Forward Attraction of Pullback Attractors and Synchronizing Behavior of Gradient-Like Systems with Nonautonomous Perturbations X

Home , Pullback attractor

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2019011 DYNAMICAL SYSTEMS SERIES B Volume 24, Number 3, March 2019 pp. 1175–1197

FORWARD ATTRACTION OF PULLBACK ATTRACTORS AND SYNCHRONIZING BEHAVIOR OF GRADIENT-LIKE SYSTEMS WITH NONAUTONOMOUS PERTURBATIONS†

Xuewei Ju Department of Mathematics, School of Science Civil Aviation University of China Tianjin 300300, China Desheng Li∗ School of Mathematics, Tianjin University Tianjin 300072, China Jinqiao Duan Department of Applied Mathematics Illinois Institute of Technology Chicago, IL 60616, USA

Abstract. We consider the nonautonomous perturbation xt + Ax = f(x) + εh(t) of a gradient-like system xt + Ax = f(x) in a Banach space X, where A is a sectorial operator with compact resolvent. Assume the non-perturbed system xt + Ax = f(x) has an attractor A . Then it can be shown that the perturbed one has a pullback attractor Aε near A . If all the equilibria of the non-perturbed system in A are hyperbolic, we also infer from [4,6] that Aε inherits the natural Morse structure of A . In this present work, we introduce the notion of nonautonomous equilibria and give a more precise description on the Morse structure of Aε and the asymptotically synchronizing behavior of the perturbed system. Based on the above result we further prove that the sections of Aε depend on time symbol continuously in the sense of Hausdorﬀ distance. Consequently, one concludes that Aε is a forward attractor of the perturbed nonautonomous system. It will also be shown that the perturbed system exhibits completely a global forward synchronizing behavior with the external force.

1. Introduction. The theory of attractors plays a fundamental role in the study of dynamical systems. This is because that if a system has an attractor, then all its long-term dynamics near the attractor will be captured by the attractor. The theory of attractors for autonomous dynamical systems has been fully developed in the past decades, in both finite and infinite dimensional cases; see e.g. [2,13,15,20,25,26,28], etc. In contrast to autonomous systems, the understanding of the dynamics of nonautonomous systems seems to be far more difficult, and even some fundamental notions such as attractors are still undergoing investigations.

2010 Mathematics Subject Classiﬁcation. 37B35, 37B55, 37E10, 34D06, 34D45, 34D06. Key words and phrases. Gradient-like system, nonautonomous perturbation, nonautonomous equilibrium, Morse structure, pullback attractor, forward attractor, synchronizing behavior. † Supported by NNSF of China under the grant 11471240 and the Foundation Research Funds for the Central Universities under the grant 3122014K008. ∗ Corresponding author.

1175 1176 XUEWEI JU, DESHENG LI AND JINQIAO DUAN

For nonautonomous dynamical systems, usually one can deﬁne two diﬀerent types of nonautonomous attractors: pullback attractor and forward attractor, corresponding to pullback attraction and forward attraction, respectively. The existence of the former one is well known and can be obtained under quite general hypotheses similar to those to guarantee the existence of attractors for autonomous systems; see e.g. [5, 12, 14, 29]. This aroused great interest in a systematic study on pullback behavior of nonautonomous systems arising in applications in the past two decades. On the other hand, in most cases we are naturally more concerned with the forward behavior of a system. However, a pullback attractor may provide little information on this aspect. This can be seen from the following easy example (see also the example in [6, (4.8)]). Example 1.1. Consider the scalar system on X = R: x0(t) = (2h(t) − 1)x, (1.1) where h is a continuous monotone function on R with lim h(t) = 0, lim h(t) = 1. t→−∞ t→∞

It is easy to verify that this system has a global pullback attractor A = {A(t)}t∈R with A(t) = {0} for all t ∈ R. On the other hand, any solution x(t) of the system goes to infinity as t → +∞ except the trivial one (hence system (1.1) has neither a uniform attractor nor a forward one). Theoretically, forward attractors sound to be quite suitable to describe the forward behavior of nonautonomous systems. Unfortunately, by far the existence of this type of attractors remains an open problem except in some particular cases such as the asymptotically autonomous and the periodic ones, and is still under investigations; see e.g. Cheban, Kloeden and Schmalfuß [9], Wang, Li and Kloe- den [31]), Kloeden [18] and Carvalho, Langa, Robinson and Surez [6, pp. 595]. It is well known that for a cocycle system Φ(t, p)x on a complete metric space X with compact base space P , if a pullback attractor A = {A(p)}p∈P has a uniform pullback attraction rate with respect to p ∈ P then it is forward attracting. Based on this fact it was shown in [9] that if the section A(p) is lower semicontinuous in p then A is a forward attractor (see also [31]). But in general the verification of the lower semicontinuity of A(p) in p seems to be a difficult task. In this paper we consider the nonautonomous perturbation

xt + Ax = f(x) + εh(t) (1.2) of a gradient-like system xt + Ax = f(x) (1.3) in a Banach space X, where A is a sectorial operator in X with compact resolvent, and h is translation compact. Suppose that all the equilibria of the non-perturbed system (1.3) are hyperbolic and that the system has an (local) attractor A . We show that the perturbed system has a pullback attractor Aε near A provided ε is sufficiently small. We then introduce the notion of nonautonomous equilibria and give a more precise description on the Morse structure of the pullback attractor Aε. Based this result we finally prove that the section of Aε depends on the time symbol continuously in the sense of Hausdorff distance. It then follows that Aε is a forward attractor. Our results demonstrate simultaneously that the perturbed system exhibits completely a forward synchronizing behavior with the external perturbation h in both point-wise and global sense. FORWARD ATTRACTORS AND SYNCHRONIZING BEHAVIOR 1177

Now let us give a more detailed description of our work. As usual we embed the system (1.2) into the following one:

xt + Ax = f(x) + εp(t), p ∈ H, (1.4) where H := H[h] is the hull of h (in some appropriate function space). Suppose f is a locally Lipschitz map from Xα to X for some α ∈ [0, 1). Then (1.4) generates α a cocycle system ϕε = ϕε(t, p)x on X with base space H and driving system θt, where θt is the shift operator on H for each t ∈ R. First, we introduce the notion of nonautonomous equilibria. Specifically, a α nonautonomous equilibrium of ϕε is a continuous map Γ ∈ C(H,X ) such that γ(t) = Γ(θtp) is a full solution of the system for each p ∈ H. We show that if E is a hyperbolic equilibrium of the non-perturbed system ϕ0, then for ε > 0 sufficiently small the perturbed system ϕε has a unique nonautonomous equilibrium Γ in a small neighborhood U of E. Moreover, the local p-section unstable manifold u WU (Γ, p) (see Section 2 for definition) of Γ depends on p continuously. It is worth mentioning that γ(t) = Γ(θth) gives a full solution of (1.2) that is completely synchronizing with the external force h. Indeed, if h is periodic (resp. quasiperiodic, almost periodic, uniformly almost automorphic), then so is σ(t) := θth. Thus the continuity of Γ(p) in p implies that γ(t) is periodic (resp. quasiperiodic, almost periodic, uniformly almost automorphic) as well. Quite similar results can be found in the nice work of Kloeden and Rodrigues [19] in a more general setting. Roughly speaking, the authors introduced a class of functions defined on the real line ex- tending periodic and almost periodic functions, and prove that if the forcing term of a perturbed nonautonomous system belongs to this class, then the system has a full solution which belongs to the same class as well. In contrast, in our case the description on the synchronizing behavior of the system given here seems to be a little more explicit. Second, we discuss Morse structures of pullback attractors of the system ϕε. This problem was addressed as early as in Carvalho et al [6]. Suppose the non-perturbed system has an attractor A which contains only a finite number of hyperbolic equilibria Ei (1 ≤ i ≤ n). Then the perturbed nonautonomous system has a pullback attractor Aε near A (see Section 4). It was proved in [6] that Aε contains the same number of full solutions which forms a Morse decomposition of Aε. This work was further developed and extended to more general cases in Carvalho and Langa [4] and Vishik, Zelik and Chepyzhov [30], and to discrete dynamical systems in Cheban, Mammana and Michetti [10]. One can even find some interesting results concerning randomly perturbed case in Caraballo, Langa and Liu [8]. Based on the existence of nonautonomous equilibria of ϕε, here we give a more precise description on the Morse structure of Aε and the point-wise asymptotic behavior of the system. We show that if ε is sufficiently small then the pullback attractor Aε contains the same number of nonautonomous equilibria Γi (1 ≤ i ≤ n), which completely characterize the Morse structure of Aε. Furthermore, the section u Aε(p) of Aε consists of precisely all the p-section unstable manifolds W (Γi, p) S (1 ≤ i ≤ n). We also show that there is a neighborhood U of Au = p∈H Aε(p) such that for any x ∈ U and p ∈ H, one has

