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Some chaotic properties of fuzzified dynamical systems
SpringerPlus volume 5, Article number: 640 (2016)
Abstract
Let X be a compact metric space and \(f:X\longrightarrow X\) a continuous map. Considering the space \({{\mathbb {F}}}(X)\) of all nonempty fuzzy sets on X endowed with the levelwise topology, we proved that its gfuzzification is turbulent or erratic if the given system f is turbulent or erratic correspondingly and f is \(\lambda\)expansive if and only if its gfuzzification is \(\lambda\)expansive.
Background
Let X be a compact metric space and f be a continuous selfmap. We denote by \({{\mathbb {F}}}(X)\) all nonempty fuzzy sets on X endowed with the levelwise topology and we define \(\widehat{f}\) as the Zadeh’s extension of the map f:
With the complexity of situations rising, the zadeh’s extension has failed to be an accurate description for discrete fuzzy systems affected by the uncertainty. The time evolution of a chosen initial state is very important. We take the fuzzy set “short people” for example. “short people” in ancient time are not defined as short at present since the average height of people is increasing. It’s not difficult to find some other examples to figure out that the zadeh’s extension can’t reflect the complex systems accurately. A very important generalization of the concept of zadeh’s extension is gfuzzification introduced in Kupka (2011a). It can be used for changing relevant membership grades and developed to describe the complex fuzzy systems in a more efficient way. The map \(\widehat{f}_{g}: {{\mathbb {F}}}(X)\longrightarrow {{\mathbb {F}}}(X)\) is called gfuzzification of f and \({{\mathbb {F}}}(X)\) denotes all nonempty and fuzzy compact subsets of X. Over the last ten or so years, since many research works have been devoted to the chaotic behaviors of the fuzzified dynamical systems, such as, topological entropy, Devaney chaos, LiYorke chaos and its connections with erratic functions which has been done by Cánovas and Kupka (2011), Kupka (2011b), Diamond (1994), Diamond and Kloeden (1994), Diamond and Pokrovskii (1994). Recently, J.Kupka have studied various chaotic behaviors (LiYorke chaos, \(\omega\)chaos, distributional chaos, topological chaos etc.) between a given dynamical system (X, f) and its gfuzzification. See more details in Kupka (2014). But it remains to be asked to study the other chaotic properties on gfuzzification. Then the turbulence and erratic properties between a crisp dynamical system and its gfuzzification are taken into our considerations in this paper.
On the other hand, the \(\lambda\)expansive property in fuzzy systems is explored in this paper. Thus, the left work is to demonstrate the \(\lambda\)expansiveness relationship between \(\widehat{f}_{g}\) and f. Meanwhile, it is not difficult to see that the \(\lambda\)expansive property can exhibit the sensitivity of \(\widehat{f}_{g}\).
Preliminaries
Metric space of fuzzy sets
Let (X, d) be a compact metric space and C(X) denote all continuous functions. \(f\in C(X)\) is a selfmap. \({{\mathcal {K}}}(X)\) denotes all nonempty and compact subsets of X with the Hausdorff metric \(d_{H}\) defined by
for any \(A, B\in {{\mathcal {K}}}\). \(\bar{f}: {{\mathcal {K}}}(X)\longrightarrow {{\mathcal {K}}}\) is defined as \(\bar{f}(A)=f(A)\) for any \(A\in {{\mathcal {K}}}\).
A fuzzy set u on the space X is a function \(u: X\longrightarrow I\) where \(I=[0,1]\). For any \(\alpha \in (0,1], [u]_{\alpha }=\{x\in X\mid u(x)\ge \alpha \}\) is called the \(\alpha\)level of u and \([u]_{0}=\{x\in X\mid u(x)\ge 0 \}\) is the support of u (shortly: supp(u)) (RománFlores et al. 2011).
Throughout this paper, \({{\mathbb {F}}}(X)\) is defined as all upper semicontinuous fuzzy sets and equipped by the following metric:
Let
and \({{\mathbb {F}}}^{1}(X)\) is denoted as the system of all normal fuzzy sets on X.
gfuzzifications
Let \(D_{m}(I)\) be the set of all nondecreasing rightcontinuous functions \(g: I\longrightarrow I, I=[0,1]\). If \(x=0\) and \(x=1\), \(g(x)=x\). A map \(\widehat{f}_{g}\) denoted by
is defined as a gfuzzification. Especially, it is the Zadeh’s extension when the function \(g(x)=1\). Also, a definition of \(\alpha\)cut \([A]^{g}_{\alpha }\) is presented by \([A]^{g}_{\alpha }=\{x\in [A]_{0}\mid g( A(x))\ge \alpha \}\). The following equation in Kupka (2011a) is indispensable.
Lemma 1
Kupka (2014): Let \(f\in C(X)\) and \(g\in D_{m}(I)\). For any \(\alpha \in (0,1]\) and nonempty fuzzy set \(A\in {{\mathbb {F}}}(X)\), if \([A]^{g}_{\alpha }\ne \emptyset\) then there is \(c\in (0,1]\) such that \([A]^{g}_{\alpha }=[A]_{c}\).
To explore some properties of gfuzzification, for any \(U\subset X\), we define
and
Clearly, \(e(U)\ne \emptyset\) if and only if \(U \ne \emptyset\). Moreover, the following lemma is essential in the paper.
Lemma 2
Kupka (2014): Let U, V be two subsets of \(X, f\in C(X)\) and \(g\in D_{m}(I)\). Then:

1)
\(e(U \cap V)=e(U)\cap e(V).\)

2)
\(\widehat{f}_{g}(e(U))\subseteq e(f(U)).\)

3)
\(\widehat{f}_{g}(e(U))= e(f(U))\ whenever\ U\ is\ closed.\)
Likewise, \(\vartheta (U)\) has the same properties compared with e(U).
Lemma 3
Kupka (2014): Let U, V be two subsets of \(X, f\in C(X)\) and \(g\in D_{m}(I)\). Then:

1)
\(\vartheta (U \cap V)=\vartheta (U)\cap \vartheta (V)\).

2)
\(\widehat{f}_{g}(\vartheta (U))\subseteq \vartheta (f(U))\).

3)
\(\widehat{f}_{g}(\vartheta (U))= \vartheta (f(U))\ whenever\ U\ is\ closed\).
Turbulence, erratic property, Block and Coppel chaos and \(\lambda\)expansiveness
Before we present the elegant results, we need to introduce some basic definitions of chaotic behavior explored in this paper.
Definition 1
RománFlores et al. (2011): Let \(f: X\longrightarrow X\) be a continuous function. We say that f is a turbulent function if there are disjoint nonempty closed subsets \(J,\ K\) of X such that:
Definition 2
RománFlores et al. (2011): A map \(f\in C(X)\) is erratic if there exists a nonempty closed set \(A\subseteq X\) satisfying the following two conditions:

1)
\(A \cap f(A) = \emptyset\).

