A Note on Some Properties of Local Random Attractors

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1 Northeast. Math. J. 24(2)(2008), A Note on Some Properties of Local Random Attractors LIU Zhen-xin ( ) (School of Mathematics, Jilin University, Changchun, ) SU Meng-long ( ) (Mathematics and Information Science College, Luoyang Normal University, Luoyang, ) ZHU Wen-zhuang ( ) (School of Mathematical Sciences, Nankai University, Tianjin, ) Abstract: In this note, we study some properties of local random pull-back attractors on compact metric spaces. We obtain some relations between attractors and their fundamental neighborhoods and basins of attraction. We also obtain some properties of omega-limit sets, as well as connectedness of random attractors. A simple deterministic example is given to illustrate some confusing problems. Key words: random dynamical system, random attractor, pull-back attractor 2000 MR subject classification: 37H99 CLC number: O175, O211 Document code: A Article ID: (2008) Introduction One important aspect of qualitative analysis of differential equations and dynamical systems is the study of asymptotic, long-term behavior of solutions/orbits. Hence the omega-limit set and the attractor have been studied by numerous authors, for instance, see [1] [5] etc. For random dynamical systems (RDS), according to the manner of convergence, there are several nonequivalent definitions of random attractors: pull-back attractors were introduced by Crauel and Flandoli [6] and Schmalfuss [7], weak random attractors were introduced by Ochs [8], and besides these two are forward random attractors. See [9] for a comparison of various concepts of random attractors. Random attractors have been studied by many authors, see [6], [7], [9] [14] etc. for instance. But in studying some problems, we also need to consider local random attractor with a random neighborhood. For example, in [8, 15], the authors introduced local weak random attractors to get Morse theory for RDS. Following [8, 15], in [16] we also considered local random attractors to study the relation Received date: Jan. 4, Foundation item: Partially Supported by the SRFDP ( ) and the Young Fund of the College of Mathematics at Jilin University.

2 164 NORTHEAST. MATH. J. VOL. 24 between attractor-repeller pair, Morse decomposition and Lyapunov function for RDS. And in [17] [19], we have to again consider local random attractors to get the random version of Conley s fundamental theorem of dynamical systems. Besides these, it is well known that local attractor is important for the study of deterministic dynamical systems, so we think that local attractor will be also important for the study of RDS. Hence we think it is useful to study some properties of local random attractors. In this note, we study some properties of local random pull-back attractors for RDS on compact metric spaces, e.g. fundamental neighborhoods of random attractors, the basins of attraction, connectedness of random attractors etc., which have been well studied for deterministic dynamical systems, see [1] [3] and [5] for instance. We also study some properties of omega-limit sets for RDS. With respect to the connectedness of random attractors, Crauel obtained a result in Proposition 3.7 of [12], which is a revision of the corresponding result of [6]. With respect to the relation between the definition of local random attractors of the present paper and that of [15], it is easy to see that the definition of [15] is weaker; with respect to the relation between the definition of present paper and that of [16] [18], they are in fact equivalent, see Remark 3.1 of [18] for details. 