lim d (ϕε(t, p)x, Γi(θtp)) = 0 (1.5) t→+∞ 1178 XUEWEI JU, DESHENG LI AND JINQIAO DUAN for some i. Since Γ(θth) is synchronizing with h,(1.5) indicates that the nonautonomous system (1.2) exhibits completely a forward asymptotically synchronizing behavior with the external force. Third, making use of the above results, we prove that the section Aε(p) is continuous in p in the sense of Hausdorﬀ distance. Consequently one concludes that Aε is uniformly forward attracting, and hence is a (uniform) forward attractor. More precisely, there is a neighborhood B of Au such that

lim H (ϕε(t, p)B, Aε(θtp)) = 0, ∀ p ∈ H, (1.6) t→∞ where H(·, ·) denotes the Hausdorff semidistance in the state space. As we have mentioned earlier, if the function h in (1.2) is periodic (resp. quasiperiodic, almost periodic, uniformly almost automorphic), then so is θtp for any fixed p ∈ H. Further by the continuity of Aε(p) in p, we deduce that A(t) = Aε(θth) is a periodic (resp. quasiperiodic, almost periodic, uniformly almost automorphic) function taking values in C (Xα), where C (Xα) denotes the metric space consisting of all the nonempty compact subsets of Xα equipped with the Hausdorff metric. This along with (1.6) then asserts that system (1.2) also exhibits a global forward synchronizing behavior with the external force. This paper is organized as follows. In Section 2 we present some basic definitions and results, and in Section 3 we study nonautonomous perturbations of hyperbolic equilibria of (1.3). In Section 4 we discuss Morse structures of pullback attractors and the point-wise asymptotically synchronizing behavior of the perturbed system (1.4). Finally in Section 5, we investigate the forward attraction of pullback attractors of (1.4) and demonstrate the global synchronizing behavior of the system.

2. Preliminaries. Let X be a complete metric space with metric d(·, ·). Given M ⊂ X, we denote M, int M and ∂M the closure, interior and boundary of M of X, respectively. A set U ⊂ X is called a neighborhood of M ⊂ X, if M ⊂ int U. For any ρ > 0, denote by

BX (M, ρ) := {x ∈ X : d(x, M) < ρ} the ρ-neighborhood of M in X, where d(x, M) = infy∈M d(x, y) The Hausdorff semidistance and the Hausdorff distance in X are defined, respectively, as HX (M,N) = sup d(x, N), ∀ M,N ⊂ X, x∈M

δX (M,N) = max{HX (M,N),HX (M,N)}, ∀ M,N ⊂ X.

2.1. Semiflow. Let R+ = [0, ∞). A continuous map S : R+ × X → X is called a semiflow on X, if it satisfies i) S(0, x) = x for all x ∈ X; and ii) S(t + s, x) = S(t, S(s, x)) for all x ∈ X and t, s ∈ R+. Remark 1. If R+ is replaced by R in the above definition, then S is called a flow on X. Let S be a given semiflow on X. As usual, we will rewrite S(t, x) as S(t)x. Henceforth we always assume S is asymptotically compact, namely,

(AC) each bounded subset B of X is admissible (that is, for any sequences xn ∈ B and tn → +∞ with S([0, tn])xn ⊂ B for all n, the sequence S(tn)xn has a convergent subsequence). FORWARD ATTRACTORS AND SYNCHRONIZING BEHAVIOR 1179

Let J ⊂ R be an interval. A solution (trajectory) γ of S on J is a map γ : J → X satisfying γ(t) = S(t − s)γ(s) for any s, t ∈ J with s ≤ t. In case J = R, we will simply call γ a full solution. A set B ⊂ X is called invariant under S if S(t)B = B for all t ≥ 0. Let B and M be subsets of X. We say that M attracts B (under S), if

lim HX (S(t)B,M) = 0. t→∞ Deﬁnition 2.1. A compact subset A ⊂ X is called an attractor for S, if it is invariant under S and attracts one of neighborhood of itself. The attraction basin of A is deﬁned as Ω(A) = {x ∈ X : lim d (S(t)x, A) = 0}. t→∞

Let A be an attractor of S. Since A is invariant, the restriction SA(t) of S on A is also a semiﬂow. A compact set A is said to be an attractor of S in A, if A is an attractor of the restricted system SA in A. Lemma 2.2. [17] An attractor A of S in A is also an attractor in X.

Deﬁnition 2.3. Let A be an attractor, and let M = {M1, ··· ,Ml} be an ordered collection of pairwise disjoint compact invariant subsets of A. We call M a Morse decomposition of A, if for every x ∈ A and every full solution γ with γ(0) = x, one of the following alternatives holds: (1) There is a Mi such that γ(t) ∈ Mi for all t ∈ R. (2) There are indices i < j such that α(γ) ⊂ Mj and ω(γ) ⊂ Mi, where α(γ) and ω(γ) denote the α-limit set and ω-limit set of γ, respectively.

Remark 2. Let M = {M1, ··· ,Ml} be a Morse decomposition of A. Set [ u Ak = W (Mi), 1 ≤ k ≤ l, 1≤i≤k where u W (Mi) = {γ(0)| γ is a full solution in A with α(γ) ⊂ Mi} is the unstable manifold of Mi. Then each Ak is an attractor of S in A; see [23], Section 2. Let A be an attractor of S with the basin of attraction Ω = Ω(A). S is called gradient-like on Ω, if there exists a continuous function L :Ω → R which decreases along each non-equilibrium solution in Ω. Remark 3. It is known that if S is a gradient-like system on Ω with only a finite number of equilibria in Ω, then A has a natural Morse decomposition consisting of precisely the equilibria of S in Ω. 2.2. Cocycle semiflows. A nonautonomous system consists of a “base flow” and a “cocycle semiflow” that is in some sense driven by the base flow. A base flow θ = θ(t)(t ∈ R) is a flow on a metric space Σ such that θ(t)Σ = Σ for all t ∈ R. Let θ be a base flow on Σ. For notational simplicity, from now on we will rewrite θ(t) as θt. Definition 2.4. A cocycle semiflow ϕ on the phase space X over θ is a continuous map ϕ : R+ × Σ × X → X satisfying • ϕ(0, σ, x) = x for all (σ, x) ∈ Σ × X, and 1180 XUEWEI JU, DESHENG LI AND JINQIAO DUAN

