A Note on Some Properties of Local Random Attractors
|
|
- Elisabeth Matthews
- 5 years ago
- Views:
Transcription
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),
10 172 NORTHEAST. MATH. J. VOL. 24 [7] Schmalfuss, B., Backward cocycles and attractors for stochastic differential equations In: V. Reitmann, T. Riedrich and N. Koksch Eds., International Seminar on Applied Mathematics- Nonlinear Dynamics: Attractor Approximation and Global Behaviour (Teubner, Leipzig), 1992, pp [8] Ochs, G., Weak Random Attractors, Institut für Dynamische Systeme, Universität Bremen Report 449, Bremen, [9] Scheutzow, M., Comparison of various concepts of a random attractor: A case study, Arch. Math., 78(2002), [10] Caraballo, T., Langa, J. A. and Robinson, J. C., Stability and random attractors for a reactiondiffusion equation with multiplicative noise, Discrete Contin. Dynam. Systems, 6(2000), [11] Caraballo, T., Langa, J. A. and Robinson, J. C., Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23(1998), [12] Crauel, H., Random point attractors versus random set attractors, J. London Math. Soc., 63(2001), [13] Schenk-Hoppé, K. R., Random attractors-general properties, existence and applications to stochastic bifurcation theory, Discrete Contin. Dynam. Systems, 4(1998), [14] Schmalfuss, B., The random attractor of the stochastic Lorenz system, Z. Angew. Math. Phys., 48(1997), [15] Crauel, H., Duc, L. H. and Siegmund, S., Towards a Morse theory for random dynamical systems, Stochastics Dynam., 4(2004), [16] Liu, Z. X., Ji, S. G. and Su, M. L., Attractor-repeller pair, Morse decomposition and Lyapunov function for random dynamical systems, [17] Liu, Z. X., The random case of Conley s theorem, Nonlinearity, 19(2006), [18] Liu, Z. X., The random case of Conley s theorem: II. The complete Lyapunov function, Nonlinearity, 20(2007), [19] Liu, Z. X., The random case of Conley s theorem: III. Random semiflow case and Morse decomposition, Nonlinearity, 20(2007), [20] Arnold, L., Random Dynamical Systems, Springer-Verlag, Berlin-Heidelberg-New York, 1998.
ATTRACTORS FOR THE STOCHASTIC 3D NAVIER-STOKES EQUATIONS
Stochastics and Dynamics c World Scientific Publishing Company ATTRACTORS FOR THE STOCHASTIC 3D NAVIER-STOKES EQUATIONS PEDRO MARÍN-RUBIO Dpto. Ecuaciones Diferenciales y Análisis Numérico Universidad
More informationLYAPUNOV FUNCTIONS IN HAUSDORFF DIMENSION ESTIMATES OF COCYCLE ATTRACTORS
PHYSCON 2011, León, Spain, September, 5-September, 8 2011 LYAPUNOV FUNCTIONS IN HAUSDORFF DIMENSION ESTIMATES OF COCYCLE ATTRACTORS Gennady A. Leonov Department of Mathematics and Mechanics Saint-Petersburg
More informationNonlinear Dynamical Systems Eighth Class
Nonlinear Dynamical Systems Eighth Class Alexandre Nolasco de Carvalho September 19, 2017 Now we exhibit a Morse decomposition for a dynamically gradient semigroup and use it to prove that a dynamically
More informationRandom attractors and the preservation of synchronization in the presence of noise
Random attractors and the preservation of synchronization in the presence of noise Peter Kloeden Institut für Mathematik Johann Wolfgang Goethe Universität Frankfurt am Main Deterministic case Consider
More informationChapter 3 Pullback and Forward Attractors of Nonautonomous Difference Equations
Chapter 3 Pullback and Forward Attractors of Nonautonomous Difference Equations Peter Kloeden and Thomas Lorenz Abstract In 1998 at the ICDEA Poznan the first author talked about pullback attractors of
More information2 GUNTER OCHS in the future, a pullback attractor takes only the dynamics of the past into account. However, in the framework of random dynamical syst