2 Preliminaries First we give the definition of continuous random dynamical systems (see [20]). Definition 2.1 Let X be a metric space. A (continuous) random dynamical system (RDS), shortly denoted by ϕ, consists of two ingredients: A model of the noise, namely, a metric dynamical system (Ω, F, P, (θ t ) t R ), where (Ω, F, P) is a probability space and (t, ω) θ t ω is a measurable flow which leaves P invariant, i.e., θ t P = P for all t R. For simplicity we also assume that θ is ergodic under P, meaning that a θ-invariant set has probability 0 or 1. A model of the system perturbed by noise, namely, a cocycle ϕ over θ, i.e., a measurable mapping ϕ : R Ω X X, (t, ω, x) ϕ(t, ω, x), such that x ϕ(t, ω, x) is continuous for all t R and ω Ω and the family ϕ(t, ω, ) = ϕ(t, ω) : X X of random self-mappings of X satisfies the cocycle property: ϕ(0, ω) = id X, ϕ(t + s, ω) = ϕ(t, θ s ω) ϕ(s, ω), t, s R, ω Ω. (2.1) It follows from (2.1) that ϕ(t, ω) is a homeomorphism of X, and we have ϕ(t, ω) 1 = ϕ( t, θ t ω). Any mapping from Ω into the collection of all subsets of X is said to be a multifunction (or a set valued mapping) from Ω into X. We now give the definition of a random set, which is a fundamental concept for RDS. Definition 2.2 Let X be a metric space with a metric d X. A set-valued map ω D(ω) taking values in the closed/compact subsets of X is said to be a random closed/compact set

3 NO. 2 LIU Z. X. et al. SOME PROPERTIES OF LOCAL RANDOM ATTRACTORS 165 if the mapping ω dist X (x, D(ω)) is measurable for any x X, where dist X (x, B) := inf y B d X(x, y). A set-valued map ω U(ω) taking values in the open subsets of X is said to be a random open set if ω U c (ω) is a random closed set, where U c denotes the complement of U. Definition 2.3 It is said to be invariant if A random set D is said to be forward invariant under the RDS ϕ if ϕ(t, ω)d(ω) D(θ t ω), t R + P-a.s.; ϕ(t, ω)d(ω) = D(θ t ω), t R P-a.s. And we denote by Ω D the omega-limit set of D, i.e., Ω D (ω) := ϕ(s, θ s ω)d(θ s ω). t 0 s t Definition 2.4 For given two random sets D and A, we say A absorbs D if for P-a.s. ω Ω there exists t(ω) such that we say A (pull-back) attracts D if ϕ(t, θ t ω)d(θ t ω) A(ω) as t t(ω); lim d(ϕ(t, θ tω)d(θ t ω) A(ω)) = 0 t holds P-a.s., where d(a B) stands for the Hausdorff semi-metric between two sets A and B, i.e., d(a B) := sup inf d(x, y); x A y B and we say A attracts D in probability or weakly attracts D if Remark 2.1 P lim t d(ϕ(t, ω)d(ω) A(θ t ω)) = 0. It is easy to see that if A absorbs/attracts/weakly attracts D, then (i) A absorbs/attracts/weakly attracts any random subsets of D; (ii) A attracts/weakly attracts the closure D of D by the definition of Hausdorff semimetric and the continuity of ϕ; if A is closed, A also absorbs D; (iii) any random set containing A also absorbs/attracts/weakly attracts D; (iv) if A absorbs/attracts/weakly attracts D 1 and D 2 respectively, then A also absorbs/attracts/weakly attracts D 1 D2 and D 1 D2. Definition 2.5 (i) An invariant random compact set A is called an (local) attractor if there exists a forward invariant random closed neighborhood N of A such that A(ω) = Ω N (ω) P-a.s. The neighborhood N is called a fundamental neighborhood of A. (ii) Assume that A is an attractor with a fundamental neighborhood N. Then we call the random set B(A), given by the basin of attraction of A. B(A)(ω) := {x ϕ(t, ω)x intn(θ t ω) for some t 0}, (2.2)