• ϕ(t + s, σ, x) = ϕ(t, θsσ, ϕ(s, σ, x)) for all (σ, x) ∈ Σ × X and t, s ≥ 0 (cocycle property). Remark 4. The flow θ and the space Σ in the above definition are usually called the driving system and base space of ϕ, respectively. Let ϕ be a given cocycle semiflow on X with driving system θ and base space Σ. As usual, we will write ϕ(t, σ, x) = ϕ(t, σ)x. For convenience, a family of subsets B = {Bσ}σ∈Σ of X is called a nonautonomous set. Let B = {Bσ}σ∈Σ be a nonautonomous set. We usually rewrite Bσ as B(σ), called the σ-section of B. We also denote [ P(B) = B(σ) × {σ}. σ∈Σ Note that P(B) is a subset of X × Σ. A nonautonomous set B is said to be closed (resp. open, compact), if P(B) is closed (resp. open, compact) in X × Σ. Let B and U be two nonautonomous sets. We say that U is a neighborhood of B, if P(U) is a neighborhood of P(B) in X × Σ. A nonautonomous set B is said to be invariant under ϕ if

ϕ(t, σ)B(σ) = B(θtσ), σ ∈ Σ. Let B and M be two nonautonomous subsets of X. We say that M pullback (resp. forward) attracts B under ϕ, if for any σ ∈ Σ,

lim HX (ϕ(t, θ−tσ)B(θ−tσ), M(σ)) = 0 t→+∞

( lim HX (ϕ(t, σ)B(σ), M(θtσ)) = 0 ). t→+∞ Deﬁnition 2.5. A nonautonomous set A is called a pullback (resp. forward) attractor, if it is invariant and pullback (forward) attracts a neighborhood of itself.

Proposition 1. Assume that the base space Σ is compact. Let A be a pullback attractor of ϕ. If the section A (σ) is continuous in σ (in the sense of Hausdorﬀ distance), then A is a forward attractor. Proof. The argument is a slight modiﬁcation of that of [9, Theorem 3.3] (see also [9, Theorem 4.3]), and is thus omitted.

2.3. Non-autonomous equilibria. Let ϕ be a given cocycle semiﬂow on X with driving system θ and base space Σ, and J ⊂ R be an interval. A map γ : J → X is called a solution of ϕ on J, if there exists σ ∈ Σ such that

γ(t) = ϕ(t − s, θsσ)γ(s), ∀ t, s ∈ J, t ≥ s.

A solution on J = R is called a full solution. Remark 5. We will also call a solution γ deﬁned as above a σ-solution of ϕ to emphasize the relation between γ and σ. Deﬁnition 2.6. A nonautonomous equilibrium of ϕ is a continuous map Γ ∈ C(Σ,X) such that γ(t) = Γ(θtσ) is a full solution of the system for each σ ∈ Σ. FORWARD ATTRACTORS AND SYNCHRONIZING BEHAVIOR 1181

Deﬁnition 2.7. Let Γ be a nonautonomous equilibrium of ϕ, and U a neighborhood of Γ. Let σ ∈ Σ. The σ-section unstable manifold and local σ-section unstable manifold of Γ in U of Γ are deﬁned, respectively, to be the sets u there is a σ-solution γ(t) on (−∞, 0] with γ(0) = x W (Γ, σ) = x ∈ X , such that limt→−∞ d(γ(t), Γ(θtσ)) = 0   there is a σ-solution γ(t) on (−∞, 0] with γ(0) = x u   W (Γ, σ) = x ∈ X and γ(t) ∈ U(θtσ) for all t ≤ 0 such that . U  limt→−∞ d(γ(t), Γ(θtσ)) = 0 

3. Nonautonomous perturbation of hyperbolic equilibria of (1.3). In this section, we ﬁrst discuss nonautonomous perturbation of hyperbolic equilibrium points of the autonomous system (1.3). Speciﬁcally, we show that the perturbed system (1.4) has a unique nonautonomous equilibrium Γε = Γε(p) near each hyperbolic equilibrium of the non-perturbed system (1.3). Furthermore, there is a nonautonomous neighborhood U of Γε such that the local section unstable manifold u WU (Γε, p) depends on p continuously. These results will play an important role in discussing persistence of the topological structures of attractors under nonautonomous perturbations. However, they may be of independent interest in their own right. 3.1. Mathematical setting. Let X be the Banach space appeared in the introduction. Denote Cb(R,X) the set of bounded continuous functions from R to X. Equip Cb(R,X) with either the uniform convergence topology generated by the metric r(h1, h2) = sup kh1(t) − h2(t)k, t∈R or the compact-open topology generated by the metric ∞ X 1 maxt∈[−n,n] kh1(t) − h2(t)k r(h , h ) = · . 1 2 2n 1 + max kh (t) − h (t)k n=1 t∈[−n,n] 1 2

Then Cb(R,X) is a complete metric space. Let h ∈ Cb(R,X) be the function in (1.2). Deﬁne the hull of h as follows C ( ,X) H := H[h] = {h(τ + ·): τ ∈ R} b R . As we have mentioned in the introduction, we assume that H is compact. Note that this covers all the cases where p is a periodic function, quasiperiodic function, almost periodic function, local almost periodic function [13, 22] or uniformly almost automorphic function [27]. One can also easily verify that if h is uniformly continuous on R, then H is compact with respect to the compact-open topology. The translation group θ on H is deﬁned as

θtp = p(t + ·), ∀ t ∈ R, p ∈ H. Let A be the sectorial operator in (1.2). Pick a number a > 0 sufficiently large so that min{Re z : z ∈ σ(Λ)} > 0, where Λ = A + aI. Then one can define fractional powers Λα of Λ for α ≥ 0; see Henry [16] for details. For each α ≥ 0, define Xα = D(Λα). As usual, Xα is quipped with the norm k · kα defined by α α kxkα = kΛ xk, x ∈ X . 1182 XUEWEI JU, DESHENG LI AND JINQIAO DUAN

It is well known that Xα is a Banach space, whose deﬁnition is independent of the choice of the number a. Now we turn our attention to the cocycle system (1.4). From now on we always assume that f is a locally Lipschitz map from Xα to X for some α ∈ [0, 1). We infer from Henry [16] that the initial value problem of (1.4) is well-posed. More precisely, we have

α Proposition 2. [16] Given x0 ∈ X , p ∈ H and t0 ∈ R, there exists T > t0 such that (1.4) has a unique (mild) solution x(t) = xε(t, t0; x0, p) on [t0,T ) with x(t0) = x0, namely, there is a unique function x ∈ C([t0,T ),X) satisfying Z t −A(t−t0) −A(t−s) x(t) = e x0 + e [f(x(s)) + εp(s)]ds, t ∈ [t0,T ). (3.1) t0

For convenience, we assume that the solution xε(t, t0; x0, p) given in the above α proposition is globally defined for all x0 ∈ X , p ∈ H and t0 ∈ R. Set α ϕε(t, p)x := xε(t, 0; x, p), x ∈ X , p ∈ H. α Then ϕε is a cocycle semiflow on X driven by the translation group θ on H. Remark 6. When ε = 0, the equation (1.4) reduces to the autonomous one. Con- α sequently ϕ0 coincides with the semiflow S on X defined by the solutions of the initial value problem of the equation (1.3). We now assume, without loss of generality, that 0 is a hyperbolic equilibrium of the non-perturbed system (1.3). Then the systems (1.4) and (1.3) can be rephrased, respectively, as follows:

xt + Lx = g(x) + εp(t), p ∈ H := H[h], (3.2)

xt + Lx = g(x), (3.3) where 0 L = A − f (0), g = o(kxkα) as kxkα → 0.

