysuch that Ny⊆Nx. Remark 2.3. Let xi∈Xi(i= 1,2,...,m) be a Levitan almost periodic point of the dynamical system (Xi,T, πi).Then, the point x:= (x1, x2,...,xm)∈X:= X1×X2×... ×Xmis also Levitan almost periodic in the product dynamical system (X, T, π),where π:T×X→Xis defined by the equality π(t, x) := (π1(t, x1), π2(t, x2), . . . , πm(t, xm)), for all t∈Tand x:= (x1, x2,...,xm)∈X. Recall (see [11]) that the point x∈Xis called asymptotically τ–periodic (respectively, asymptotically quasi periodic,asymptotically Bohr almost periodic,asymptotically almost automorphic,asymptotically recurrent,asymptotically pseudo recurrent) if there exists a τ-periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent, pseudo recurrent) point p∈Xsuch that lim t→+∞ρ(π(t, x), π(t, p)) = 0. Thus, we can prove the following result. Lemma 2.4. Let h(X, T1, π),(Y, T2, σ), hibe a convergent non-autonomous dynamical system, and JX(respectively, JY) be the Levinson center of the dynamical system (X, T1, π)(respectively, (Y, T2, σ)), and y0∈JYbe a τ–periodic (respectively, quasi periodic, Bohr almost periodic, recurrent, pseudo recurrent) point. Then, the following statements hold: (i) the point x0∈JXTXy0is also τ–periodic (respectively, quasi periodic, Bohr almost periodic, recurrent, pseudo recurrent); (ii) every point x∈Xy0is asymptotically τ–periodic (respectively, asymptotically quasi periodic, asymptotically Bohr almost periodic, asymptotically recurrent, asymptotically pseudo recurrent). ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 7 Proof. The first statement follows directly from the corresponding definition. Let x∈Xy0be an arbitrary point. We will show that lim t→+∞ρ(π(t, x), π(t, x0)) = 0. Indeed, if we suppose that it is not true, then there are ε0>0 and tn→+∞ ({tn} ⊆ T1) such that ρ(π(tn, x), π(tn, x0)) ≥ε0.(7) Since the dynamical system (X, T1, π) is compact dissipative, we may suppose that the sequences {π(tn, x)}and {π(tn, x0)}are convergent. Denote by p:= lim n→∞π(tn, x), and p0:= lim n→∞π(tn, x0). Then, p, p0∈Xq⊆JX, where q:= lim n→∞σ(tn, y0). Since the non-autonomous dynamical system h(X, T1, π),(Y, T2, σ), hi is convergent, then p=p0. On the other hand, passing to the limit in (7), we obtain ρ(p, p0)≥ε0>0. This contradiction proves our statement. 3. Non-Autonomous Dynamical Systems with Weak Convergence. In this section we will study a class of non-autonomous dynamical systems which is very close to convergent systems, but possessing a non-trivial global attractor. This means that this class of non-autonomous systems will conserve almost all properties of convergent systems, but will have a “non-trivial” global attractor JX, i.e., there exists at least one point y∈JYsuch that the set JXTXycontains more than one point. A non-autonomous dynamical system h(X, T1, π),(Y, T2, σ), hiis said to be weak convergent, if the following conditions hold: (i) the dynamical systems (X, T1, π) and (Y, T2, σ) are compact dissipative with Levinson centers JXand JYrespectively; (ii) it follows that lim t→+∞ρ(π(t, x1), π(t, x2)) = 0, for all x1, x2∈JXwith h(x1) = h(x2). Remark 3.1. It is clear that every convergent non-autonomous dynamical system is weak convergent. The opposite statement is not true in general. Indeed, the last statement can be confirmed by the following example. Let (X, T, π) be an autonomous dynamical system with compact global attractor J, which possesses a unique stationary attracting point p(i.e., lim |t|→+∞ρ(π(t, x), p) = 0 for all x∈J) and J6={p}). For example, consider the dynamical system (X, R, π) on the space X=R2, generated by following system of differential equations            x′=x2(2y−x) + 2y5 (x2+y2)[1 + (x2+y2)2] y′=8y2(y−x) (x2+y2)[1 + (x2+y2)2]. (8) The