4 166 NORTHEAST. MATH. J. VOL. 24 Remark 2.2 (i) The basins of attractors are well defined, i.e., they do not depend on the choice of their fundamental neighborhoods. The readers can refer to [15] and [17] for details. (ii) The basin of attraction B(A) is an invariant random open set, see [15], [16] and [18] for details. (iii) A pull-back attracts the random closed subsets of N, and it also pull-back attracts the random closed subsets of B(A). See Lemma 4.3 in [16] for details. Throughout the paper, we assume that X is a compact metric space with a metric d. 3 Properties of Ω-limit Sets and Attractors for RDS First we give some properties of Ω-limit sets. For some other properties, see [6]. Proposition 3.1 have Assume that D, D 1 and D 2 are arbitrary given random sets. Then we (i) If the random set D is forward invariant, then Ω D (ω) D(ω) P-a.s.; specially, we have Ω D = D if D is invariant; (ii) If D 1 (ω) D 2 (ω) P-a.s., then Ω D1 (ω) Ω D2 (ω) P-a.s.; (iii) Ω D (ω) = Ω D (ω) P-a.s.; (iv) Ω D1 D 2 (ω) Ω D1 (ω) Ω D2 (ω) P-a.s.; (v) If Ω Di (ω) D i (ω) P-a.s., i = 1, 2, then Ω D1 D 2 (ω) = Ω D1 (ω) Ω D2 (ω) P-a.s.; (vi) Ω D1 D 2 (ω) = Ω D1 (ω) Ω D2 (ω) P-a.s. Proof. Recall that, for all ω Ω, Ω D (ω) = ϕ(s, θ s ω)d(θ s ω), t 0 s t so (i) and (ii) hold. By the proof of Lemma 3.3 in [17], for all t R we have ϕ(s, θ s ω)d(θ s ω), ϕ(s, θ s ω)d(θ s ω) = s t s t which verifies (iii). Note that (iv) follows directly from (ii). Now we verify (v). By (iv) we only need to prove On the one hand, since we have Ω D1 D 2 (ω) Ω D1 (ω) Ω D2 (ω) P-a.s. (3.1) Ω Di (ω) D i (ω) P-a.s., i = 1, 2, Ω D1 D 2 (ω) Ω D1 (ω) Ω D2 (ω) D 1 (ω) D 2 (ω) P-a.s. On the other hand, by the definition of omega-limit sets we know that Ω D1 D 2 is the maximal invariant random compact set inside D 1 D 2. Note that Ω D1 Ω D2 is an invariant random compact set inside D 1 D 2 by the fact that Ω Di, i = 1, 2, are invariant random compact sets. Hence (3.1) holds.

5 NO. 2 LIU Z. X. et al. SOME PROPERTIES OF LOCAL RANDOM ATTRACTORS 167 Finally we verify (vi). By (ii) we have Ω D1 D 2 (ω) Ω D1 (ω) Ω D2 (ω) P-a.s., so we only need to prove that the converse inclusion also holds. To see this, for arbitrary x Ω D1 D 2 (ω), there exist sequences t n and x n D 1 (θ tn ω) D 2 (θ tn ω) such that ϕ(t n, θ tn ω)x n x, n. Therefore, there exists a subsequence such that x nk D 1 (θ tnk ω) or x nk D 2 (θ tnk ω) holds for all k = 1, 2, and ϕ(t nk, θ tnk ω)x nk x, k. That is, Hence we get This completes the proof of the proposition. x Ω D1 (ω) or x Ω D2 (ω). Ω D1 D 2 (ω) Ω D1 (ω) Ω D2 (ω). Remark 3.1 (i) Note that the properties (i) (iv) and (vi) in Proposition 3.1 also hold when the state space X is Polish and the RDS ϕ is one-sided. (ii) It is easy to see that for any random set D, we have Ω ΩD (ω) = Ω D (ω) by the invariance of Ω D and (i) of Proposition 3.1. P-a.s. (iii) In contrast to (v) of Proposition 3.1, if D 1 and D 2 are two forward invariant random sets, we cannot, in general, obtain For example, in the Example 3.1, we set Ω D1 D 2 (ω) = Ω D1 (ω) Ω D2 (ω) P-a.s. D 1 = {0}, D 2 = (0, 1 2 ). Then it is easy to see that D 1, D 2 are forward invariant. Ω D1 D 2 =. But Ω D1 = Ω D2 = {0}. Hence Since D 1 D 2 =, we have Ω D1 D 2 Ω D1 Ω D2. (iv) It is easy to see that if a forward invariant random closed set U satisfies Ω U (ω) U(ω) P-a.s., then the maximal invariant random compact set inside U, denoted by A, equals to Ω U P-a.s., which is a random attractor and U is a fundamental neighborhood of A. The following theorem shows the relation between the union (the intersection) of two (hence finite) random attractors and the union (the intersection) of their fundamental neighborhoods and their basins of attraction. Theorem 3.1 Assume that A i are random attractors and U i, B(A i ) are the corresponding fundamental neighborhoods and basins of attraction respectively, where i = 1, 2. Then A 1 A 2, A 1 A 2 are also random attractors with corresponding fundamental neighborhoods U 1 U 2, U 1 U 2 and corresponding basins B(A 1 ) B(A 2 ), B(A 1 ) B(A 2 ) respectively.