Denote k(ρ) the Lipschitz constant of g in BXα (ρ), where BXα (r) denotes the ball α in X centered at 0 with radius r. Then limρ→0 k(ρ) = 0. Note that

kg(x1) − g(x2)k ≤ k(ρ)kx1 − x2kα, ∀ x1, x2 ∈ BXα (ρ). (3.4)

The spectrum σ(L) of L has a decomposition σ(L) = σ1 ∪ σ2, where

σ1 = σ(L) ∩ {Re λ < 0}, σ2 = σ(L) ∩ {Re λ > 0}.

Accordingly, the space X has a direct sum decomposition: X = X1 ⊕ X2. Let

Πi : X → Xi, i = 1, 2 be the projection from X to Xi. Set α α Xi := Xi ∩ X , i = 1, 2. α α α α Then X = X1 ⊕ X2 . Note that X1 coincides with X1 since dim(X1) < ∞.

Denote L1 = L|X1 and L2 = L|X2 . By the basic knowledge on sectorial operators (see Henry [16]), we know that there exist M ≥ 1, β > 0 kΛαe−L1tk ≤ Meβt, ke−L1tk ≤ Meβt, t ≤ 0, (3.5)

α −L2t −α −βt α −L2t −α −βt kΛ e Π2Λ k ≤ Me , kΛ e k ≤ Mt e , t > 0, (3.6) where Λ = A + aI. FORWARD ATTRACTORS AND SYNCHRONIZING BEHAVIOR 1183

3.2. A basic lemma. This subsection consists of a fundamental lemma which will play a crucial role in our discussion. Lemma 3.1. The following assertions hold: (a) Let x :(−∞, 0] → Xα be continuous and bounded. Then x is the solution of (1.4) if and only if it solves the following integral equation Z t −L1t −L1(t−s) x(t) = e Π1x(0) + e Π1[g(x(s)) + εp(s)]ds 0 (3.7) Z t −L2(t−s) + e Π2[g(x(s)) + εp(s)]ds. −∞ (b) Let x : R → Xα be continuous and bounded. Then x is the solution of (1.4) if and only if it solves the integral equation Z t −L2(t−s) x(t) = e Π2[g(x(s)) + εp(s)]ds −∞ (3.8) Z ∞ −L1(t−s) − e Π1[g(x(s)) + εp(s)]ds. t Proof. (a) Let x :(−∞, 0] → Xα be a bounded solution of (1.4). We write x(t) = x1(t) + x2(t), where xi(t) = Πi x(t)(i = 1, 2). Then Z t −L1t −L1(t−s) x1(t) = e x1(0) + e Π1[g(x(s)) + εp(s)]ds, (3.9) 0 Z t −L2(t−t0) −L2(t−s) x2(t) = e x2(t0) + e Π2[g(x(s)) + p(s)]ds, (3.10) t0 α for t ≥ t0. By the boundedness of x in X , we have

−A2(t−t0) −β(t−t0) ke x2(t0)kα ≤ Me sup kx(t)kα → 0, as t0 → −∞. t≤0 It follows that Z t −L2(t−s) x2(t) = e Π2[g(x(s)) + εp(s)]ds. −∞ Therefore x(t) = x1(t) + x2(t) Z t −L1t −L1(t−s) = e x1(0) + e Π1[g(x(s)) + εp(s)]ds 0 Z t −L2(t−s) + e Π2[g(x(s)) + εp(s)]ds. −∞ This is precisely what we desired in (3.7). If x satisﬁes (3.7), then it clearly solves (1.4) on (−∞, 0]. The proof of (a) is complete. b α a ( ) Let x : R → X be a bounded solution of (1.4). Let t0 ∈ R. Similar to ( ), x(t)(t ≤ t0) solves the following integral equation: Z t −L1(t−t0) −L1(t−s) x(t) = e Π1x(t0) + e Π1[g(x(s)) + εp(s)]ds t0 Z t −L2(t−s) + e Π2[g(x(s)) + εp(s)]ds. −∞ 1184 XUEWEI JU, DESHENG LI AND JINQIAO DUAN

−L1(t−t0) By the boundedness of x, we have e Π1x(t0) → 0, as t0 → ∞. It follows that x solves (3.8). If x satisfies (3.8), then it clearly solves (1.4) on R. This completes the proof of (b). 3.3. Main results. The main results in this section are summarized in the following theorem. Theorem 3.2. Assume that 0 is a hyperbolic equilibrium of (1.3). Then there exist ρ, ε0 > 0 such that the following assertions hold: 0 (a) ϕε has a nonautonomous equilibrium Γ := Γε ∈ C(H, BXα (ρ)) for ε ≤ ε . (b) There exists % > 0 (independent of ε ∈ [0, ε0]) and a family of Lipschitz continuous maps α ξ : B α (%) → X , p ∈ H p X1 2 such that u W (Γ, p) = {y + ξ (y): y ∈ B α (%)}. (3.11) U p X1 Here U is a nonautonomous neighborhood of Γ defined by α U(p) = {x ∈ X : kΠ1xkα ≤ %} ∩ BXα (ρ), p ∈ H. α (c) The map Ξ: H → C B α (%),X defined by X1 2

Ξ(p) = ξp, p ∈ H is continuous.

Remark 7. The continuous dependence of ξp on p implies that u u lim δXα (WU (Γ, p),WU (Γ, q)) = 0, ∀ q ∈ H. (3.12) p→q Remark 8. As we have mentioned in the introduction, the non-autonomous perturbation problem has already been investigated comprehensively in the work of Kloeden and Rodrigues [19] in a more general setting. But comparing with the results in [19], our description on the synchronizing behavior of system (3.2) seems to be a little more explicit. Proof of Theorem 3.2. (a). Set Z ∞ −α −βs Mβ := M (1 + s )e ds, 0 where M, β are the constants in (3.5)-(3.6). Fix a number ρ > 0 with

Mβk(ρ) < 1, (3.13) and choose a number ε0 > 0 suﬃciently small so that 0 ∞ Mβ k(ρ) + ε khkL (R,X)/ρ < 1. (3.14) For each p ∈ H, one can use the righthand side of equation (3.8) to deﬁne a map T = Tp on the space α ∞ α X := {x ∈ C(R,X ): kxkL (R,X ) ≤ ρ} as follows: Z t −L2(t−s) T x(t) = e Π2[g(x(s)) + εp(s)]ds −∞ Z ∞ −L1(t−s) − e Π1[g(x(s)) + εp(s)]ds, ∀x ∈ X . t FORWARD ATTRACTORS AND SYNCHRONIZING BEHAVIOR 1185

We show that T X ⊂ X and is a contraction map. For every t ∈ R, simple computations show that Z ∞ β(t−s) kT x(t)kα ≤ M e (k(ρ)kx(s)kα + εkp(s)k) ds t Z t −α −β(t−s) + M (t − s) e (k(ρ)kx(s)kα + εkp(s)k) ds −∞ Z ∞ −βs (3.15) ∞ ≤ M e ds k(ρ)ρ + εkpkL (R,X) 0 Z ∞ −α −βs ∞ + M s e ds k(ρ)ρ + εkpkL (R,X) 0 ∞ ≤ Mβ k(ρ)ρ + εkhkL (R,X) ≤ (by (3.14)) ≤ ρ. Here we have used the simple fact that

∞ ∞ kpkL (R,X) ≤ khkL (R,X) < ∞, ∀ p ∈ H.