2)
\(A \cup f(A) \subseteq f^{2}(A)\).
Definition 3
RománFlores et al. (2011): Let \(f: X\longrightarrow X\) be a continuous function.We say that f is chaotic in the sense of Block and Coppel (in short: BC chaos) if and only if its iterates is turbulent, i.e., there exists \(n\ge 1\) and two disjoint nonempty compact subsets J, K of X such that
It should be remarkable that the erratic property is stronger than BC chaos.
Definition 4
Let \(f: X\longrightarrow X\) be a function. We claim that f is expansive if and only if there exists a real constant \(\lambda >1\) such that
In this case we claim that f is \(\lambda\)expansive RománFlores and ChalcoCano (2005).
Main results
To prove the turbulent and erratic properties, firstly, a theorem on e(U) is given here.
Theorem 1
Let U be closed subset of X and \(e(U)=\{\hbox { B }\in {{\mathbb {F}}}(X)\mid \hbox { supp}(\hbox {B})\subseteq U\}, f\in C(X)\) and \(g\in D_{m}(I)\). Then,
Proof
It can be proved by the mathematical induction.
When \(n=1\), Left = Right = \(\widehat{f}_{g}(e(U))\). Clearly, the theorem holds.
Assume that the theorem is true for \(n=k\), i.e.,
By the Lemma 2 in Kupka (2014), we have
When \(n=k+1\),
Consequently, the statement is proved completely. \(\square\)
Likewise, we can achieve the similar result on \(\vartheta (U)\).
Theorem 2
Let U be closed subset of X and \(\vartheta (U)=e_{1}(U)\cap e(U)\). \(f\in C(X)\) and \(g\in D_{m}(I)\). Then,
Here, we make a point that \(\forall \alpha \in (0,1]\),
See Lan and Mu (2014) for more details.
RománFlores et al. (2011) explored the dynamics of \(\widehat{f}\) and f. These conclusions can be listed as follows:
More precisely, RománFlores et al. (2011) present some examples in order to show the irreversible links between f and \(\widehat{f}\). We generalizes the results with \(\widehat{f_{g}}\). As the Zadeh’s extension is a spacial case of the gfuzzification, we can make a conclusion:
The left work is to prove the following implication:
Theorem 3
If \(f\in C(X), f: X\longrightarrow X\) is a turbulent function, then \(\widehat{f}_{g}: {{\mathbb {F}}}(X)\longrightarrow {{\mathbb {F}}}(X)\) is a turbulent function.
Proof
Since f is turbulent, there exists nonempty and closed \(U, V\subseteq X\) and \(U\cap V=\emptyset\) such that \(U\cup V \subseteq f(U)\cap f(V)\).
By the definition of e(U), clearly, e(U) and e(V) are two disjoint nonempty closed subsets of \({{\mathbb {F}}}(X)\). Applying the Lemma 2, we have
Finally, we can conclude that \(\widehat{f}_{g}\) is a turbulent function. \(\square\)
Theorem 4
Let \(f\in C(X)\) be erratic, then, \(\widehat{f}_{g}\) is a erratic function.
Proof
Since f is erratic, there exists a nonempty closed subset \(U\subseteq X\) such that \(U\cap f(U)=\emptyset\) and \(U\cup f(U)\subseteq f^{2}(U)\). Thus, e(U) is nonempty closed subset of \({{\mathbb {F}}}(X)\) and
On the other hand, using the Theorem 1 and the Lemma 2, we have
Consequently, \(\widehat{f}_{g}\) is erratic. \(\square\)
Corollary 1
Let \(f\in C(X)\) be erratic, then, \(\widehat{f}_{g}\) is a BC chaos function.
Remark 1
Because \(\vartheta (U)\) has similar properties to the e(U), it can be verified with \(\vartheta (U)\) that the statements are true. Comparing Theorem 3 with Theorem 4, the fuzzy set containing e(U) at least is perfect for the two theorems to make sense.
Next, we shall discuss the \(\lambda\)expansive property of \(\widehat{f}_{g}\).
Theorem 5
Let \(f: X\longrightarrow X\) be a continuous function and \(g\in D_{m}(I)\). Then f is \(\lambda\)expansive if and only if \(\widehat{f}_{g}\) is \(\lambda\)expansive.
Proof
(\(\Rightarrow\)) Since f is \(\lambda\)expansive, for any \(u,v\in {{\mathbb {F}}}(X)\) and \(\alpha \in (0,1]\), it follows that
Applying \(f([A]_{\alpha }^{g})=[\widehat{f}_{g}(A)]_{\alpha }\), it follows that
By the Lemma 1, there is \(c\in (0,1]\) such that
which implies \(\widehat{f}_{g}\) is \(\lambda\)expansive.
(\(\Leftarrow\)) Based on the two following equations:
and
it is obvious that f is \(\lambda\)expansive. \(\square\)
Corollary 2
If \(\widehat{f}_{g}\) is \(\lambda\)expansive, \(\lambda >1\), then \(\widehat{f}_{g}\) is sensitively dependent.
Proof
Applying the method of induction, it is obvious that
Thus, \(\widehat{f}_{g}\) is sensitively dependent. \(\square\)
Corollary 3
If f is \(\lambda\)expansive, then \(\widehat{f}_{g}\) is sensitively dependent.
Conclusions
In this paper, exploiting the turbulent and erratic properties, we develop the ideas of RománFlores et al. (2011) and present some properties of e(U) and \(\vartheta (U)\) for \(\widehat{f}_{g}\) , which can be applied to the proof of Theorem 3 and Theorem 4. Moreover, the \(\lambda\)expansive property between f and \(\widehat{f}_{g}\) is studied and exhibit the sensitivity. Inducing sensitivity on fuzzy systems contains asymptotic sensitive, LiYorke sensitive and spatialtemporal sensitive etc, which will be further investigated and solved in a later work.
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Authors’ contributions
This work was carried out in collaboration among the authors. CM, PZ and TL have a good contribution to design the study, and to perform the analysis of this research work. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by National Natural Science Foundation of China (11501391) and the Scientific Research Project of Sichuan University of Science and Engineering (2014RC02).
Competing interests
The authors declare that they have no competing interests.
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Ma, C., Zhu, P. & Lu, T. Some chaotic properties of fuzzified dynamical systems. SpringerPlus 5, 640 (2016) doi:10.1186/s400640162297z
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Keywords
 gFuzzification
 Turbulent
 Erratic
 λExpansive