6 168 NORTHEAST. MATH. J. VOL. 24 Proof. Case 1: A 1 A 2. Denote U = U 1 U 2. Then U is forward invariant by the forward invariance of U 1, U 2. Denote by A the maximal invariant random compact set inside U. By (v) of Proposition 3.1 we easily obtain that A is a random attractor with U being one of its fundamental neighborhoods and A = A 1 A 2. Next we show that B(A) = B(A 1 ) B(A 2 ). By the fact B(A) B(A i ), i = 1, 2, we obtain that B(A) B(A 1 ) B(A 2 ). Hence we only need to show that the converse inclusion also holds. By the definition of basin of attraction we know that B(A i ) denotes the set of points which enter U i in finite time. And by the forward invariance of U i we know that once a point enters U i, it will permanently stay inside U i. Hence B(A 1 ) B(A 2 ) is the set of points which enter U = U 1 U 2 in finite time, and hence we have B(A 1 ) B(A 2 ) B(A) by the definition of basin of attraction again. Case 2: A 1 A 2. Denote Ũ = U 1 U 2. It is clear that Ũ is forward invariant by the forward invariance of U 1, U 2. Denote by à the maximal invariant random compact set inside Ũ, and we need to show that à is a random attractor with Ũ being one of its fundamental neighborhoods and à = A 1 A 2. This follows directly from (vi) of Proposition 3.1. Finally we show that B(Ã) = B(A 1) B(A 2 ). By the fact B(A i ) B(A 1 A 2 ), i = 1, 2, we obtain that B(A 1 ) B(A 2 ) B(A 1 A 2 ) = B(Ã). And the converse inclusion is easy to verify by the definition of basin of attraction as in Case 1. This completes the proof of the theorem. By Proposition 3.6 in [6], we know that for a given random set D, the omega-limit set Ω D pull-back attracts D. The following proposition is about the minimal random attracting set. Proposition 3.2 Assume that E is a random closed set. Then E attracts another random set D if and only if Ω D (ω) E(ω) P-a.s., i.e. Ω D is the minimal random closed set which attracts D. Proof. The sufficiency follows from the Proposition 3.6 of [6] and (iii) of Remark 2.1. Assume that the necessity were false, i.e., the set ˆΩ = {ω Ω D (ω) E(ω)} had positive probability. For ω ˆΩ, assume x Ω D (ω)\e(ω). Then there exists t n, x n D(θ tn ω) such that ϕ(t n, θ tn ω)x n x, n by the definition of Ω-limit set of D. Hence by the definition of Hausdorff semi-metric and the fact that E attracts D we have 0 < d({x} E(ω)) = lim n d(ϕ(t n, θ tn ω)x n E(ω)) lim n d(ϕ(t n, θ tn ω)d(θ tn ω) E(ω)) = 0, a contradiction. This completes the proof of the proposition. Remark 3.2 By Proposition 3.2, it is easy to see that attraction has the property of transitivity. That is, if three random closed sets D, E and F satisfy that D attracts E abd E attracts F, then D attracts F. In fact, since D attracts E and E attracts F, we have Ω E (ω) D(ω) P-a.s., Ω F (ω) E(ω) P-a.s. (3.2)

