Attractors of dynamical systems in locally compact spaces

Abstract In this article the properties of attractors of dynamical systems in locally compact metric space are discussed. Existing conditions of attractors and related results are obtained by the near isolating block which we present.


Introduction
The dynamical system theory studies the rules of changes in the state which depends on time. In the investigation of dynamical systems, one of very interesting topics is the study of attractors (see [ − ] and the references therein). In [ − ], the authors gave some de nitions of attractors and also made further investigations about the properties of them. In [5], the limit set of a neighborhood was used in the de nition of an attractor, and in [6] Hale and Waterman also emphasized the importance of the limit set of a set in the analysis of the limiting behavior of a dynamical system. In [ − ], the authors de ned intertwining attractor of dynamical systems in metric space and obtained some existing conditions of intertwining attractor. In [ ], the author studied the properties of limit sets of subsets and attractors in a compact metric space. In [11], the author studied a positively bounded dynamical system in the plane, and obtained the conditions of compactness of the set of all bounded solutions. In [ ], the uniform attractor was de ned as the minimal compact uniformly pullback attracting random set, several existence criteria for uniform attractors were given and the relationship between uniform and co-cycle attractors was carefully studied. In [ ], the authors established conditions for the existence and stability of invariant sets for dynamical systems de ned on metric space of fuzzy subsets of R n . In [ ], the authors studied the recurrence and the gradient-like structure of a ow and gave some properties of connecting orbit. The paper [ ] de ned an attractor of a dynamical system on a locally compact metric space and investigated topological properties of the attraction domain of dynamical systems. In [ ], the authors obtained equivalent conditions for the existence of global attractors for transformation semigroups on principal bundles. The article [ ] investigated the structure of a global attractor for an abstract evolutionary system and obtained weak and strong uniform tracking properties of omega-limits and global attractors.
Motivated by the above discussion, in this paper we study properties of attractors of dynamical systems. Main results are as follows. First of all, we de ne the near isolating block, then we present some basic properties about attractors, give the existing conditions of attractors of dynamical systems and obtain the relation between the attractor and the attraction neighborhood in a locally compact metric space. At the end of this article, as byproduct, we obtain some results about the compactness of the set formed by bounded orbits using near isolating block. To the best of our knowledge, results similar to these presented in our work have never been reported.

Preliminaries
Let X be a locally compact metric space with a metric ρ. In X, there is a ow π : X×R → X. Denote x·t = π(x, t), x · R, x · R + and x · R − denote the trajectory, the positive trajectory and the negative trajectory of A point x ∈ X, respectively. The ω-limit set of x(or the trajectory x·R + ), ω(x, π) = {y ∈ X| there exists a sequence {tn} +∞ n= ⊂ R + such that tn → +∞ and x · tn → y as n → +∞}. The α-limit set of x(or the trajectory For the set Y ⊂ X, Cl{Y}, ∂Y, IntY and ExtY denote the closure, the boundary, the interior and the complement of Y, respectively. The ω-limit set of Y ⊂ X de nes ω(Y , π) = ∩ t≥ Cl{Y · [t, +∞)}. In [ ], the authors have proven that ω(Y , π) is the maximal invariant subset in Cl{Y · [ , +∞)}. For r > , B(x, r) = {y | ρ(x, y) < r} denotes the ball with the center x and the radius r.
To avoid confusion, we rst x some notations and de nitions. De nition 2.1. For a set A ⊂ X, A is called an attractor for the ow π if A admits a neighborhood N such that A = ω(N, π).
In [18], the author considered the di erential system de ned in the plane In [5], [11], [14] and [18,19], the properties of attractors and connecting orbits of dynamical systems were discussed by an isolating block. Similar to De nition 2.2, we give the following de nition which is very useful to our main results.
De nition 2.3. For N ⊂ X, the set N is called a near isolating block of a dynamical system if x·R − ∩ExtN ≠ for any point x ∈ ∂N.