phase plane of this dynamical system (8) is described in Figure 1. For more details see [31] and also [3, Ch.II, pp.94-98] and [21, Ch.V, pp.191-194]. For a similar example the reader is referred to [1, Ch.V, p.59] (Example 1.7.8). We can now prove the following result which will be crucial in the proof of one of our main results (namely, Theorem 3.5). Lemma 3.2. Let h(X, T1, π),(Y, T2, σ), hibe a non-autonomous dynamical system, and assume that the following conditions hold: 8 TOM´ AS CARABALLO AND DAVID CHEBAN Figure 1. (i) Yis a compact minimal set; (ii) there exists a point x0∈Xwith relatively compact positive semi-trajectory Σ+ x0={π(t, x0)|t≥0}; (iii) lim t→+∞ρ(π(t, x1), π(t, x2)) = 0, for all x1, x2∈Xwith h(x1) = h(x2). Then, there exists a unique compact minimal set M⊆Xsuch that: (i) the section MTXyof the set Mconsists of a single point myfor all y∈Y; (ii) every positive semi-trajectory Σ+ xis relatively compact; (iii) lim t→+∞ρ(π(t, x), mσ(t,h(x))) = 0,(9) for all x∈X. Proof. Since the positive semi-trajectory Σ+ x0of x0is relatively compact, the ω–limit set ωx0of the point x0is nonempty, compact, invariant and contains at least one minimal subset M⊆ωx0. We will prove that the dynamical system (X, T1, π) has at most one minimal set. Indeed, if we suppose that M1and M2are two different minimal sets of (X, T1, π), then M1TM2=∅and, in particular, M1yTM2y=∅ for all y∈Y, where Miy := h−1(y)TMi(i= 1,2). Let xi∈Miy and tn→+∞ such that σ(tn, y)→yand π(tn, xi)→¯xi∈Miy (i= 1,2) as n→ ∞; lim n→∞ρ(π(tn, x1), π(tn, x2)) = 0.(10) It is now easy to see that there exists such a sequence. From the equality (10) we have ¯x1= ¯x2∈M1yTM2y. This is a contradiction and, therefore, Mis the unique compact minimal set of the dynamical system (X, T1, π). Let y∈Ybe an arbitrary point, then it is recurrent. By Lemma 6.5.19 in [12, Ch.VI, p.226], there exists a unique recurrent point my∈Mysuch that the equality (9) holds for all x∈My. Now, we will prove that My={my}. Indeed, if x∈My then, there exists a sequence tn→+∞such that π(tn, my)→xbecause Mis minimal. On the other hand, σ(tn, y)→yand, since the point myis uniformly compatible by the character of recurrence with the point y, then π(tn, my)→my. Thus, we have x=myand, consequently, My={my}for all y∈Y. ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 9 To finish the proof it is sufficient to note that lim t→+∞ρ(π(t, x), mσ(t,h(x))) = 0, for all x∈X. Let (X, T, π) be a dynamical system. Denote by ΩX:= ∪{ωx|x∈X}and by D+(M) := Tε>0St≥0π(t, B(M, ε)), where M⊆X. Corollary 3.3. Under the assumptions in Lemma 3.2, the dynamical system (X, T1, π)is point dissipative and ΩXis a compact minimal set. Below we give an example of point dissipative, but not compact dissipative, dynamical system with weak convergence. Example 3.4. Let ϕ∈C(R,R) be a function possessing the following properties: 1. ϕ(0) = 0; 2. supp(ϕ) = [0,2]; 3. ϕ∈C∞(R,R); 4. ϕ(1) = 1; 5. the function ϕis monotone increasing from 0 to 1 and it is decreasing from 1 to 2; 6. xϕ(x−1)→0 as x→+∞. A function ϕwith properties 1.−6.can be constructed as follows. Let ϕ0(t) := exp [t2−1]−1+ 1,|t|<1 0,|t| ≥ 1. Then, the function ϕ(t) := ϕ0(t−1) is as desired. We set X:= {aϕ t a+h|h∈ R, a > 0}∪{θ}, where θis the function from C(R,R) identically equal to 0. It is possible to show that the set Xis closed in C(R,R), and it is invariant with respect to shifts. Thus, on the set Xis induced a dynamical system (on C(R,R) is defined the dynamical system of translations or Bebutov’s dynamical system), which we denote by (X, R, σ). We will indicate some properties of this system: (i) for every function ψ∈Xthe set {σ(t, ψ) : t∈R}is relatively compact and ωψ=αψ={θ}, where αψdenotes the α-limit