∞ α Since t is arbitrary, we see that kT (x)kL (R,X ) ≤ ρ. Hence T (x) ∈ X . Now let x, z ∈ X . We have Z ∞ β(t−s) kT x(t) − T z(t)kα ≤Mk(ρ) e kx(s) − z(s)kαds t Z t −α −β(t−s) + Mk(ρ) (t − s) e kx(s) − z(s)kαds −∞

∞ α ≤Mβk(ρ)kx − zkL (R,X ), t ∈ R. Hence

∞ α ∞ α kT x − T zkL (R,X ) ≤ Mβk(ρ)kx − zkL (R,X ). Since Mβk(ρ) < 1, we conclude that T is a contraction map on X . Thanks to the classical Banach fixed-point theorem, T = Tp has a unique fixed point xp ∈ X . It is easy to see that xp is precisely a full solution of (1.4) satisfying α (3.8). Define Γ := Γε : H → X as

Γ(p) = xp(0), p ∈ H.

One easily veriﬁes that xp(t) = Γ(θtp), t ∈ R. To prove that Γ is a nonautonomous equilibrium of ϕε, there remains to check the continuity of Γ. For notational convenience, given µ, β ≥ 0, we denote

± −µ|t| ± β |||x|||µ,β = sup e kx(t)kβ , ∀ x ∈ C(R ,X ) ± t∈R and −µ|t| β |||x|||µ,β = sup e kx(t)kβ , ∀ x ∈ C(R,X ). t∈R 0 By (3.13), we can pick a β ∈ (0, β) so that Mβ0 k(ρ) < 1, where Z ∞ −α −β0s Mβ0 := M (1 + s )e ds. 0 Set µ = β − β0. One observes that −µ|t| −µ|s+(t−s)| −µ(|s|−|(t−s)|) −µ|s| µ|t−s| e = e ≤ e = e e , ∀ t, s ∈ R. (3.16) 1186 XUEWEI JU, DESHENG LI AND JINQIAO DUAN

Recall that xp(t) = Γ(θtp) solves (3.8). Let p, q ∈ H. Then for t ∈ R, by (3.8) and (3.16) we deduce that

−µ|t| e kxp(t) − xq(t)kα Z t −α −β(t−s) µ|t−s| −µ|s| ≤ Mk(ρ) (t − s) e e e kxp(s) − xq(s)kα ds −∞ Z ∞ −β(s−t) µ|t−s| −µ|s| + Mk(ρ) e e e kxp(s) − xq(s)kα ds t Z t + Mε (t − s)−αe−β(t−s)eµ|t−s| e−µ|s|kp(s) − q(s)k ds −∞ Z ∞ −β(s−t) µ|t−s| −µ|s| + Mε e e e kp(s) − q(s)k ds (3.17) t Z t −α −β0(t−s) −µ|s| = Mk(ρ) (t − s) e e kxp(s) − xq(s)kα ds −∞ Z ∞ −β0(s−t) −µ|s| + Mk(ρ) e e kxp(s) − xq(s)kα ds t t Z 0 + Mε (t − s)−αe−β (t−s) e−µ|s|kp(s) − q(s)k ds −∞ ∞ Z 0 + Mε e−β (s−t) e−µ|s|kp(s) − q(s)k ds. t Hence −µ|t| e kxp(t) − xq(t)kα Z t Z ∞ −α −β0(t−s) −β0(s−t) ≤ Mk(ρ) (t − s) e ds + e ds |||xp − xq|||µ,α −∞ t (3.18) Z t Z ∞ −α −β0(t−s) −β0(s−t) + Mε (t − s) e ds + e ds |||p − q|||µ,0 −∞ t ≤ Mβ0 k(ρ)|||xp − xq|||µ,α + Mβ0 |||p − q|||µ,0, ∀ t ∈ R. Thus Mβ0 |||xp − xq|||µ,α ≤ |||p − q|||µ,0. 1 − Mβ0 k(ρ) Therefore

Mβ0 kΓ(p) − Γ(q)kα = kxp(0) − xq(0)kα ≤ |||p − q|||µ,0. (3.19) 1 − Mβ0 k(ρ)

Since p → q (in H) implies that |||p − q|||µ,0 → 0, we conclude from (3.19) that

kΓ(p) − Γ(q)kα → 0, as p → q, which completes the proof of the continuity of Γ. (b) and (c). Take a % > 0 small enough so that 0 ∞ M%/ρ + Mβ k(ρ) + ε khkL (R,X)/ρ < 1, (3.20) where ε0 is the number in (3.14). Let − α ∞ − α Y := {x ∈ C(R ,X ): kxkL (R ,X ) ≤ ρ}. FORWARD ATTRACTORS AND SYNCHRONIZING BEHAVIOR 1187

For each p ∈ H and y ∈ B α (%), using the right hand side of equation (3.7) we can X1 ˆ ˆ deﬁne a map T = Tp,y on Y as follows: Z t −L1t −L1(t−s) Tˆx(t) = e y + e Π1[g(x(s)) + εp(s)]ds 0 Z t −L2(t−s) + e Π2[g(x(s)) + εp(s)]ds. −∞ We ﬁrst show that Tˆ maps Y to itself. Indeed, let x ∈ Y . Then for any t ≤ 0, Z 0 βt β(t−s) kTˆx(t)kα ≤ Me kykα + M e (k(ρ)kx(s)kα + εkp(s)k) ds t Z t −α −β(t−s) + M (t − s) e (k(ρ)kx(s)kα + εkp(s)k) ds −∞ Z −t Z ∞ −βs −α −βs ∞ ≤ Mkykα + M e ds + s e ds k(ρ)ρ + εkhkL (R,X) 0 0 ∞ ≤ Mkykα + Mβ k(ρ)ρ + εkhkL (R,X) ≤ (by 3.20) ≤ ρ, from which it immediately follows that Tˆ(x) ∈ Y . Now we check that Tˆ is contracting on Y . For x, z ∈ Y , we have

kTˆx(t) − Tˆz(t)kα Z 0 β(t−s) ≤ Mk(ρ) e kx(s) − z(s)kαds t Z t −α −β(t−s) + Mk(ρ) (t − s) e kx(s) − z(s)kαds −∞ Z −t Z ∞ −βs −α −βs ∞ − α ≤ Mk(ρ) e ds + s e ds kx − zkL (R ,X ) 0 0

∞ − α = (Mβk(ρ)) kx − zkL (R ,X ), t ≤ 0.

Since Mβk(ρ) < 1, we see from the above estimate that Tˆ is indeed a contraction map. ˆ By the Banach ﬁxed-point theorem, T has a unique ﬁxed point xp,y in Y . x(t) = − xp,y(t) is precisely a bounded solution of (1.4) on R with Π1x(0) = y, which equivalently solves the integral equation Z t −L1t −L1(t−s) x(t) = e y + e Π1[g(x(s)) + εp(s)]ds 0 (3.21) Z t −L2(t−s) − + e Π2[g(x(s)) + εp(s)]ds, t ∈ R . −∞

We claim that xp,y(0) is Lipschitz continuous in y uniformly with respect to p ∈ H. Indeed, let y, z ∈ B α (%). Then for t ≤ 0, simple computations show that X1

kxp,y(t) − xp,z(t)kα Z 0 βt β(t−s) ≤ Me ky − zkα + Mk(ρ) e kxp,y(s) − xp,z(s)kαds t Z t −α −β(t−s) + Mk(ρ) (t − s) e kxp,y(s) − xp,z(s)kαds −∞ 1188 XUEWEI JU, DESHENG LI AND JINQIAO DUAN Z −t Z ∞ −βs −α −βs ∞ − α ≤ Mky − zkα + Mk(ρ) e ds + s e ds kxp,y − xp,zkL (R ,X ) 0 0