7 NO. 2 LIU Z. X. et al. SOME PROPERTIES OF LOCAL RANDOM ATTRACTORS 169 by Proposition 3.2. By (ii) of Remark 3.1, (ii) of Proposition 3.1 and (3.2) we have Ω F (ω) = Ω ΩF (ω) Ω E (ω) D(ω) i.e., D attracts F by Proposition 3.2 again. P-a.s., With respect to fundamental neighborhoods of random attractors, we have the following result. Theorem 3.2 Assume that A is a random attractor with a fundamental neighborhood U. Then for any T > 0, the random set U T, given by is also a fundamental neighborhood of A. U T (ω) := ϕ( T, θ T ω)u(θ T ω), Proof. Note that U U T by the forward invariance of U, so U T is a random closed neighborhood of A. We now show that U T is forward invariant. In fact, for any s > 0, we have ϕ(s, ω)u T (ω) = ϕ(s, ω)ϕ( T, θ T ω)u(θ T ω) = ϕ(s, ω)ϕ( T, θ s θ T θ s ω)u(θ s θ T θ s ω) = ϕ( T + s, θ s θ T θ s ω)u(θ s θ T θ s ω) = ϕ( T, θ s+t ω)ϕ(s, θ s θ T θ s ω)u(θ s θ T θ s ω) ϕ( T, θ T θ s ω)u(θ T θ s ω) = U T (θ s ω) P-a.s., where holds by the forward invariance of U. We can also obtain that Ω UT (ω) = ϕ(t, θ t ω)u T (θ t ω) τ 0 t τ = t 0 ϕ(t, θ t ω)u T (θ t ω) = t 0 ϕ(t, θ t ω)ϕ( T, θ T θ t ω)u(θ T θ t ω) = t 0 ϕ(t T, θ T t ω)u(θ T t ω) [ = 0 t T ] [ ϕ(t T, θ T t ω)u(θ T t ω) = ϕ(t T, θ T t ω)u(θ T t ω) t T = s 0 ϕ(s, θ s ω)u(θ s ω) = Ω U (ω) = A(ω) P-a.s., where * holds because for t s, we have t T ϕ(t, θ t ω)u T (θ t ω) ϕ(s, θ s ω)u T (θ s ω) ] ϕ(t T, θ T t ω)u(θ T t ω) by the forward invariance of U T (ω); ** holds because for any t > 0, we have ϕ( t, θ t ω)u(θ t ω) U(ω)

8 170 NORTHEAST. MATH. J. VOL. 24 by the forward invariance of U. This completes the proof of the theorem. Remark 3.3 (i) It is obvious that for any T < 0, the result of Theorem 3.2 also holds. (ii) For P-almost all ω Ω, we have B(A)(ω) = lim U T (ω); T See Lemma 4.2 in [15] for details. Assume that D B(A) is a random closed set and there exists a T > 0 such that D(ω) U T (ω) P-a.s. Then by Theorem 3.2 we obtain that A attracts D. Actually, if there is no such T, A also attracts D; see Lemma 4.3 in [16]. Proposition 3.3 Assume that A is a random attractor with a fundamental neighborhood U. If there exists a random closed set D which is connected with positive probability and satisfies A(ω) D(ω) U(ω) P-a.s., then A is connected P-a.s. Proof. The idea of proof is similar to that of Proposition 3.7 of [12] and Proposition 3.13 of [6]. Assume that the assertion were false, i.e., A were non-connected with positive probability. Then by ϕ(t, ω)a(ω) = A(θ t ω) and the fact that ϕ(t, ω) : X X is a homeomorphism for any (t, ω) R Ω, we know that the set Ω nc := {ω A(ω) is non-connected} is a θ-invariant subset of Ω. Hence we have P(Ω nc ) = 1 by the ergodicity of θ under P. If we define α(ω) := inf{d(c(ω) A(ω)) A(ω) C(ω), C(ω) connected}, then we have α(ω) > 0 with positive probability by Lemma 3.12 of [6]. And by the definition of α(ω), it is easy to see that d(ϕ(t, θ t ω)d(θ t ω) A(ω)) α(ω) > 0 (3.3) with positive probability independent of t. By the fact that A is an attractor we have lim d(ϕ(t, θ tω)d(θ t ω) A(ω)) = 0 P-a.s., t a contradiction to (3.3). This completes the proof of the proposition. Remark 3.4 More generally, if D denotes a universe of random sets and A D is the random pull-back attractor in D, that is, lim d(ϕ(t, θ tω)d(θ t ω) A(ω)) = 0 P-a.s., D D, t then the similar result to that of Proposition 3.3 holds. That is, if there exists a random set D satisfying A D D and D(ω) is connected with positive probability, then A(ω) is connected P-a.s. In fact, it is obvious that, in Proposition 3.3, the attractor A is in fact the random pull-back attractor in the universe D = {D D(ω) U(ω) P-a.s. and D is a random closed set}. Example 3.1 Consider the differential equation ẋ = x 2 (x 2 1) on the interval [ 1, 1] and assume that ϕ is the flow generated by it; see Figure 3.1.