WEAK RANDOM ATTRACTORS GUNTER OCHS Abstract. We dene point attractors and set attractors for random dynamical systems via convergence in probability forward in time. This denition of a set attractor is
More information(x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively);
STABILITY OF EQUILIBRIA AND LIAPUNOV FUNCTIONS. By topological properties in general we mean qualitative geometric properties (of subsets of R n or of functions in R n ), that is, those that don t depend
More informationTHE SET OF RECURRENT POINTS OF A CONTINUOUS SELF-MAP ON AN INTERVAL AND STRONG CHAOS
J. Appl. Math. & Computing Vol. 4(2004), No. - 2, pp. 277-288 THE SET OF RECURRENT POINTS OF A CONTINUOUS SELF-MAP ON AN INTERVAL AND STRONG CHAOS LIDONG WANG, GONGFU LIAO, ZHENYAN CHU AND XIAODONG DUAN
More informationA Lyapunov function for pullback attractors of nonautonomous differential equations
Nonlinear Differential Equations, Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 91 102 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp) A Lyapunov
More informationON THE CONTINUITY OF GLOBAL ATTRACTORS
ON THE CONTINUITY OF GLOBAL ATTRACTORS LUAN T. HOANG, ERIC J. OLSON, AND JAMES C. ROBINSON Abstract. Let Λ be a complete metric space, and let {S λ ( ) : λ Λ} be a parametrised family of semigroups with
More informationDYNAMICS OF STOCHASTIC NONCLASSICAL DIFFUSION EQUATIONS ON UNBOUNDED DOMAINS
Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 282, pp. 1 22. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu DYNAMICS OF STOCHASTIC
More informationWHAT IS A CHAOTIC ATTRACTOR?
WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties
More informationTHE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM
Bull. Aust. Math. Soc. 88 (2013), 267 279 doi:10.1017/s0004972713000348 THE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM MICHAEL F. BARNSLEY and ANDREW VINCE (Received 15 August 2012; accepted 21 February
More informationNonlinear Dynamical Systems Ninth Class
Nonlinear Dynamical Systems Ninth Class Alexandre Nolasco de Carvalho September 21, 2017 Lemma Let {T (t) : t 0} be a dynamically gradient semigroup in a metric space X, with a global attractor A and a
More informationMATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1
MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION ELEMENTS OF SOLUTION Problem 1 1. Let X be a Hausdorff space and K 1, K 2 disjoint compact subsets of X. Prove that there exist disjoint open sets U 1 and
More informationTopologic Conjugation and Asymptotic Stability in Impulsive Semidynamical Systems
CADERNOS DE MATEMÁTICA 06, 45 60 May (2005) ARTIGO NÚMERO SMA#211 Topologic Conjugation and Asymptotic Stability in Impulsive Semidynamical Systems Everaldo de Mello Bonotto * Departamento de Matemática,
More informationPullback permanence for non-autonomous partial differential equations
Electronic Journal of Differential Equations, Vol. 2002(2002), No. 72, pp. 1 20. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Pullback permanence
More informationThe small ball property in Banach spaces (quantitative results)
The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More informationThe Conley Index and Rigorous Numerics of Attracting Periodic Orbits
The Conley Index and Rigorous Numerics of Attracting Periodic Orbits Marian Mrozek Pawe l Pilarczyk Conference on Variational and Topological Methods in the Study of Nonlinear Phenomena (Pisa, 2000) 1
More informationSYNCHRONIZATION OF NONAUTONOMOUS DYNAMICAL SYSTEMS
Electronic Journal of Differential Equations, Vol. 003003, No. 39, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp SYNCHRONIZATION OF
More informationAbstract The construction of a Lyapunov function characterizing the pullback attraction of a cocycle attractor of a nonautonomous discrete time dynami
