Uncertain Dynamical Systems: Analysis and ApplicationsView this Special Issue
Some Chaotic Properties of Discrete Fuzzy Dynamical Systems
Letting be a metric space, a continuous map, and the space of nonempty fuzzy compact subsets of with the Hausdorff metric, one may study the dynamical properties of the Zadeh's extension . In this paper, we present, as a response to the question proposed by Román-Flores and Chalco-Cano 2008, some chaotic relations between and . More specifically, we study the transitivity, weakly mixing, periodic density in system , and its connections with the same ones in its fuzzified system.
Throughout this paper, let be a compact metric space with metric and let be continuous. A discrete dynamical system is a pair . For other notions and notations mentioned in this section, we refer to Section 2. The main goal of the theory of discrete dynamical system is to understand the dynamics of individuals (points) in the state space . However, in many cases such as biological species and migration phenomenon, it is not enough to know how the points move, but it is necessary to know how the subsets of move, which is so-called collective dynamics. When studying the chaotic dynamics of individual members in a certain ecosystem, the natural question that arise is what the relationship between individual chaos and collective chaos is.
Motivated by this question, the study of set-valued discrete systems has recently become active [1–6]. Moreover, when the available datas are uncertain, the fuzzy system should be considered: where is a continuous map on a metric space and is the Zadeh's extension of to , the class of all nonempty compact fuzzy subsets of . Consequently, the fundamental question here is to analyze relations between and : When does the chaoticity of imply the chaoticity of ? And conversely?
As a partial response to this question, in the case of Devaney chaos, Román-Flores and Chalco-Cano  investigate the discrete fuzzy dynamical system associated to given by , and then obtain the following results: transitive transitive, transitive transitive, sensitive sensitive, has periodic density has periodic density.
In addition, by analyzing connections between the fuzzified dynamical system related to the original one, the authors have pointed out that this kind of investigation should be useful in many real problems, such as in ecological modelling and demographic sciences. Some recent works along these lines appear, for example, we refer to [8, 9], where different dynamical properties were demonstrated. In , Kupka shows that the dynamical behavior of the set-valued and fuzzy extension of the original system mutually inherits some global characteristics. In particular, the author proves that there exists a transitive fuzzification on the space of normal fuzzy sets, which contains the solution of the problem that was partially solved in . Specifically, the author considers a symbolic dynamical system as the original system and then shows that Zadeh's extension of the shift map is transitive. As regards periodic density, a concept of piecewise constant fuzzy set is introduced in , and then period density equivalence of and is proposed. Together, Kupka presents a complete solution of the open problem that has been established in .
In this paper, unless otherwise stated a chaotic map is always Devaney chaotic. We investigate relations between and and other dynamical concepts that describe chaos. Although the problem has been completely solved in , we propose some different methods to solve the problem. Following [8, 10] and references therein, the space of fuzzy sets does not admit a transitive fuzzification, thus it cannot be mixing, weakly mixing, and so forth. Hence, it makes sense to consider the space of normal fuzzy set on . We show that in case where is weakly mixing, is chaotic provided that is chaotic. In the converse direction of the question mentioned above, we prove that, in certain conditions, chaotic implies chaotic. Moreover, we also prove that a totally transitive map with dense small period set is weakly mixing.
Below, Section 2 gives some known definitions and notations. In Section 3 and Section 4 the main results are presented. We conclude this paper with some discussions in Section 5.
In this section, we complete notations and recall some known definitions. Let be the class of all nonempty and compact subset of . If we define the -neighbourhood of as the set where .
The Hausdorff separation of is defined by
The Hausdorff metric on is defined by letting
Define as the class of all upper semicontinuous fuzzy sets such that , where -cuts and the support of are defined by respectively.
Moreover, let denote the space of all normal fuzzy sets on and denote the empty fuzzy set ( for all ).
A levelwise metric on is defined by for all , where . It is well known that if is complete, then is also complete but is not compact and is not separable (see [9, 11, 12]).