Remark 2.1. The condition of an isolating block is di erent to that of a near isolating block.  ( . ) The point O = ( , ) is only an attractor of the dynamical system. Any subset N ⊂ R , which does not contain O = ( , ), is an attractor neighborhood, but ω(N, π) ⊂ N does not hold. At the end of this paper, we also use the two following de nitions. De nition 2.5. A simple closed curve is called a singular closed orbit if it is the union of alternating nonclosed whole orbits and equilibrium points, and is contained in the ω-(or α-)limit set of an orbit.
De nition 2.6. For an equilibrium point p, De nition 2.7. If there exists a point p ∈ R n such that lim t→−∞ p · t = a and lim t→+∞ p · t = b, then the set π(p, R) = {p · t | t ∈ R} is called a connecting orbit from a to b.
To give the existing conditions of attractors of dynamical systems, we need rst give the properties of the limit set, which have been proven in [ ], [ ] and will be used in our main theorems.
In the article, for simpler, we mainly discuss properties of the attractor and related results of the dynamical system in the space X = R n in the next section. Of course, the results given still hold in the general locally compact metric space. At the end of this section, we give an example which explains a fact that ∪ x∈M ω(x) = ω(M, π) does not hold in general. Example 2.3. Consider the dynamical system de ned by di erential equations as follows

Main results
In this section we give main results in the article. First of all, based on the properties of a near isolating block, we shall give the existing condition of an attractor.
Because ω(M, π) is the maximal invariant subset in Cl{M · [ , +∞)}, we know ω(M, π) = A. By De nition 2.1, we need only prove that M is a neighborhood of A. Now we prove it by contradiction. If M is not a neighborhood of A, then A ∩ ∂M ≠ . Choose x ∈ A ∩ ∂M. For one thing, since x ∈ A, x · R ⊂ IntN; For other, for x ∈ ∂M, there exists a sequence {xn} +∞ n= such that xn ∈ N \ M, limn→+∞ xn = x . Let tn = max{t ≥ : xn · [ , t] ⊂ N}, and then limn→+∞ tn = +∞ because x · R + ⊂ IntN by the continuous dependence on initial values. By the de nition of tn, xn · tn ∈ ∂N and tn is nite for xn ∈ N \ M. Since N is a bounded subset of R n , Cl{N} and ∂N are compact. The compactness of ∂N implies that there exists a convergent subsequence of {xn · tn} +∞ n= . Without loss of generalization, {xn · tn} +∞ n= also denotes its convergent subsequence, and then limn→+∞ xn · tn = y ∈ ∂N. For y ∈ ∂N, there exists a Ty > such that y · (−Ty) ∈ ExtN; An enough large number n > is chosen such that xn · tn · (−Ty) ∈ ExtN and tn > Ty. Hence xn · (tn − Ty) ∈ ExtN and < tn − Ty < tn, which contradict with the de nition of tn.  N such that ω(N, π) = A, and ω(Cl{N}, π) The Next we shall give an equivalent condition of existence of a near isolating block of a dynamical system. Theorem 3.3. Assume that the set N is a bounded subset of R n , then for each x ∈ ∂N, α(x, π) ∩ ExtN ≠ or x · R − ∩ ExtN ≠ hold if and only if N is a bounded near isolating block of R n .
Proof. Now we prove the necessity. If x · R − ∩ ExtN ≠ for x ∈ ∂N, it is obvious to satisfy the condition of a near isolating block. If α(x, π) ≠ and α(x, π) ∩ ExtN ≠ hold for x ∈ ∂N, there is a δ > such that B(y, δ) ⊂ ExtN for y ∈ α(x, π) ∩ ExtN. From the de nition of α(x, π), it follows that x · R − ∩ ExtN ≠ .
