# Some chaotic properties of fuzzified dynamical systems

- Cuina Ma
^{1}Email author, - Peiyong Zhu
^{1}and - Tianxiu Lu
^{2}

**Received: **20 January 2016

**Accepted: **6 May 2016

**Published: **17 May 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 *g*-fuzzification is turbulent or erratic if the given system *f* is turbulent or erratic correspondingly and *f* is \(\lambda\)-expansive if and only if its *g*-fuzzification is \(\lambda\)-expansive.

## Keywords

*g*-FuzzificationTurbulentErratic

*λ*-Expansive

## Background

*X*be a compact metric space and

*f*be a continuous self-map. 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*:

*g*-fuzzification 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

*g*-fuzzification 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, Li-Yorke 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 (Li-Yorke chaos, \(\omega\)-chaos, distributional chaos, topological chaos etc.) between a given dynamical system (

*X*,

*f*) and its

*g*-fuzzification. See more details in Kupka (2014). But it remains to be asked to study the other chaotic properties on

*g*-fuzzification. Then the turbulence and erratic properties between a crisp dynamical system and its

*g*-fuzzification 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

*X*,

*d*) be a compact metric space and

*C*(

*X*) denote all continuous functions. \(f\in C(X)\) is a self-map. \({{\mathcal {K}}}(X)\) denotes all nonempty and compact subsets of

*X*with the Hausdorff metric \(d_{H}\) defined by

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)) (Román-Flores et al. 2011).

###
*g*-fuzzifications

*g*-fuzzification. 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*
\([A]^{g}_{\alpha }=[A]_{c}\).

*g*-fuzzification, for any \(U\subset X\), we define

###
**Lemma 2**

*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**

*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**

*f*is a turbulent function if there are disjoint nonempty closed subsets \(J,\ K\) of

*X*such that:

###
**Definition 2**

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

- 2)
\(A \cup f(A) \subseteq f^{2}(A)\).

###
**Definition 3**

*f*is chaotic in the sense of Block and Coppel (in short: B-C 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 B-C chaos.

###
**Definition 4**

*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 Román-Flores and Chalco-Cano (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.

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,*

*f*. These conclusions can be listed as follows:

*f*and \(\widehat{f}\). We generalizes the results with \(\widehat{f_{g}}\). As the Zadeh’s extension is a spacial case of the g-fuzzification, we can make a conclusion:

###
**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 \(U\cup V \subseteq f(U)\cap f(V)\).

*e*(

*U*), clearly,

*e*(

*U*) and

*e*(

*V*) are two disjoint nonempty closed subsets of \({{\mathbb {F}}}(X)\). Applying the Lemma 2, we have

###
**Theorem 4**

*Let*
\(f\in C(X)\)
*be erratic, then,*
\(\widehat{f}_{g}\)
*is a erratic function*.

###
*Proof*

*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

###
**Corollary 1**

*Let*
\(f\in C(X)\)
*be erratic, then*, \(\widehat{f}_{g}\)
*is a B-C 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*

*f*is \(\lambda\)-expansive, for any \(u,v\in {{\mathbb {F}}}(X)\) and \(\alpha \in (0,1]\), it follows 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*

###
**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 Román-Flores 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, Li-Yorke sensitive and spatial-temporal sensitive etc, which will be further investigated and solved in a later work.

## Declarations

### 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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## Authors’ Affiliations

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