We say that is transitive if for any pair of nonempty open sets and there exists such that ; is totally transitive if all its iterates is transitive; is weakly mixing if for all nonempty open sets , , , and there exists such that and ; is mixing if for any pair of nonempty open sets and there exists such that for all one has . We say that has dense small period sets if for every nonempty open set there is a closed subset and such that .
A point is periodic if for some . We say that has sensitive dependence on initial conditions if there is a constant such that for every point and every neighborhood about , there is a and a such that . A map that is transitive, has a dense set of periodic points, and has sensitive dependence on initial conditions is called Devaney chaotic . However, sensitive dependence on initial conditions is a consequence of transitivity together with a dense set of periodic points [14, 15]. More precisely, sensitivity is redundant in the definition if the state space is infinite. This fact reveals the topological, rather than metric, nature of chaos. In this paper, we say that a map is Devaney chaotic (chaotic for short), if it is transitive and has dense set of periodic points.
Proposition 2.1 (see ). The family satisfies the following properties:(1), for all ,(2) if and only if , for all ,(3), for all ,(4).
Proposition 2.2 (see [7, 8]). Let be an open subset of . Define , then is an open subset of .
Proposition 2.3 (see [7, 8]). If has periodic density on , then has periodic density on .
Proposition 2.4 (see ). For any continuous map , is weakly mixing if and only if for any nonempty open sets and there is a such that and .
3. Chaotic Implies Chaotic
In this section, some conditions are discussed, under which chaotic implies chaotic. Let be a subspace of . Notice that for all . We say that a topological space has the fixed point property (in short, .) if every continuous map has a fixed point. We will denote the family of all nonempty compact subsets of which have the . by . Define . Theorem 3.2 below shows that when chaotic will imply chaotic. Note that in [7, 8], transitive implies transitive, to prove is chaotic, it suffices to prove that has periodic density.
Remark 3.1. Let be a subset of and let . We can conclude that if is an open subset of , then is an open subset of . Consequently, we obtain transitive implies transitive, the proof is similar to the Theorem 3 in .
Theorem 3.2. Let be continuous, the Zadeh's extension of , and a subspace of such that for all . If , then chaotic implies chaotic.
Proof. Since is transitive, it follows, by Remark 3.1, that is transitive. Therefore, it suffices to show that is periodically dense.
If and , then and, by periodic density of , there exist and such that (a), (b).
On one hand, by Proposition 2.1 (3) and (4), we have . Thus, combing (a) and (b), we have for all .
On the other hand, the map given by for every is a continuous map. Since has the . (recall that ), it follows that has a fixed point such that , that is to say, is a periodic point of contained in . Thus, due to (3.1), we obtain for all . Consequently, has periodic density on . The theorem is proved.
Remark 3.3. The conditions on in Theorem 3.2 are restrictive. In fact, we could consider with different conditions, for instance being the subspace of compact convex fuzzy set. More specifically, let be the family of all fuzzy sets that satisfy(i), which means that is upper semicontinuous and is compact,(ii) is fuzzy convex, that is, ,(iii) is nonempty.
If is endowed with the topology generated by the levelwise metric , then each compact convex subset of has the . . Hence, the conditions in Theorem 3.2 could be slightly changed.
Proposition 3.4. If is transitive, then is weakly mixing.
Proof. Suppose is transitive. By Proposition 2.4, it suffices to show that for any nonempty open sets and , there is a such that
Due to Proposition 2.2, and are open subsets of and so, is open. Thus, by transitivity of , there is a such that
Therefore, we have
On one hand, since , there exist and such that . By Proposition 2.1(3) and (4), we have For any , we have . On the other hand, we can choose such that , it would then follow by induction that .
It is known that a totally transitive system having dense period points is weakly mixing. The following proposition shows that a totally transitive map with dense small period sets is also weakly mixing.
Proposition 3.5. If is a totally transitive map with dense small period sets, then is weakly mixing.