associated to ψ; (ii) the dynamical system (X, R, σ) is pointwise dissipative, and ΩX={θ}; (iii) D+(ΩX) = Xand, consequently, the dynamical system (X, R, σ) is not compact dissipative because the set X, evidently, is not compact; (iv) the dynamical system (X, R, σ) does not admit a maximal compact invariant set. The necessary example is therefore constructed. The subset A⊆Xis said to be chain transitive (see [15, 22]) if for any a, b ∈A, and any ε > 0 and L > 0, there are finite sequences x1, x2,...,xm∈Awith a= x1, b =xm, and t1, t2,...,tm≥Lsuch that ρ(π(ti, xi), xi+1)< ε (1 ≤i≤m−1). The sequence {x1, x2,...,xm}is called an ε–chain in Aconnecting aand b. We can now establish and prove the main results of our paper. Theorem 3.5. Let h(X, T1, π),(Y, T2, σ), hibe a non-autonomous dynamical system satisfying the following conditions: (i) Yis a compact minimal set; (ii) the dynamical system (X, T1, π)is compact dissipative with Levinson center J; 16 TOM´ AS CARABALLO AND DAVID CHEBAN In this section we suppose that equation (18) is regular. Equation (18) is called dissipative (see [9]), if there exists a positive number rsuch that lim sup t→+∞|ϕ(t, u, y)|< r for all u∈Rnand y∈Y, where |·| is a norm in Rn. It is well known (see [19, 33]) that a dissipative equation with almost periodic coefficients (Yis an almost periodic minimal set) does not have, in general, an almost periodic solution. For certain classes of dissipative equations of the form (18), in the works [5]–[8] one can find sufficient conditions for the existence of at least one almost periodic solution. In this subsection we give a simple geometric condition which guarantees existence of a unique almost periodic solution, and this solution, in general, is not the unique solution of equation (18) which is bounded on R. We can now establish the following interesting result. Theorem 4.2. Suppose that the following conditions are fulfilled: (i) equation (18) is regular and dissipative; (ii) the space Yis compact, and the dynamical system (Y, R, σ)is minimal; (iii) for all y∈Y lim t→+∞|ϕ(t, u1, y)−ϕ(t, u2, y)|= 0,(21) where ϕ(t, ui, y)(i= 1,2) is the solution of equation (18) passing through ui at the initial moment t= 0, which is bounded on R. Then, (i) if the point yis τ–periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent), then equation (18) admits a unique τ–periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent) solution ϕ(t, uy, y)(uy∈Rn); (ii) every solution ϕ(t, x, y)is asymptotically τ–periodic (respectively, asymptotically quasi periodic, asymptotically Bohr almost periodic, asymptotically almost automorphic, asymptotically recurrent). Proof. Let hRn, ϕ, (Y, R, σ)ibe the cocycle associated to equation (18). Denote by (X, R+, π) the skew-product dynamical system, where X:= Rn×Yand π:= (ϕ, σ) (i.e., π(t, (u, y)) := (ϕ(t, u, y), σ(t, y)) for all x:= (u, y)∈Rn×Yand t∈R+). Consider the non-autonomous dynamical system h(X, R+, π),(Y, R, σ), higenerated by the cocycle ϕ(respectively, by equation (18)), where h:= pr2:X→Yis the projection with respect to the second variable, i.e., pr2(α, y) = y. Since Yis compact, it is evident that the dynamical system (Y, R, σ) is compact dissipative and its Levinson center JYcoincides with Y. By Theorem 2.23 in [9], the skewproduct dynamical system (X, R+, π) is compact dissipative. Denote by JXits Levinson center and by Iy:= pr1(JXTXy) for all y∈Y, where Xy:= {x∈X: h(x) = y}, and pr1is the projection function with respect to the first variable, i.e. pr1(α, y) = α. According to the definition of the set Iy⊆Rn, and by Theorem 2.24 in [9], u∈Iyif and only if the solution ϕ(t, u, y) is defined on Rand bounded (i.e., the set ϕ(R, u, y)⊆Rnis compact). Thus, Iy={u∈Rn|(u, y)∈JX}. It is easy to see the condition (21) means that the non-autonomous dynamical system h(X, R+, π),(Y, R, σ), hiis weak convergent. To finish the proof, it is sufficient to apply Lemma 6.5.19 in [12, Ch.VI, p.226] and Corollary 3.10 for the non-autonomous system h(X, R+, π),(Y, R, σ), higenerated by equation (18). ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 17 Remark 4.3. Under the conditions of Theorem 4.2 there exists a unique almost periodic solution of equation (18), but equation (18) has, generally speaking, more than one solution defined and bounded on R. Below, we will give an example which confirms this statement. Example 4.4. Consider the following almost periodic system of two differential equations        u′=(u−sin t)2(2v−u+2 sin √2t−sin t)+2(v−sin √2t)5 ((u−sin t)2+(v−sin √2t)2)[1+((u−sin t)2+(v−sin √2t)2)2]+ cos t v′=8(v−sin √2t)2(v−u+sin √2t−sin t) ((u−sin t)2+(v−sin √2t)2)[1+((u−sin t)2+(v−sin √2t)2)2]+√2 cos √2t . (22) It is easy to check that the almost periodic function ϕ:R→R2defined by the equality ϕ(t) := (sin t, sin √2t) is a solution of system (22). Let now x:= u−sin t and y:= v−sin √2t. Then, the system (22) reduces to (8). Thus, every solution φ of the system (22) possesses the form φ=ϕ+ψ, where ψis some solution of the system (8). Since (8) is weak convergent and admits more than one solution which is bounded on R, the system (22) possesses the same property. 4.2. Uniform compatible solutions of strict dissipative equations. In this section we consider equation (18) when the driving system (Y, R, σ) is pseudo recurrent, and the function f∈C(Y×Rn,Rn) is strict dissipative with respect to its second variable x∈Rn, i.e., hf(y, x1)−f(y, x2), x1−x2i<0 (23) for all x1, x2∈Rn(x16=x2) and y∈Y. Recall (see [28, 29, 30]) that the point x∈Xis called comparable (respectively, uniformly comparable) by the character of recurrence with the point y∈Yif Ny⊆ Nx(respectively, My⊆Mx). Let us now recall a result which plays an important role in the proof of our main result in this subsection. Theorem 4.5. (See [28, 30]) Let (X, T, π)and (Y, T, σ)be two dynamical systems, x∈Xand y∈Y. Then, the following statements hold: (i) If xis comparable by the character of recurrence with y, and yis τ–periodic (respectively, Levitan almost periodic, almost recurrent, Poisson stable), then so is the point x. (ii) If xis uniformly comparable by the character of recurrence with y, and yis τ–periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent), then so is the point x. Following B. A. Shcherbakov [28, 30], a solution ϕ(t, u, y) of equation (18) is said to be compatible (respectively, uniformly compatible) by the character of recurrence with the right hand-side if Ny⊆Nϕ(·,u,y)(respectively, My⊆Mϕ(·,u,y)). Theorem 4.6. Let (Y, R, σ)be pseudo recurrent, f∈C(Y×Rn,Rn)be strict dissipative with respect to the variable x, and assume that there exists at least one solution ϕ(t, x0, y)of equation (18) which is bounded on R+. Then, (i) equation (18) is convergent, i.e., the cocycle ϕassociated to equation (18) is convergent; 18 TOM´ AS CARABALLO AND DAVID CHEBAN (ii) for all y∈Y, equation (18) admits a unique solution ϕ(t, xy, y)which is bounded on Rand uniformly compatible, i.e., My⊆Mϕ(·,xy,y); (iii) if the point yis τ–periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent), then (a) equation (18) has a unique τ–periodic (respectively, quasi periodic, Bohr almost periodic, almost automorphic, recurrent) solution; (b) every solution ϕ(t, x, y)is asymptotically τ–periodic (respectively, asymptotically quasi periodic, asymptotically Bohr almost periodic, asymptotically almost automorphic, asymptotically recurrent); (c) lim t→∞|ϕ(t, x, y)−ϕ(t, xy, y)|= 0 for all