Let denote a compact metric space and let be a continuous map. It is known that a discrete dynamical system () naturally induces its fuzzified counterpart, that is, a discrete dynamical system on the space of fuzzy compact subsets of . In 2011, a new generalized form of Zadeh’s extension principle, so-called -fuzzification, had been introduced by Kupka 2011. In this paper, we study the relations between Martelli’s chaotic properties of the original and -fuzzified system. More specifically, we study the transitivity, sensitivity, and stability of the orbits in system () and its connections with the same ones in its -fuzzified system.

1. Introduction

The main goal of the theory of discrete dynamical system is to understand the asymptotic properties and topological structures of the orbits. In certain sense, the study of the orbits in discrete dynamical system is to investigate the movement of the points in the base space. In many cases, however, it is not sufficient to know how the points move, but it is necessary to understand the motion of the subsets of base space (e.g., in migration phenomenon), and this leads us to the problem of analyzing the dynamics of the set-valued discrete dynamical systems. In this direction, many elegant results have been obtained (please see [17] and the references cited therein).

As the complexity of research subjects increased, an accurate description for systems becomes more and more difficult, and the situation would become more complicated when the systems are affected by the uncertainty. In this case, the fuzzy system should be considered. It is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart, that is, a discrete system on the space of fuzzy sets. It is natural to ask the following question: what is the relation between dynamical properties of the original and fuzzified systems?

Motivated by this question, the study of discrete fuzzy dynamical systems has recently become active [812]. As a partial response to the question above, in the case of Devaney chaos [13], the transitivity, periodic density, and sensitivity between two systems have been analyzed in 2008 [14]. In addition, by analyzing connections between the fuzzified dynamical systems 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. In 2011, Kupka 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 [14]. 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 the periodic density, a concept of piecewise constant fuzzy set is introduced, and then period density equivalence of and is proposed. Consequently, the question has been completely solved [15]. And then we discuss this issue by using the weakly mixing property [16].

Among the methods of fuzzification, Zadeh’s extension [17] is often used, but it can lose information that is carried by the original system. Therefore, more general extension principles have been developed [18, 19]. Recently, a concept of -fuzzification, which allows us to modify the membership grades of points in each iteration, has been introduced [19]. The use of -fuzzification is quite natural but differs slightly from Zadeh’s extension principle, and it can be useful in several situations. Take a fuzzy set “old people” for example; old people in ancient time are not considered as old at present since the average age of people is increasing. We also can find several examples to illustrate such kinds of fuzzy sets with variable membership grades. Zadeh’s extension, however, does not reflect this fact. On the other hand, the situation becomes more complicated in fuzzy control. In [20], the authors show that a chaotic function on , for its fuzzification in the sense of Zadeh, is degenerate, because the iterates are asymptotically crisp and, ultimately, we obtain chaos of a mapping of ordinary sets rather than of fuzzy output. In this case, the usual fuzzification is inadequate to describe complexities which may arise in fuzzy control. Consequently, a concept of -fuzzification has been developed in [18], which does not degenerate under chaotic iteration. Now, as a new generalized extension principle, -fuzzification includes the usual fuzzification (Zadeh’s extension) and -fuzzification as two special cases of it, and the developed methods enable us to study the dynamics of discrete fuzzy systems in a more efficient way.

Chaotic dynamics has been hailed as the third great scientific revolution of the 20th century, along with relativity and quantum mechanics. But there is not a generally accepted definition of chaos yet. The different definitions of chaos being around have been designed to meet different purposes and they are based on very different backgrounds and levels of mathematical sophistication. To compare various kinds of definitions of chaos naturally attracts the interest of many researchers. In 2002, Huang and Ye showed that chaos in the sense of Devaney is stronger than that of Li-Yorke [21]. The conclusion stimulates the study of the relations between different definitions of chaos [2224].