Lyapunov functions for cocycle attractors in nonautonomous dierence equations P.E. Kloeden Weierstra{Institut fur Angewandte Analysis und Stochastik, Mohrenstrae 39, 10117 Berlin, Germany kloeden@wias-berlin.de
More informationOn Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q)
On Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q) Andreas Löhne May 2, 2005 (last update: November 22, 2005) Abstract We investigate two types of semicontinuity for set-valued
More informationSHADOWING AND INTERNAL CHAIN TRANSITIVITY
SHADOWING AND INTERNAL CHAIN TRANSITIVITY JONATHAN MEDDAUGH AND BRIAN E. RAINES Abstract. The main result of this paper is that a map f : X X which has shadowing and for which the space of ω-limits sets
More informationChain transitivity, attractivity and strong repellors for semidynamical systems
Chain transitivity, attractivity and strong repellors for semidynamical systems Morris W. Hirsch Department of Mathematics University of California Berkeley, CA, 9472 Hal L. Smith and Xiao-Qiang Zhao Department
More informationExponential stability of families of linear delay systems
Exponential stability of families of linear delay systems F. Wirth Zentrum für Technomathematik Universität Bremen 28334 Bremen, Germany fabian@math.uni-bremen.de Keywords: Abstract Stability, delay systems,
More informationTakens embedding theorem for infinite-dimensional dynamical systems
Takens embedding theorem for infinite-dimensional dynamical systems James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. E-mail: jcr@maths.warwick.ac.uk Abstract. Takens
More informationHYPERBOLIC SETS WITH NONEMPTY INTERIOR
HYPERBOLIC SETS WITH NONEMPTY INTERIOR TODD FISHER, UNIVERSITY OF MARYLAND Abstract. In this paper we study hyperbolic sets with nonempty interior. We prove the folklore theorem that every transitive hyperbolic
More informationThe Stochastic Bifurcation Behaviour of Speculative Financial Markets
The Stochastic Bifurcation Behaviour of Speculative Financial Markets Carl Chiarella 1, Xue-Zhong He 1, Duo Wang 2 and Min Zheng 1,2 1 School of Finance and Economics, University of Technology, Sydney
More informationORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY
ORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY AND ω-limit SETS CHRIS GOOD AND JONATHAN MEDDAUGH Abstract. Let f : X X be a continuous map on a compact metric space, let ω f be the collection of ω-limit
More informationSAMPLE PATH PROPERIES OF RANDOM TRANSFORMATIONS.
SAMPLE PATH PROPERIES OF RANDOM TRANSFORMATIONS. DMITRY DOLGOPYAT 1. Models. Let M be a smooth compact manifold of dimension N and X 0, X 1... X d, d 2, be smooth vectorfields on M. (I) Let {w j } + j=
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More informationLYAPUNOV STABILITY OF CLOSED SETS IN IMPULSIVE SEMIDYNAMICAL SYSTEMS
Electronic Journal of Differential Equations, Vol. 2010(2010, No. 78, pp. 1 18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LYAPUNOV STABILITY
More informationGlobal compact attractors and their tripartition under persistence
Global compact attractors and their tripartition under persistence Horst R. Thieme (joint work with Hal L. Smith) School of Mathematical and Statistical Science Arizona State University GCOE, September
More informationYuqing Chen, Yeol Je Cho, and Li Yang
Bull. Korean Math. Soc. 39 (2002), No. 4, pp. 535 541 NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY Yuqing Chen, Yeol Je Cho, and Li Yang Abstract. In this paper, we introduce the concept of lower semicontinuous
More informationErrata and additional material for Infinite-Dimensional Dynamical Systems
Errata and additional material for Infinite-Dimensional Dynamical Systems Many thanks to all who have sent me errata, including: Marco Cabral, Carmen Chicone, Marcus Garvie, Grzegorz Lukaszewicz, Edgardo