The su ciency. Assume that the set N is a bounded near isolating block of R n , i.e., for each x ∈ ∂N, x · R − ∩ ExtN ≠ , so we only prove if α(x, π) ≠ for x ∈ ∂N, then α(x, π) ∩ ExtN ≠ . Since N is a bounded near isolating block, there exists a t < such that x · t ∈ ExtN. If x · (−∞, t ] ∩ N = , it follows that α(x, π) ⊂ Cl{ExtN}; Otherwise, by the connectedness of x · (−∞, t ], there exists a t (< t ) such that x · t ∈ ∂N. Thus there is a negative number t (< t ) such that x · t ∈ ExtN since N is a bounded near isolating block of R n . Now we consider two cases as follows: Case I: If there exists a Tx < such that x · (−∞, Tx] ∩ N = for x ∈ ∂N, then α(x, π) ⊂ Cl{ExtN}. So α(x, π) ∩ ExtN ≠ . Otherwise, since the set α(x, π) is invariant, it follows that α(x, π) ⊂ ∂N, which contradicts with x · R − ∩ ExtN ≠ for each x ∈ ∂N.
Proof. Let N = {x | x · R + ⊂ N}, then by the proof of Theorem 3.1, N is a neighborhood of A and ω(N , π) = A. By ω(N , π) = A, for any neighborhood U of A there exists a t U > such that N · [t U , +∞) ⊂ U. Let U = IntN , then there exists a t N > such that N · [t N , +∞) ⊂ IntN . Now we show that there exists a tx > such that x · (−∞, −tx] ∩ N = for any x ∈ ∂N . Otherwise, there is a P ∈ ∂N , for any t P > , P · (−∞, −t P ] ∩ N ≠ . Choose t P > t N and Q ∈ P · (−∞, −t P ] ∩ N , and then there exists a t Q > t P > t N such that Q = P · (−t Q ) ∈ N . Hence, Q · [t N , +∞) = P · (−t Q ) · [t N , +∞) = P · [t N − t Q , +∞) ⊂ IntN . By t N − t Q < , it implies P ∈ P · [t N − t Q , +∞) ⊂ IntN , which contradicts with P ∈ ∂N . Let M = N · t N . Since A is an invariant subset of IntN , it follows that A = A · t N ⊂ N · t N = M ⊂ IntN and the map π(·, −t N ) : ∂M → ∂N is a homeomorphism. Thus for any x ∈ ∂M, x · (−∞, −tx − t N ] ∩ N = holds. Hence α(x, π) ⊂ Cl{ExtN } and α(x, π) ∩ M = when α(x, π) ≠ . The proof is completed. Remark 3.3. According to the argument above, M is a neighborhood of A satisfying ω(M, π) ⊂ M. However, ω(M, π) ⊂ M holds for an attractor neighborhood N of A, and ω(N, π) ⊂ N may not hold yet.
Example 3.1 Consider the following planar dynamical system de ned by the di erential equations: Proof. Now we consider the necessity. Suppose ω(N, π) ⊂ N.
Remark 3.4. Under the conditions above, if N is a closed set and N ⊂ A(M, π), we get α(x, π) ⊂ ExtN for any x ∈ ∂N. Otherwise, there is a x ∈ ∂N such that α(x, π) ∩ N ≠ . We can choose a y ∈ α(x, π) ∩ N and ω(y) ⊂ α(x, π). Hence ω(y) ∩ M = , which is a contradiction with x ∈ N ⊂ A(M, π). Remark 3.5. In some literature, A(M, π) = {x ∈ X | ≠ ω(x, π) ⊂ M} also denotes the attraction region of the attractor M. So Theorem 3.5 explains the relations between two kinds of attraction regions. In fact, from the proofs above, we easily see that the results of Theorems 3.1-3.5 still hold in generally locally compact metric spaces.
Next, we can get the condition of the compactness of the set formed by bounded orbits by a near isolating block. For simpler, we only consider the dynamical systems de ned in the space R .