∞ − α ≤ Mky − zkα + Mβk(ρ)kxp,y − xp,zkL (R ,X ). Hence

∞ − α ∞ − α kxp,y − xp,zkL (R ,X ) ≤ Mky − zkα + Mβk(ρ)kxp,y − xp,zkL (R ,X ). Therefore ∞ − α kxp,y(0) − xp,z(0)kα ≤ kxp,y − xp,zkL (R ,X ) M ≤ ky − zkα, 1−Mβ k(ρ) which justiﬁes our claim. Deﬁne Z 0 L s ξ (y) := e 2 Π [g(x (s)) + εp(s)]ds, y ∈ B α (%). (3.22) p 2 p,y X1 −∞ Setting t = 0 in (3.21) leads to

x (0) = y + ξ (y), y ∈ B α (%). (3.23) p,y p X1 α The Lipschitz continuity of x (0) in y then implies ξ : B α (%) → X is a Lipschitz p,y p X1 2 continuous map. Set u M (p) = {y + ξ (y): y ∈ B α (%)}. % p X1 u Clearly M (p) is homeomorphic to B α (%). Let U = {U } , where % X1 p p∈H α Up = {x ∈ X : kΠ1xkα ≤ %} ∩ BXα (ρ) := U, p ∈ H. (3.24) Then U is a (nonautonomous) neighborhood of Γ. We show that u u WU (Γ, p) = M% (p). (3.25) u u u Note that WU (Γ, p) ⊂ M% (p). Thus we only need to check that M% (p) ⊂ u W (Γ, p). For this purpose, it suﬃces to verify that for each y ∈ B α (%), U X1

lim kxp,y(t) − Γ(θtp)kα = 0. (3.26) t→−∞ − (It is clear that xp,y is a p-solution of ϕε on R .) We argue by contradiction and suppose that (3.26) was false. Then there would exist δ > 0 and a sequence 0 ≥ tn → −∞ such that

kxp,y(tn) − Γ(θtn p)kα ≥ δ. Let xn(t) = xp,y(t + tn), t ∈ [tn, −tn], and

yn(t) = Γ(θt+tn p), t ∈ [tn, −tn].

Then both xn(t) and yn(t) are solutions of the equation

xt + Ax = f(x) + εpn(t) (3.27) on [tn, −tn], where pn = θtn p. Note that

kxn(0) − yn(0)kα ≥ δ > 0 (3.28) for all n. By the compactness of H we may assume that θtn p → p0 ∈ H. Thus there exists a subsequence nk of n such that xnk (t) and ynk (t) converge uniformly FORWARD ATTRACTORS AND SYNCHRONIZING BEHAVIOR 1189 on every compact interval, respectively, to bounded full solutions x(t) and y(t) of the equation: xt + Ax = f(x) + εp0(t). (3.29) It is obvious that kx(t)kα, ky(t)kα ≤ ρ (3.30) for all t ∈ R. By (3.28) we also have kx(0)−y(0)kα ≥ δ. Moreover, we infer from the proof of (a) that for ﬁxed p0 the bounded full solutions of (3.29) satisfying (3.30) are unique. Hence x(t) ≡ y(t) on R, which leads to a contradiction and completes the proof of (3.25). Finally, we verify that ξp is continuous in p, which ﬁnishes the proof of (c). 0 Set µ = β − β . Let y ∈ B α (%), and p, q ∈ H. Then similar computations as in X1 (3.17) and (3.18) apply to show that − − − |||xp,y − xq,y|||µ,α ≤ Mβ0 k(ρ)|||xp,y − xq,y|||µ,α + Mβ0 |||p − q|||µ,0. Therefore

− Mβ0 − kxp,y(0) − xq,y(0)kα ≤ |||xp,y − xq,y|||µ,α ≤ |||p − q|||µ,0. 1 − Mβ0 k(ρ) It then follows by (3.23) that

Mβ0 − kξp(y) − ξq(y)kα ≤ |||p − q|||µ,0. (3.31) 1 − Mβ0 k(ρ) − It is trivial to check that if p → q in H then |||p − q|||µ,0 → 0. By (3.31) one immediately concludes that

lim sup kξp(y) − ξq(y)kα = 0. p→q y∈B α (%) X1 Remark 9. Let U ⊂ Xα be the set given in (3.24). We infer from the proof of Theorem3 .2 that if γ(t)(t ≤ 0) is a backward p-solution of ϕε satisfying γ(t) ⊂ U (t ≤ −t0) for some t0 > 0, then

lim kγ(t) − Γ(θtp)kα = 0. t→−∞ Using a similar argument as in verifying (3.26), it can be shown that if γ(t) = ϕε(t, p)x is a forward p-solution satisfying γ(t) ⊂ U (t ≥ t0) for some t0 > 0, then

lim kγ(t) − Γ(θtp)kα = 0. t→∞

4. Morse structures of pullback attractors of the perturbed system. It is known that for an autonomous gradient-like system with only a ﬁnite number of equilibria, an attractor can be completely characterized via Morse decomposition- s, and the internal dynamics in the attractor is explicit: each full solution in the attractor goes forward (backward) to a single equilibrium and there is no homo- clinic structure associated to the equilibria. If all the equilibria are hyperbolic, this structure maintains stable under both autonomous and nonautonomous perturbations [4,6, 11, 15]. In this present work, based on the results in the previous section we give a more precise description on the stability of Morse structures of attractors for gradient-like systems under nonautonomous perturbations. 1190 XUEWEI JU, DESHENG LI AND JINQIAO DUAN

4.1. Stability of attractors under nonautonomous perturbations. Let us ﬁrst present a very basic result concerning the stability of attractors under nonautonomous perturbations. Speciﬁcally, suppose the non-perturbed system S = ϕ0 has an attractor A. We show that the perturbed system ϕε possesses a pullback attractor Aε with lim HXα (Aε(p), A) = 0 ε→0 uniformly with respect to p ∈ H. First, we have the following fundamental result. Lemma 4.1. For every T > 0 and bounded subset B of Xα, we have

lim kϕε(t, p)x − ϕε0 (t, p)xkα = 0, (4.1) ε→ε0 uniformly with respect to (t, x) ∈ [0,T ] × B and p ∈ H. Proof. Since A is sectorial, there exist M ≥ 1 and b ∈ R such that kΛαe−Atk ≤ Mt−αe−bt.

Denote xε(t) = ϕε(t, p)x. Then Z t 0 −α −b(t−s) kxε(t) − xε0 (t)kα ≤ |ε − ε |M (t − s) e kp(s)kds 0 (4.2) Z t −α −b(t−s) + Mk(ρ) (t − s) e kxε(s) − xε0 (s)kαds. 0 Let Z t −α −b(t−s) ∞ a(t) = MkhkL (R,X) (t − s) e ds. 0 Applying the Gronwall inequality (see e.g. [16, Lemma 7.1.1]) to (4.2) it yields Z t 0 0 −b(t−s) kxε(t) − xε0 (t)kα ≤ |ε − ε | a(t) + η Z1−α(η(t − s))e a(s)ds , (4.3) 0 1 where η = [Mk(ρ)Γ(1 − α)] 1−α , and ∞ Z ∞ X sn(1−α) Γ(1 − α) = s−αe−sds, Z (s) = . 1−α Γ[n(1 − α) + 1] 0 n=0 The conclusion then immediately follows. Lemma 4.2. Assume that A ⊂ Xα is an attractor of S. Let N be a neighborhood of A which is such that A attracts N. Then for every δ > 0, there exist 0 < ε0 < 1 and T > 0 such that if ε ∈ [0, ε0] then