More informationDYNAMICAL PROPERTIES OF MONOTONE DENDRITE MAPS
DYNAMICAL PROPERTIES OF MONOTONE DENDRITE MAPS Issam Naghmouchi To cite this version: Issam Naghmouchi. DYNAMICAL PROPERTIES OF MONOTONE DENDRITE MAPS. 2010. HAL Id: hal-00593321 https://hal.archives-ouvertes.fr/hal-00593321v2
More informationREGULARITY OF PULLBACK ATTRACTORS FOR NON-AUTONOMOUS STOCHASTIC COUPLED REACTION-DIFFUSION SYSTEMS
Journal of Applied Analysis and Computation Volume 7, Number 3, August 2017, 884 898 Website:http://jaac-online.com/ DOI:10.11948/2017056 REGULARITY OF PULLBACK ATTRACTORS FOR NON-AUTONOMOUS STOCHASTIC
More informationarxiv: v3 [math.ds] 9 Nov 2012
Positive expansive flows Alfonso Artigue November 13, 2018 arxiv:1210.3202v3 [math.ds] 9 Nov 2012 Abstract We show that every positive expansive flow on a compact metric space consists of a finite number
More informationUNIFORM NONAUTONOMOUS ATTRACTORS UNDER DISCRETIZATION. Peter E. Kloeden. and Victor S. Kozyakin
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 10, Numbers1&2, January & March 2004 pp. 423 433 UNIFORM NONAUTONOMOUS ATTRACTORS UNDER DISCRETIZATION Peter E. Kloeden
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationEQUIVALENCE OF TOPOLOGIES AND BOREL FIELDS FOR COUNTABLY-HILBERT SPACES
EQUIVALENCE OF TOPOLOGIES AND BOREL FIELDS FOR COUNTABLY-HILBERT SPACES JEREMY J. BECNEL Abstract. We examine the main topologies wea, strong, and inductive placed on the dual of a countably-normed space
More informationAn Algorithmic Approach to Chain Recurrence
An Algorithmic Approach to Chain Recurrence W.D. Kalies, K. Mischaikow, and R.C.A.M. VanderVorst October 10, 2005 ABSTRACT. In this paper we give a new definition of the chain recurrent set of a continuous
More informationInput to state Stability
Input to state Stability Mini course, Universität Stuttgart, November 2004 Lars Grüne, Mathematisches Institut, Universität Bayreuth Part IV: Applications ISS Consider with solutions ϕ(t, x, w) ẋ(t) =
More informationCOMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS
Series Logo Volume 00, Number 00, Xxxx 19xx COMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS RICH STANKEWITZ Abstract. Let G be a semigroup of rational functions of degree at least two, under composition
More informationPassivity-based Stabilization of Non-Compact Sets
Passivity-based Stabilization of Non-Compact Sets Mohamed I. El-Hawwary and Manfredi Maggiore Abstract We investigate the stabilization of closed sets for passive nonlinear systems which are contained
More informationON FRACTAL DIMENSION OF INVARIANT SETS
ON FRACTAL DIMENSION OF INVARIANT SETS R. MIRZAIE We give an upper bound for the box dimension of an invariant set of a differentiable function f : U M. Here U is an open subset of a Riemannian manifold
More informationLYAPUNOV EXPONENTS AND STABILITY FOR THE STOCHASTIC DUFFING-VAN DER POL OSCILLATOR
LYAPUNOV EXPONENTS AND STABILITY FOR THE STOCHASTIC DUFFING-VAN DER POL OSCILLATOR Peter H. Baxendale Department of Mathematics University of Southern California Los Angeles, CA 90089-3 USA baxendal@math.usc.edu
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationPreface Issue B H.-Ch. Grunau. Hans-Christoph Grunau 1
Jahresber Dtsch Math-Ver (2015) 117:171 DOI 10.1365/s13291-015-0118-x PREFACE Preface Issue 3-2015 Hans-Christoph Grunau 1 Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2015 Autonomous
More informationPart III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
More informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More informationHARTMAN-GROBMAN THEOREMS ALONG HYPERBOLIC STATIONARY TRAJECTORIES. Edson A. Coayla-Teran. Salah-Eldin A. Mohammed. Paulo Régis C.