x∈Rnand y∈Y. Proof. Let V:X˙ ×X→R+be the mapping defined by the equality V((x1, y), (x1, y)) := |x1−x2|2for all x1, x2∈Rnand y∈Y, where | · |2:= h·,·i and hξ1, ξ2i:= n P i=1 ξ1 iξ2 i(ξj:= (ξj 1, ξj 2,...,ξj n)∈Rn(j= 1,2)). Let (X, R+, π) be the skew-product dynamical system associated to the cocycle ϕ. Then, dV (π(t, (u1, y)), π(t, (u2, y))) dt t=0 =hf(y, u1)−f(y, u2), u1−u2i<0 (24) for all u1, u2∈Rn(u16=u2) and y∈Y. From (24) it follows that V(π(t, (u1, y)), π(t(u2, y)) < V ((u1, y),(u2, y)) for all u1, u2∈Rn(u16=u2) and y∈Y. Now to prove the first statement it is sufficient to apply Theorem 3.11. According to the first statement of the theorem, the skew-product dynamical system (X, R+, π) is compact dissipative and, if JXis its Levinson center, then JXTXy consists of a single point xy= (uy, y) and ϕ(t, uy, y) is the unique solution of equation (18) defined and bounded on R. Now, we will prove that the solution ϕ(·, uy, y) is uniformly compatible by the character of recurrence, i.e., My⊆Mϕ(·,uy,y). It is easy to see that the last statement is equivalent to the inclusion My⊆Mxy. Let {tk} ∈ My. Then, there exists a point q∈Ysuch that σ(tn, y)→qas n→ ∞. Consider the sequence {π(tk, xy)}. Since xy∈JX, the sequence {π(tk, xy)}is relatively compact. Let p1and p2be two points of accumulation of this sequence. Then, there exist two subsequences {t(i) k} ⊆ {tk}(i= 1,2) such that pi= lim k→∞π(ti k, xy) (i= 1,2). Since JXis a compact invariant set, then pi∈JX(i= 1,2). On the other hand, π(ti k, xy) = (ϕ(ti k, uy, y), σ(ti k, y)) →(¯ui, q) = piand, consequently, pi∈Xq. Thus pi∈JXTXq(i= 1,2) and, consequently, p1=p2. This means that the sequence {π(tk, xy)}is convergent. The second statement is therefore proved. Taking into account the first and second statements, to finish the proof of the third statement it is sufficient to apply Theorem 4.5. The theorem is completely proved. Remark 4.7. 1. Theorem 4.6 remains true if we replace the standard scalar product h·,·i on the space Rnby an arbitrary scalar product hu, uiW:= hWu, ui, where W= (wij )n i,j=1 (wij ∈R) is a symmetric and positive defined n×n–matrix. 2. If we replace condition (23) by a stronger condition, then Theorem 4.6 is also true without the requirement that there exists at least one solution which is bounded on R+. Namely, if there exists a function ζ∈ K such that hf(y, u1)−f(y, u2), u1−u2i ≤ −ζ(|u1−u2|2) for all u1, u2∈Rnand y∈Y, where ζpossesses some additional properties (see, for example, [14]). ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 19 3. It is easy to see that Theorem 4.6 remains true also for equation (18) in an arbitrary Hilbert space H, if we suppose that the cocycle ϕ, generated by equation (18), is asymptotically compact (i.e., the corresponding skew-product dynamical system is asymptotically compact), and we replace the condition about the existence of at least one solution which is bounded on R+, by the existence of a relatively compact solution ϕ0on R+(this means that ϕ(R+) is a relatively compact subset from H). Acknowledgements. We would like to thank the anonymous referee for the helpful suggestions which allowed us to improve the presentation of this article. This paper was written while the second author was visiting the University of Sevilla (February–September 2010) under the Programa de Movilidad de Profesores Universitarios y Extranjeros (Ministerio de Educaci´on, Spain) grant SAB2009-0078. He would like to thank people of this university for their very kind hospitality. 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Zubov, “Theory of Oscillations”, Nauka, Moscow, 1979. (in Russian) E-mail address, T. Caraballo: [email protected] E-mail address, D. Cheban: [email protected]