Among various definitions of chaos, Martelli’s chaos is one of the definitions of chaos which are suitable for easy and reliable numerical verification [25]. The authors make comparison of different definitions of chaos and point out that Martelli’s chaos embodies the essential features which all other definitions are trying to capture [26]. It is worth noting that although formulated in a different way, Martelli’s chaos is practically equivalent to chaos in the sense of Wiggins [27]. There remains, however, a difference between the two definitions. Wiggins does not require sensitivity with respect to the base space, while Martelli requires instability with respect to the base space.

In this paper, we focus on relations between Martelli’s chaotic properties of the original and -fuzzified dynamical systems. Below, Section 2 gives basic notions and definitions. Section 3 discusses the relation between Martelli’s chaotic properties of the original and -fuzzified systems. A brief conclusion concludes the paper.

2. Preliminaries

In this section, we complete notations and recall some known definitions. Let be a continuous map acting on a compact metric space . An orbit of a point is the set , denoted by or simply when the function is clearly specified. A point is a limit point of if a subsequence of converges to . The set of limit points of is denoted by .

We say that is transitive if for any pair of nonempty open sets and there exists such that ; is point transitive if there exists a point such that the orbit of is dense in ; that is, , and is called a transitive point of .

We say that is unstable if there exists such that, for any neighborhood of , there exist and such that . An orbit which is not unstable is said to be stable.

We say that has sensitive dependence on initial conditions if there is a constant such that for every point and every neighborhood about there are a and a such that . Hence, every orbit with is unstable with the same constant . Consequently, sensitive dependence on initial conditions is stronger than instability.

Definition 1 (see [25]). Let be a compact metric space and let be continuous. Then, is Martelli chaotic provided that there exists such that(i);(ii) is unstable.

In this research, we call a Martelli chaotic map M-chaotic for short.

Below, we present some definitions from fuzzy theory. Let be the class of all nonempty and compact subsets 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 nonempty fuzzy sets on and let denote the empty fuzzy set ( for all ).

A level-wise metric on is defined by for all . It is well known that if is complete, then is also complete but is not compact and is not separable (see [19, 28, 29]).

Lemma 2 (see [9, 14]). Let be an open subset of . Define , and then is an open subset of .

Let be continuous. A usual fuzzification (often called Zadeh’s extension) is defined by for any and .

Now let us introduce -fuzzification. Denote as the set of all nondecreasing right continuous functions for which if and . Let be the set of all continuous maps from . For any , a -fuzzification is defined by An -cut of a fuzzy set with respect to is

Lemma 3 (see [19]). Let be continuous and let be -fuzzification. Then, holds for any , , and .

Lemma 4 (see [19]). Let , , and . If . then there is such that .

3. -Chaotic Relations between and

In this section, we study the relations between Martelli’s chaotic properties of the original system and -fuzzified system , where is equipped with the level-wise topology, that is, the metric topology induced by .

On the one hand, some conditions are discussed, under which -chaotic implies -chaotic. On the other hand, several examples are presented to illustrate that, in general, -chaotic does not imply -chaotic.

Proposition 5. Define . Then holds for every and .

Proof. The inclusion follows directly from the definition of . If , then, because is nondecreasing, we have , which implies , and, consequently, holds.

Proposition 6. Let be subset of and let be continuous. Then, .

Proof. If , then there exists such that . Hence, due to Lemma 3 and Proposition 5, we have that , since and ; thus , and the inclusion follows.

Proposition 7. Let , , and , . Then, there exists an such that .

Proof. We do the proof by mathematical induction.
When , by Lemmas 3 and 4, the formula gives us ; therefore, the statement holds for .
Assume that the statement is true for ; that is, Note that .
When , On the other hand, .
This completes the proof.

Remark 8. The proof of Proposition 7 can also be done as follows.
Due to Lemmas 3 and 4, we have that On the other hand, by Lemma 3 again, we obtain , and, consequently, holds.

Theorem 9. Let be a transitive point of . Then, every is a transitive point of for .

Proof. Since is a transitive point of , there exists such that for any and . By using Proposition 7 and Lemma 3, we obtain for some . Hence, for each , there exists such that , which means that every is a transitive point of .

Theorem 10. If , then there exists such that .

Proof. It follows directly from Theorem 9.