Theorem 3.6. The set of the bounded orbits of the dynamical system is compact if there exists a bounded near isolating block N ⊂ R such that all equilibrium points and closed orbits of the dynamical system are contained in the region N.
Proof. Assume that p · R + is bounded for p ∈ R , so the positive limit set ω(p) contains an equilibrium point, or a closed orbit, or a connected set composed of some equilibrium points and some orbits whose positive semi-orbit and negative semi-orbit tend to a singular point respectively by Poincaré-Bendixson Theorem(see [ ]). Similarly, if p · R − is bounded for p ∈ R , so the negative limit set α(p) contains an equilibrium point, or a closed orbit, or a connected set composed of some singular points and some orbits whose positive semi-orbit and negative semi-orbit tend to a singular point, respectively. Since the set N ⊂ R is a near isolating block and contains all equilibrium points and closed orbits of the dynamical system, it contains all bounded orbits of the dynamical system. From Theorem . , we know that in N, the set of the bounded orbits is an attractor. That is, the attractor is formed by all bounded orbits of the dynamical system, and it is only one attractor. Hence, the set of all bounded orbits is compact by the boundedness of N. The proof is completed. Theorem 3.7. Assume that ω(x, π) ≠ for x ∈ R . Then the set of bounded orbits of the dynamical system is compact if and only if there exists a bounded near isolating block N ⊂ R such that all equilibrium points and closed orbits of the dynamical system are contained in the region N.
Proof. From the proof of Theorem . , we know that the su ciency holds. The necessity. Let M denote the set of bounded orbits of the dynamical system. If M is compact, M is bounded. There exists a bounded subset N ⊂ R such that M ⊂ N. Since ω(x, π) ≠ for x ∈ R , any bounded neighborhood of M is a near isolating block containing all equilibrium points and closed orbits of the dynamical system. In fact, ω(x, π) ⊂ M for x ∈ N, and N contains all bounded orbits. The proof is completed.
Theorem 3.8. If the dynamical system has a bounded near isolating block N which has only two equilibrium points and no homoclinic orbit or closed orbit, then there exist one or uncountable connecting orbits in N.
Proof. From Theorem 3.6, the set of bounded orbits of a dynamical system in N is compact. By Theorem 3.1, it is the attractor A which has exactly two equilibrium points, and there exist no closed orbit and homoclinic orbit. Then the set of all bounded orbits is composed of the equilibrium points and connecting orbits by the connectedness of the limit set and the Poincaré-Bendixson Theorem, and the possible numbers of the connecting orbits are one or uncountable. The proof is completed.
Corollary 3.1. If the dynamical system has a bounded near isolating block N which has at least two equilibrium points but no homoclinic orbit or closed orbit, there exist one or uncountable connecting orbits in N.
Example 3.2. Consider the dissipative dynamical system de ned by the deferential equations in the plane: Obviously, the dynamical system ( . ) has a saddle point O = ( , ) and two stable foci P = (− , ), Q = ( , ). There exist two connecting orbits γ , γ which connect O,P and O, Q, respectively. Each orbit through a point x in the plane goes to P or Q provided that x does not lie on the stable or unstable manifolds of O. That is, {O, P, Q} ∪ γ ∪ γ is an attractor, and we can choose the set N such that {O, P, Q} ∪ γ ∪ γ ⊂ N, which is an attractor neighborhood of the attractor {O, P, Q} ∪ γ ∪ γ and a near isolating block.

Conclusion
In this article, we investigate the properties of attractors of the dynamical system. The dynamical system considered here may not be the positively bounded system in the locally compact space, where near isolating block de ned may or may not be a closed set. The results include mainly two aspects: (1) The existing conditions of attractors of the dynamical system are given by near isolating block; (2) The equivalent condition of compactness of the set which is formed by the bounded orbits of the planar dynamical system is presented. This paper shows that near isolating block plays an important role in investigating the properties of dynamical systems. Thus, the applications of near isolating block require further studies.