- Research
- Open Access
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-Fuzzification
- Turbulent
- Erratic
- λ-Expansive
Background
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
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
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 \([A]^{g}_{\alpha }=[A]_{c}\).
Lemma 2
- 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
- 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
Definition 2
- 1)
\(A \cap f(A) = \emptyset\).
- 2)
\(A \cup f(A) \subseteq f^{2}(A)\).
Definition 3
It should be remarkable that the erratic property is stronger than B-C chaos.
Definition 4
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
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
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 \(U\cup V \subseteq f(U)\cap f(V)\).
Theorem 4
Let \(f\in C(X)\) be erratic, then, \(\widehat{f}_{g}\) is a erratic function.
Proof
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
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.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Cánovas JS, Kupka J (2011) Topological entropy of fuzzified dynamical systems. Fuzzy Sets Syst 165(1):37–49View ArticleGoogle Scholar
- Diamond P (1994) Chaos in iterated fuzzy systems. J Math Anal Appl 184(3):472–484View ArticleGoogle Scholar
- Diamond P, Kloeden PE (1994) Metric spaces of fuzzy sets: theory and applications. World scientific, SingaporeGoogle Scholar
- Diamond P, Pokrovskii A (1994) Chaos, entropy and a generalized extension principle. Fuzzy Sets Syst 61(3):277–283View ArticleGoogle Scholar
- Kupka J (2011a) On fuzzifications of discrete dynamical systems. Inf Sci 181(13):2858–2872View ArticleGoogle Scholar
- Kupka J (2011b) On devaney chaotic induced fuzzy and set-valued dynamical systems. Fuzzy Sets Syst 177(1):34–44View ArticleGoogle Scholar
- Kupka J (2014) Some chaotic and mixing properties of fuzzified dynamical systems. Inf Sci 279:642–653View ArticleGoogle Scholar
- Lan Y, Mu C (2014) Martelli chaotic properties of a generalized form of Zadeh’s extension principle. J Appl Math 2014. doi:10.1155/2014/956467
- Román-Flores H, Chalco-Cano Y, Silva GN, Kupka J (2011) On turbulent, erratic and other dynamical properties of Zadeh’s extensions. Chaos Solitons Fractals 44(11):990–994View ArticleGoogle Scholar
- Román-Flores H, Chalco-Cano Y (2005) Robinson’s chaos in set-valued discrete systems. Chaos Solitons Fractals 25(1):33–42View ArticleGoogle Scholar