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 17, Number 2, February 27 pp. 281 292 HARTMAN-GROBMAN THEOREMS ALONG HYPERBOLIC STATIONARY TRAJECTORIES Edson A. Coayla-Teran
More informationNonlinear Dynamical Systems Lecture - 01
Nonlinear Dynamical Systems Lecture - 01 Alexandre Nolasco de Carvalho August 08, 2017 Presentation Course contents Aims and purpose of the course Bibliography Motivation To explain what is a dynamical
More informationA CONCISE PROOF OF THE GEOMETRIC CONSTRUCTION OF INERTIAL MANIFOLDS
This paper has appeared in Physics Letters A 200 (1995) 415 417 A CONCISE PROOF OF THE GEOMETRIC CONSTRUCTION OF INERTIAL MANIFOLDS James C. Robinson Department of Applied Mathematics and Theoretical Physics,
More informationInfinite-Dimensional Dynamical Systems in Mechanics and Physics
Roger Temam Infinite-Dimensional Dynamical Systems in Mechanics and Physics Second Edition With 13 Illustrations Springer Contents Preface to the Second Edition Preface to the First Edition vii ix GENERAL
More informationA COMMENT ON FREE GROUP FACTORS
A COMMENT ON FREE GROUP FACTORS NARUTAKA OZAWA Abstract. Let M be a finite von Neumann algebra acting on the standard Hilbert space L 2 (M). We look at the space of those bounded operators on L 2 (M) that
More informationTECHNIQUES FOR ESTABLISHING DOMINATED SPLITTINGS
TECHNIQUES FOR ESTABLISHING DOMINATED SPLITTINGS ANDY HAMMERLINDL ABSTRACT. We give theorems which establish the existence of a dominated splitting and further properties, such as partial hyperbolicity.
More informationFixed Points & Fatou Components
Definitions 1-3 are from [3]. Definition 1 - A sequence of functions {f n } n, f n : A B is said to diverge locally uniformly from B if for every compact K A A and K B B, there is an n 0 such that f n
More informationON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay
ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS G. Makay Student in Mathematics, University of Szeged, Szeged, H-6726, Hungary Key words and phrases: Lyapunov
More informationPart V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory
Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite
More informationLocally Convex Vector Spaces II: Natural Constructions in the Locally Convex Category
LCVS II c Gabriel Nagy Locally Convex Vector Spaces II: Natural Constructions in the Locally Convex Category Notes from the Functional Analysis Course (Fall 07 - Spring 08) Convention. Throughout this
More informationMEAN DIMENSION AND AN EMBEDDING PROBLEM: AN EXAMPLE
MEAN DIMENSION AND AN EMBEDDING PROBLEM: AN EXAMPLE ELON LINDENSTRAUSS, MASAKI TSUKAMOTO Abstract. For any positive integer D, we construct a minimal dynamical system with mean dimension equal to D/2 that
More informationMAS3706 Topology. Revision Lectures, May I do not answer enquiries as to what material will be in the exam.
MAS3706 Topology Revision Lectures, May 208 Z.A.Lykova It is essential that you read and try to understand the lecture notes from the beginning to the end. Many questions from the exam paper will be similar
More informationITERATED FUNCTION SYSTEMS WITH CONTINUOUS PLACE DEPENDENT PROBABILITIES
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XL 2002 ITERATED FUNCTION SYSTEMS WITH CONTINUOUS PLACE DEPENDENT PROBABILITIES by Joanna Jaroszewska Abstract. We study the asymptotic behaviour
More informationA NEW LINDELOF SPACE WITH POINTS G δ
A NEW LINDELOF SPACE WITH POINTS G δ ALAN DOW Abstract. We prove that implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality 2 ℵ1 which has points G δ. In addition, this space has
More informationON µ-compact SETS IN µ-spaces
Questions and Answers in General Topology 31 (2013), pp. 49 57 ON µ-compact SETS IN µ-spaces MOHAMMAD S. SARSAK (Communicated by Yasunao Hattori) Abstract. The primary purpose of this paper is to introduce
More informationGlobal Attractors in PDE
CHAPTER 14 Global Attractors in PDE A.V. Babin Department of Mathematics, University of California, Irvine, CA 92697-3875, USA E-mail: ababine@math.uci.edu Contents 0. Introduction.............. 985 1.