ϕε(t, p)N ⊂ BXα (A, δ), t ≥ T, p ∈ H. (4.4) Proof. The proof can be obtained by modifying some argument in the proof of [21, Theorem 4.1]. We omit the details. Now we state and prove the following stability result concerning attractors under nonautonomous perturbations. α Theorem 4.3. Assume that A ⊂ X is an attractor of S. Then there exists ε0 > 0 such that if ε ≤ ε0, then ϕε has a pullback attractor Aε; furthermore,

lim dH (Aε(p), A) = 0 (4.5) ε→0 uniformly with respect to p ∈ H. FORWARD ATTRACTORS AND SYNCHRONIZING BEHAVIOR 1191

Proof. Take a δ > 0 suﬃciently small so that BXα (A, 2δ) ⊂ N, where N is the neighborhood of A given in Lemma 4.2. Then by Lemma 4.2 there exist ε0 > 0 and T > 0 such that (4.4) holds for all ε ∈ [0, ε0]. Set \ [ Aε(p) = ϕε(t, θ−tp)N. τ>0 t≥τ

Then Aε = {Aε(p)}p∈H is a pullback attractor of ϕε. Clearly

Aε(p) ⊂ BXα (A, δ), ∀ p ∈ H. The conclusion in (4.5) is a simple consequence of the fact that δ can be chosen arbitrarily small.

4.2. Morse structures of perturbed pullback attractors. Let S be the autonomous system generated by (1.3), and let A be an attractor of S with the attraction basin Ω = Ω(A). Assume that S is a gradient-like system on the basin of attraction Ω of an attractor A, with a ﬁnite number of hyperbolic equilibria n E = {Ei}i=1. Then E forms a natural Morse decomposition of A. We have 0 0 Theorem 4.4. There exists ε > 0 such that if ε < ε , the perturbed system ϕε has a pullback attractor Aε = {Aε(p)}p∈H near A; furthermore, the following assertions hold. i i (1) ϕε has a ﬁnite number of nonautonomous equilibria Γε = Γε(p)(i = 1, 2, ··· , i n) with each Γε being close to Ei. i (2) For each p ∈ H, Aε(p) is the union of the section unstable manifolds of Γε (i = 1, 2, ··· , n) at p, namely, n [ u i Aε(p) = W (Γε, p). i=1 (3) There is a neighborhood N of A (independent of ε) such that for each x ∈ N, j there is a Γε such that j lim d ϕε(t, p)x, Γε(θtp) = 0. t→∞

Proof. The existence part of the pullback attractor Aε is contained in Theorem 4.3, and the conclusion in (1) is contained in Theorem 3.2. (2) The argument below is adapted from the proof of [6, Theorem 2.11]. 0 0 Take ε > 0 sufficiently small so that (1) holds for ε ≤ ε ; furthermore, ϕε possesses a pullback attractor Aε closed to A. We infer from the proof of Theorem α 3.2 and Remark9 that for each 1 ≤ i ≤ n, there is a neighborhood Ui ⊂ X of Ei − such that for any solution γ of ϕε on R in Ui, it holds that i lim kγ(t) − Γε(θtp)kα = 0 (4.6) t→−∞ for some p ∈ H. It can be assumed that Ui ∩ Uj = ∅ for i 6= j. To prove the second conclusion (2), it now suffices to check if ε > 0 is sufficiently small, then for any full solution γ in Aε, there exist 1 ≤ i ≤ n and t0 > 0 such that

γ(t) ∈ Ui, t ≤ −t0. (4.7)

Since M = {E1, ··· ,En} is a Morse decomposition of the attractor A of the non-perturbed system S = ϕ0, for every full solution γ of S in A, we have either 1192 XUEWEI JU, DESHENG LI AND JINQIAO DUAN

γ(R) ≡ Ei for some i, or else there are indices i < j such that

lim γ(t) = Ej, lim γ(t) = Ei. (4.8) t→−∞ t→∞ S u Furthermore, Ak = 1≤i≤k W (Ei) is an attractor of S in A for each 1 ≤ k ≤ n. α By [17, Lemma 2.5] we also deduce that each Ak is an attractor of S in X . Note that {E1} = A1 A2 ··· An = A.

By Lemma 4.2 there exists δ, εˆ > 0 such that Nk = BXα (Ak, δ) uniformly attracts itself under the perturbed system ϕε for all ε ≤ εˆ and 1 ≤ k ≤ n. More precisely, there exists T > 0 such that if ε ≤ εˆ, then

ϕε(t, p)Nk ⊂ Nk, t > T, p ∈ H (4.9) for all ε ≤ εˆ and 1 ≤ k ≤ n. For convenience, we assign N0 := ∅. We may assume that δ is suﬃciently small so that

Uk+1 ∩ Nk = ∅ for all k (recall that Uk+1 is a neighborhood of Ek+1 and Ek+1 6∈ Ak). We ﬁrst claim that for each full solution γ of ϕε in Aε (ε ≤ εˆ), there is a 1 ≤ k ≤ n such that

γ(t) ∈ Nk \ Nk−1 (4.10) for all t < 0 suﬃciently large. Indeed, by (4.9) we see that γ(R) ⊂ Nn. Let k be the smallest number such that γ(R) ⊂ Nk. We show that (4.10) holds true, which completes our claim. Suppose the contrary. There would exist a sequence tm → −∞ such that γ(tm) ∈ Nk−1. For any t ∈ R, pick a tm such that t − tm > T (T is the number in (4.9)). Then for some p ∈ H,

γ(t) = ϕε(t − tm, p)γ(tm) ∈ Nk−1.

Hence γ(R) ⊂ Nk−1. This leads to a contradiction. In what follows, we again argue by contradiction. Suppose that (4.7) failed to be true. Then there would existε ˆ ≥ εm → 0 such that for each εm, ϕεm has a full m solution γm in Aεm and a sequence 0 ≥ tl → −∞ (as l → ∞) such that n m [ γm(tl ) ∈/ Uj, l ≥ 1. (4.11) j=1 0 Since there are only a ﬁnite number of Nk s, by the claim above we deduce that there is a k ∈ {1, 2, ··· , n} and a subsequence of γm (m = 1, 2, ··· ), still denoted by γm, such that for each m ≥ 1,

γm(t) ∈ Nk \ Nk−1, ∀ t ≤ Tm m m for some Tm < 0. For each m, pick two numbers tl and tl0 with m m m m tl , tl0 < Tm, and tl0 − tl > −m. m m Write τm = tl − tl0 . Then τm → +∞. Deﬁne m xm(t) = γm(t + tl0 ), t ∈ Jm = [−τm, τm]. It is clear that n m [ xm(0) = γm(tl0 ) ∈/ Uj, and xm(Jm) ⊂ Nk \ Nk−1 (4.12) j=1 FORWARD ATTRACTORS AND SYNCHRONIZING BEHAVIOR 1193 for all m. Now by very standard argument, it can be shown that the sequence xm(t) has a subsequence, still denoted by xm(t), such that xm(t) converges uniformly on any compact interval of R to a bounded full solution x(t) of S in A (recall that Sn εm → 0 as m → ∞). By (4.12) we conclude that x(0) ∈/ j=1 Uj and

x(R) ⊂ Nk \ Nk−1. Sn It follows by x(0) ∈/ j=1 Uj that x(t) is not an equilibrium solution. Thus by (4.8) there exist two distinct equilibria Ei,Ej such that

lim x(t) = Ej, and lim x(t) = Ei. t→−∞ t→∞

This leads to a contradiction, as there is only one equilibrium in Nk \ Nk−1.

(3) Let Nk (1 ≤ k ≤ n) andε ˆ > 0 be the same as above, and let N = Nn. Then for each x ∈ N,

ϕε(t, p)x ∈ N = Nn, t ≥ T.