The following example shows that, in general, the converse of Theorem 10 is not true.

Example 11 (irrational rotation of circle). Let be an irrational number and is defined by . It is well known that, for each , the orbit of is dense in and, consequently, . Nevertheless, it is not necessary for some to exist such that . In fact, assume that and . Given that , let and , and by Proposition 5, we obtain since for . Hence, , which means that there exists no such that for some , and, consequently, .

Theorem 12. Let be continuous, let be the -extension of , and let . If the orbit of is unstable in , then there exists such that the orbit of is unstable in , where .

Proof. Let the assumptions be satisfied. Then, there exists such that for every we can find and satisfying and Thus, there exist and such that . Since , we have . This proves that there exists such that the orbit of is unstable in with instable constant .

Example 13. Consider the foregoing example (Example 11); because is isometric, it does not exhibit sensitive dependence on initial conditions and hence the orbit of each is stable, which implies that, by Theorem 12, there exists no orbit of that is unstable.

By combining Theorems 9, 10, and 12, we obtain the following theorem.

Theorem 14. If is -chaotic, then is -chaotic.

We will need some notions from Denjoy map [13]. Recall that the circle can be considered as the quotient space , where and are the sets of real numbers and integers, respectively. The irrational rotation of the circle is then given by where is irrational. Recall that a Denjoy map can be constructed as follows. Take any point . We cut out each point on the orbit of and replace it with a small interval . For ,(a), , , and , where denotes the length of the interval ;(b).

Consequently, a new circle has been constructed. The Denjoy homeomorphism is an orientation preserving homeomorphism of . There exists a Cantor set on which acts minimally. It is known that there exists a continuous surjection that semiconjugates with . In [30], the authors show that the system is not sensitive.

Proposition 15. Let ; then the orbit of is unstable in with the constant .

Proof. Suppose that and for any . Since the orbit of is dense in , there exist some such that , where is the closed arc in . Thus, we have . Consequently, due to the construction of Denjoy map, we obtain , which means that .

The following proposition shows that the instability of the orbit in cannot be inherited by its -fuzzification. More specifically, there exist points arbitrarily close to which eventually also close to under iteration of , although there exist some such that the orbits of these points are unstable in . It should be mentioned that our approach was inspired by the idea in [8] where a continuous map was defined.

Define by for any and any , where is the characteristic function of (that is to say, if and if ). Hence, . Note that is continuous.

Proposition 16. Let ; then there exist some and such that .

Proof. Since is not sensitive, for and , there exist and such that, for all , Suppose that (recall that ), and by continuity of and (19), we have
Without loss of generality, assume that . This completes the proof.

Remark 17. Theorem 9 together with Theorem 10 shows that -chaotic implies -chaotic, but generally speaking, the converse is not true, which has been discussed in Example 11 and Proposition 16.

4. Conclusions and Discussions

In this present investigation, we discuss relations between Martelli chaotic properties of the original and -fuzzified dynamical systems. More specifically, we study stability of the orbits and transitivity and present several examples to illustrate the relations between two dynamical systems. We show that the dynamical properties of the original system and its fuzzy extension mutually inherit some global characteristics. The following main results are obtained.(a)If , then there exists such that (Theorem 10).(b)The instability of implies the instability of , where , , and (Theorem 12).(c)-chaotic implies -chaotic (Theorem 14).(d)-chaotic does not imply -chaotic (Example 11 and Proposition 16).

It is worth noting that any -fuzzification is connected to a crisp discrete dynamical system in two different ways [19]. One way is to connect two systems via -cut, and another approach is to consider -fuzzified discrete dynamical system as a crisp system that is induced by a certain product map. We develop, in this present paper, the first method. It would be interesting to use the second approach to study the relations between dynamical properties of the original and -fuzzified dynamical systems, and this will be one aspect of our future works.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


This work was partially supported by the National Natural Science Foundation of China (Grant no. 11226268), Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ131219), Postdoctoral Science Foundation of Chongqing (Grant no. Xm201328), and Program for Innovation Team Building at Institutions of Higher Education in Chongqing (Grant no. KJTD201321).