More informationarxiv: v1 [math.ap] 31 May 2007
ARMA manuscript No. (will be inserted by the editor) arxiv:75.4531v1 [math.ap] 31 May 27 Attractors for gradient flows of non convex functionals and applications Riccarda Rossi, Antonio Segatti, Ulisse
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationObservability and forward-backward observability of discrete-time nonlinear systems
Observability and forward-backward observability of discrete-time nonlinear systems Francesca Albertini and Domenico D Alessandro Dipartimento di Matematica pura a applicata Universitá di Padova, 35100
More informationHAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
More informationSome Properties of the Augmented Lagrangian in Cone Constrained Optimization
MATHEMATICS OF OPERATIONS RESEARCH Vol. 29, No. 3, August 2004, pp. 479 491 issn 0364-765X eissn 1526-5471 04 2903 0479 informs doi 10.1287/moor.1040.0103 2004 INFORMS Some Properties of the Augmented
More informationPHY411 Lecture notes Part 5
PHY411 Lecture notes Part 5 Alice Quillen January 27, 2016 Contents 0.1 Introduction.................................... 1 1 Symbolic Dynamics 2 1.1 The Shift map.................................. 3 1.2
More informationWe will begin our study of topology from a set-theoretic point of view. As the subject
p. 1 Math 490-01 Notes 5 Topology We will begin our study of topology from a set-theoretic point of view. As the subject expands, we will encounter various notions from analysis such as compactness, continuity,
More informationLocally Lipschitzian Guiding Function Method for ODEs.
Locally Lipschitzian Guiding Function Method for ODEs. Marta Lewicka International School for Advanced Studies, SISSA, via Beirut 2-4, 3414 Trieste, Italy. E-mail: lewicka@sissa.it 1 Introduction Let f
More informationASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of
More informationNEW NORMALITY AXIOMS AND DECOMPOSITIONS OF NORMALITY. J. K. Kohli and A. K. Das University of Delhi, India
GLASNIK MATEMATIČKI Vol. 37(57)(2002), 163 173 NEW NORMALITY AXIOMS AND DECOMPOSITIONS OF NORMALITY J. K. Kohli and A. K. Das University of Delhi, India Abstract. Generalizations of normality, called(weakly)(functionally)
More informationTOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS OF INTERVALS
Chung, Y-.M. Osaka J. Math. 38 (200), 2 TOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS OF INTERVALS YONG MOO CHUNG (Received February 9, 998) Let Á be a compact interval of the real line. For a continuous
More informationUniquely Universal Sets
Uniquely Universal Sets 1 Uniquely Universal Sets Abstract 1 Arnold W. Miller We say that X Y satisfies the Uniquely Universal property (UU) iff there exists an open set U X Y such that for every open
More informationEuler Characteristic of Two-Dimensional Manifolds
Euler Characteristic of Two-Dimensional Manifolds M. Hafiz Khusyairi August 2008 In this work we will discuss an important notion from topology, namely Euler Characteristic and we will discuss several
More informationSiegel Discs in Complex Dynamics
Siegel Discs in Complex Dynamics Tarakanta Nayak, Research Scholar Department of Mathematics, IIT Guwahati Email: tarakanta@iitg.ernet.in 1 Introduction and Definitions A dynamical system is a physical
More informationDynamical Systems and Ergodic Theory PhD Exam Spring Topics: Topological Dynamics Definitions and basic results about the following for maps and
Dynamical Systems and Ergodic Theory PhD Exam Spring 2012 Introduction: This is the first year for the Dynamical Systems and Ergodic Theory PhD exam. As such the material on the exam will conform fairly
More information2. The Concept of Convergence: Ultrafilters and Nets
2. The Concept of Convergence: Ultrafilters and Nets NOTE: AS OF 2008, SOME OF THIS STUFF IS A BIT OUT- DATED AND HAS A FEW TYPOS. I WILL REVISE THIS MATE- RIAL SOMETIME. In this lecture we discuss two
More informationLattice Structures for Attractors II
Lattice Structures for Attractors II WILLIAM D. KALIES Florida Atlantic University 777 Glades Road Boca Raton, FL 33431, USA KONSTANTIN MISCHAIKOW Rutgers University 110 Frelinghusen Road Piscataway, NJ
More information(Non-)Existence of periodic orbits in dynamical systems
(Non-)Existence of periodic orbits in dynamical systems Konstantin Athanassopoulos Department of Mathematics and Applied Mathematics University of Crete June 3, 2014 onstantin Athanassopoulos (Univ. of
More informationABSOLUTE CONTINUITY OF FOLIATIONS
ABSOLUTE CONTINUITY OF FOLIATIONS C. PUGH, M. VIANA, A. WILKINSON 1. Introduction In what follows, U is an open neighborhood in a compact Riemannian manifold M, and F is a local foliation of U. By this
More informationATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N.
ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS Emerson A. M. de Abreu Alexandre N. Carvalho Abstract Under fairly general conditions one can prove that
More informationSHADOWING PROPERTY FOR INDUCED SET-VALUED DYNAMICAL SYSTEMS OF SOME EXPANSIVE MAPS
Dynamic Systems and Applications 19 (2010) 405-414 SHADOWING PROPERTY FOR INDUCED SET-VALUED DYNAMICAL SYSTEMS OF SOME EXPANSIVE MAPS YUHU WU 1,2 AND XIAOPING XUE 1 1 Department of Mathematics, Harbin
More informationLATTICE PROPERTIES OF T 1 -L TOPOLOGIES
RAJI GEORGE AND T. P. JOHNSON Abstract. We study the lattice structure of the set Ω(X) of all T 1 -L topologies on a given set X. It is proved that Ω(X) has dual atoms (anti atoms) if and only if membership
More informationPERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS. A. M. Blokh. Department of Mathematics, Wesleyan University Middletown, CT , USA
PERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS A. M. Blokh Department of Mathematics, Wesleyan University Middletown, CT 06459-0128, USA August 1991, revised May 1992 Abstract. Let X be a compact tree,
More informationMeasurability of Intersections of Measurable Multifunctions
preprint Mathematics No. 3/1995, 1-10 Dep. of Mathematics, Univ. of Tr.heim Measurability of Intersections of Measurable Multifunctions Gunnar Taraldsen We prove universal compact-measurability of the
More informationBloch radius, normal families and quasiregular mappings
Bloch radius, normal families and quasiregular mappings Alexandre Eremenko Abstract Bloch s Theorem is extended to K-quasiregular maps R n S n, where S n is the standard n-dimensional sphere. An example
More informationΓ Convergence of Hausdorff Measures
Journal of Convex Analysis Volume 12 (2005), No. 1, 239 253 Γ Convergence of Hausdorff Measures G. Buttazzo Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy buttazzo@dm.unipi.it
More informationNOWHERE LOCALLY UNIFORMLY CONTINUOUS FUNCTIONS
NOWHERE LOCALLY UNIFORMLY CONTINUOUS FUNCTIONS K. Jarosz Southern Illinois University at Edwardsville, IL 606, and Bowling Green State University, OH 43403 kjarosz@siue.edu September, 995 Abstract. Suppose
More informationLocally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem
56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi
More informationA classification of explosions in dimension one
Ergod. Th. & Dynam. Sys. (29), 29, 715 731 doi:1.117/s143385788486 c 28 Cambridge University Press Printed in the United Kingdom A classification of explosions in dimension one E. SANDER and J. A. YORKE
More informationMathematics for Economists
Mathematics for Economists Victor Filipe Sao Paulo School of Economics FGV Metric Spaces: Basic Definitions Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb. 2017 1 / 34 Definitions and Examples
More informationMathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang
Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions
More informationA GENERAL THEOREM ON APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATION. Miljenko Huzak University of Zagreb,Croatia
GLASNIK MATEMATIČKI Vol. 36(56)(2001), 139 153 A GENERAL THEOREM ON APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATION Miljenko Huzak University of Zagreb,Croatia Abstract. In this paper a version of the general
More information