9 NO. 2 LIU Z. X. et al. SOME PROPERTIES OF LOCAL RANDOM ATTRACTORS Figure 3.1 The flow generated by ẋ = x 2 (x 2 1) (i) By (i) of Proposition 3.1, we know Ω D (ω) D(ω) P-a.s. for any forward invariant random set D. But if D is a forward invariant but non-invariant random open set, we still cannot obtain Ω D (ω) D(ω) P-a.s. generally. For example, set D = [ 1, 0) (0, 1 2 ). Then it is easy to see that for any t > 0 we have ϕ(t)d D. But it is obvious that Ω D = [ 1, 0] D. (ii) Assume that A is a random attractor with the basin of attraction B(A). Then we know that A attracts any random closed sets inside B(A). But we cannot, in general, obtain that A attracts B(A). It is obvious that A = { 1} is an attractor with B(A) = [ 1, 0), but A does not attract B(A) in spite that it attracts any closed subsets of B(A). (iii) Assume that U is a forward invariant random open set and A is the maximal invariant random compact set inside U, but A is not necessarily a random attractor. For instance, U = ( 1, 1 ) is a forward invariant set and A = {0} is the maximal invariant compact set 2 inside U, but it is obvious that A is not an attractor. (iv) Assume that A is a random attractor and the corresponding basin is B(A). We know that B(A) is an invariant random open set and A is the maximal invariant random compact set inside B(A). Conversely, assume that D is an invariant random open set, A is the maximal invariant random compact set inside D, and also assume that A is a random attractor. Now the question is whether D is necessarily the basin of attraction of A? The answer is negative. It is easy to see that D = [ 1, 0) (0, 1) is an invariant open set (notice that X = [ 1, 1], and hence D is open relative to X), A = { 1} is the maximal invariant compact set inside D and A is also an attractor. But any closed subsets of (0, 1) is not attracted by A. Hence D is not the basin of attraction of A. In fact, it is obvious that [ 1, 0) is the basin of attraction of A. Acknowledgment The authors sincerely thank Professor Li Yong for helpful suggestions and invaluable comments. References [1] Bhatia, N. and Szegö, G., Stability Theory of Dynamical Systems, Springer-Verlag, Berlin- Heidelberg-New York, [2] Conley, C., Isolated Invariant Sets and the Morse Index, Conf. Board Math. Sci. Vol. 38, American Mathematical Society, Providence, RI, [3] Conley, C., The gradient structure of a flow: I, Ergodic Theory Dyn. Syst., 8(1988), [4] Hale, J., Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, [5] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Springer-Verlag, Berlin Heidelberg New York, [6] Crauel, H. and Flandoli, F., Attractors for random dynamical systems, Probab. Theory Related Fields, 100(1994),

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