Let k be the smallest number such that ϕε(t, p)x ∈ Nk for all t > 0 sufficiently large. Then as in (4.10) it can be shown that 0 ϕε(t, p)x ∈ Nk \ Nk−1, ∀ t ≥ T for some T 0 > 0. Based on this result, one can prove by a fully analogous argument as in the verification of (4.7) that if ε > 0 is sufficiently small, then for any x ∈ N and p ∈ H, there exists 1 ≤ i ≤ n and t0 > 0 such that

ϕε(t, p)x ∈ Ui, t ≥ −t0. (4.13)

It is clear that x is attracted by an equilibrium Ei under S for some i. Similar to the proof of Lemma4 .2, there is a T > 0 such that for suﬃciently small ε and p ∈ H,

ϕε(t, p)x ∈ Ui, t ≥ T. It then follows by Remark9 that i lim d(ϕε(t, p)x, Γε(θtp)) = 0. t→∞ This completes the proof.

Remark 10. The interested reader is referred to [1,7, 24] etc. for some general theories on Morse decompositions of nonautonomous dynamical systems.

5. Forward attraction of the perturbed attractors and synchronizing behavior of the perturbed nonautonomous system. As we have mentioned in the introduction, in general the existence of forward attractors of nonautonomous systems remains an open problem. In this section we show that the pullback attractor Aε of the perturbed system ϕε given in Theorem 4.4 is a forward attractor, using the Morse structures of pullback attractors. For the sake of clarity, in this section we drop the subscript “ε” in Aε and rewrite Aε = A .

Theorem 5.1. Assume the hypotheses in Theorem 4.4, and let A = {A (p)}p∈H be the pullback attractor of the perturbed system ϕε given in Theorem 4.4. Then (1) A (p) is continuous in p (in the sense of Hausdorﬀ distance); and (2) A is a forward attractor of ϕε. 1194 XUEWEI JU, DESHENG LI AND JINQIAO DUAN

Remark 11. The forward attraction of A means that there is a neighborhood B S of Au := p∈H A (p) such that

lim HXα (ϕε(t, p)B, A (θtp)) = 0. (5.1) t→∞ Remark 12. Recall that if the external force h in (1.2) is periodic (resp. quasiperiodic, almost periodic, uniformly almost automorphic), then θtp is periodic (resp. quasiperiodic, almost periodic, uniformly almost automorphic) for any ﬁxed p ∈ H := H[h]. Thus under the assumptions of Theorem 5.1, A (p) is continuous in p and A (θtp) is periodic (resp. quasiperiodic, almost periodic, uniformly almost automorphic). The convergence (5.1) then indicates that the system exhibits a global synchronizing behavior with the external force h.

Proof of Theorem 5.1. (1) The verification of the upper semicontinuity of A (p) can be performed easily. Since the argument is quite standard, we omit the details. In what follows we only check the lower semicontinuity of A (p) in p. For this purpose, by virtue of [5, Lemma 3.2, pp. 75] it suffices to verify that for any p0 ∈ H, x0 ∈ A (p0) and sequence pk → p0, there is a sequence xk with xk ∈ A (pk) (for each k) such that xk → x0 as k → ∞. We may assume that ε > 0 is sufficiently small so that Theorem 4.4 holds. Sn u i Since x0 ∈ A (p0) = i=1 W (Γi, p0), where Γi := Γε (1 ≤ i ≤ n) are the nonautonomous equilibria of ϕε given by Theorem 4.4, there is a j ∈ {1, ··· , n} such u that x0 ∈ W (Γj, p0). Thus we can find a full solution γ with γ(0) = x0 such that

d(γ(t), Γj(θtp0)) → 0 as t → −∞.

Let Uj be the neighborhood of the equilibrium point Ej of the autonomous system S = ϕ0 given in the proof of Theorem 4.4, and let Uj = Uj × H. Choose a τ > 0 such that z = γ(−τ) ∈ W u (Γ , q ), where q = θ p . Uj j 0 0 −τ 0 Let q = θ p . Clearly q → q . Thus by the continuity of W u (Γ , p) in p (see k −τ k k 0 Uj j Remark7) we can choose a sequence {zk}n∈N with z ∈ W u (Γ , q ) ⊂ (q ) k Uj j k A k

(for each k) such that zk → z as k → ∞. Note that by the continuity property of ϕε, we have

lim kϕε(τ, qk)zk − ϕε(τ, q0)zkα = lim kϕε(τ, qk)zk − x0kα = 0. (5.2) n→∞ n→∞

Let xk := ϕε(τ, qk)zk. Then by the invariance property of A we have

xk = ϕε(τ, qk)zk ∈ ϕε(τ, qk)A (qk) = A (θτ qk) = A (pk).

Then (5.2) asserts that xk → x0 as k → ∞. This completes the proof. (2) The forward attraction property of A follows from the continuity of A (p) in p and Proposition1. Finally, let us give an example to illustrate our results.

Example 5.1. Consider the nonautonomous system: ( ut − ∆u = f(u) + εh(x, t), t > 0, x ∈ Ω; (5.3) u = 0, t > 0, x ∈ ∂Ω, FORWARD ATTRACTORS AND SYNCHRONIZING BEHAVIOR 1195 where Ω is a bounded domain in R3 with smooth boundary, ε > 0, and 3 2 f(s) = −b3s + b2s + b1s + b0, b3 > 0. Denote by A the operator −∆ associated with the homogeneous Dirichlet boundary condition. Then A is a sectorial operator on X = L2(Ω) with compact resolvent, 2 T 1 and D(A) = H (Ω) H0 (Ω). The system (5.3) can be written into an abstract equation on X:

ut + Au = f(u) + εh(t), where h(t) = h(·, t). The non-perturbed system reads

ut + Au = f(u), (5.4) which is a gradient-like system, with energy functional 1 Z L(u) = |∇u|2 − F (u) dx, 2 Ω where F is a primitive of f. It is well known that (5.4) has a global attractor A (see e.g. [28]). We also infer from [3] (see Sect. 3) that all equilibria of (5.4) are 4 hyperbolic generically for (b0, ··· , b3) ∈ R . Thus we can reasonably assume that (5.4) has only a ﬁnite number of equilibria E1, ··· ,En. Consider the cocycle system:

ut + Au = f(u) + εp(t), p ∈ H. (5.5)

1 Denote ϕε := ϕε(t, p)u the cocycle semiflow on H0 (Ω) driven by the base flow (translation group) θ on H. Assume that h ∈ C(R,X) is translation compact with hull H := H[h]. Then the system is dissipative and hence has a global pullback attractor A = {A (p)}p∈H such that A (p) approaches A as ε → 0 for each p ∈ H. Since the hypotheses in the main theorems are fulfilled, we obtain some interesting results concerning the dynamics of the perturbed system as follows.

Theorem 5.2. If ε is suﬃciently small then the following assertions about the pullback attractor A = {A (p)}p∈H of the perturbed system hold true. i i (1) ϕε has a ﬁnite number of nonautonomous equilibria Γ = Γ (p)(i = 1, 2, ··· , n) such that for each p ∈ H,

n [ A (p) = W u(Γi, p). i=1

1 j (2) For each u ∈ H0 (Ω), there is a Γ such that j lim d ϕε(t, p)u, Γ (θtp) = 0. t→∞ (3) A is a uniform forward attractor.

Acknowledgments. We would like to express our gratitude to the anonymous ref- erees for their valuable comments and suggestions which helped us greatly improve the quality of the paper, and also for bringing into our attention the nice work of Kloeden and Rodrigues [19]. 1196 XUEWEI JU, DESHENG LI AND JINQIAO DUAN

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