Proof. Let , , , and be any nonempty open subsets of . It suffices to show that for some . Since is transitive, there is a such that is open and nonempty. Hence contains a closed subset of period . For , we have , whence . Since is nonempty and open and is transitive, there is a such that whence . Setting implies that is weakly mixing.
4. Weakly Chaotic Implies Chaotic
Concerning the transitivity of fuzzy dynamical systems, the authors in  has proved that transitive implies transitive, but the converse is not true. In [8, 10] the author presents that no fuzzification can be transitive on the whole , but there exists a transitive fuzzification on the space of normal fuzzy sets. In this section, we propose another method to prove that is weakly mixing implies is weakly mixing and thus transitive. It should be mentioned that our approach was inspired by the idea presented in [8, 10].
We say that a map is weakly Devaney chaotic (weakly chaotic for short) if it is weakly mixing and periodically dense. Let be a subset of . Set
Proposition 4.1. Let , , and be subsets of ,(1) if and only if , where is the empty fuzzy set ( for each ),(2)suppose that implies , then ,(3),(4)if is a nonempty open subset of , then is a nonempty open subset of .
Proof. Follows directly from the definitions.
If , then there exists such that . Then and . Therefore, the inclusion follows. Conversely, let . Then there exist and such that and , respectively. Hence, by hypothesis, which means that and so, . Consequently, there is such that and the inclusion is true.
If , then there exists such that . Thus, by Proposition 2.1 (3), we have , consequently, .
Suppose that is not open. For any and , there exists open -neighborhood of such that and . Consider a fuzzy set . Since and we obtain , where is an open ball in . However, , and consequently, . That is to say, no -neighborhood of contains in , this contradicts the fact that is open in .
Proposition 4.2. If is weakly mixing then is weakly mixing.
Proof. Suppose is weakly mixing. By Proposition 2.4, it suffices to show that for any nonempty open sets and of , there is a such that Since and are open, by Proposition 4.1(4), and are also open sets. Due to is weakly mixing, there is a such that By Propositions 4.1(3) and 2.1(4), we have Thus, using Proposition 4.1(2), it follows that Therefore,
Theorem 4.3 below is an immediate consequence, it shows that in case when is weakly mixing, will be chaotic provided is chaotic.
Theorem 4.3. Let be continuous and the Zadeh's extension of . If is weakly chaotic, then is chaotic.
Proof. By Proposition 4.2, is weakly mixing and hence transitive, combine this assertion and Proposition 2.3, we can conclude that is chaotic.
5. Conclusions and Discussions
In this present investigation, we discuss relations between dynamical properties of the original and fuzzified dynamical systems. More specifically, we study transitivity, periodic density, and weakly mixing and so forth. And we show that the dynamical properties of the original system and its fuzzy extension mutually inherits some global characteristics. More precisely, the following implications are obtained(a) chaotic implies chaotic (Theorem 3.2),(b) is transitive is weakly mixing (Proposition 3.4),(c) is weakly mixing is weakly mixing (Proposition 4.2),(d) is weakly chaotic is chaotic (Theorem 4.3).
Actually the open question raised in  has been completely solved in , here we propose another approaches to answer the question. It is worth noting that to complete the generalization of the system , we need to endow with a metric. When considering the practical interpretation and computer realization, a question can be addressed here: Is there any other metric available? In , the author discusses some other metrics on this subject.
On the other hand, it is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart, that is, a discrete fuzzy dynamical system. There have been various attempt to “fuzzify” the discrete dynamical systems. One of these methods appeared in  where either a -norm or -conorm was used to fuzzify discrete systems and then elaborated and specified in  in a more general way, namely, -fuzzification. Therefore, on the basis of the idea presented in , it would be interesting to study the relations between dynamical properties of the original and -fuzzified dynamical systems. And this will be one aspect of our future works.
This work was partially supported by the National Natural Science Foundation of China (no. 11226268, no. 11071061, no. 61202349), the National Basic Research Program of China (no. 2011CB311808), NNSF of Hunan Province (no. 10JJ2001), and NSF of CUAS (no. Z2010ST14).
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