# Martelli Chaotic Properties of a Generalized Form of  Zadeh’s Extension Principle

Authors: Yaoyao Lan; Chunlai Mu
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

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


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## Body

## 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 [1–7] 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 [8–12]. 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 f and f^ 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 g-fuzzification, which allows us to modify the membership grades of points in each iteration, has been introduced [19]. The use of g-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 asold 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 ℝn, 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 Γ-fuzzification has been developed in [18], which does not degenerate under chaotic iteration. Now, as a new generalized extension principle, g-fuzzification includes the usual fuzzification (Zadeh’s extension) and Γ-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 [22–24].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 andg-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 g-fuzzified systems. A brief conclusion concludes the paper.


## 2. Preliminaries

In this section, we complete notations and recall some known definitions. Letf:X→X be a continuous map acting on a compact metric space (X,d). An orbit of a point x0∈X is the set {fn(x0):n≥0}, denoted by orb(x0,f) or simply orb(x0) when the function f is clearly specified. A point y is alimit point of orb(x0) if a subsequence of orb(x0) converges to y. The set of limit points of orb(x0) is denoted by L(x0).We say thatf istransitive if for any pair of nonempty open sets U and V there exists n≥1 such that fn(U)∩V≠∅; f ispoint transitive if there exists a point x0∈X such that the orbit of x0 is dense in X; that is, orb(x0)¯=X, and x0 is called atransitive point of X.We say thatorb(x) isunstable if there exists δx such that, for any neighborhood U of x, there exist y∈U and n≥0 such that d(fn(x),fn(y))>δx. An orbit which is not unstable is said to bestable.We say thatf hassensitive dependence on initial conditions if there is a constant δ>0 such that for every point x and every neighborhood U about x there are a y∈U and a k≥1 such that d(fk(x),fk(y))≥δ. Hence, every orbit orb(x) with x∈X is unstable with the same constant δ. Consequently, sensitive dependence on initial conditions is stronger than instability.Definition 1 (see [25]).
Let(X,d) be a compact metric space and let f:X→X be continuous. Then, f is Martelli chaotic provided that there exists x0∈X such that(i)
L(x0)=X;(ii)
orb(x0) is unstable.In this research, we call aMartelli chaotic mapM-chaotic for short.Below, we present some definitions from fuzzy theory. Let𝒦(X) be the class of all nonempty and compact subsets of X. If A∈𝒦(X), we define the ɛ-neighbourhood of A as the set
(1)N(A,ɛ)={x∈X∣d(x,A)<ɛ},
where d(x,A)=inf⁡a∈A∥x-a∥.The Hausdorff separationρ(A,B) of A,B∈𝒦(X) is defined by
(2)ρ(A,B)=inf⁡{ɛ>0∣A⊆N(B,ɛ)}.The Hausdorff metric on𝒦(X) is defined by letting
(3)H(A,B)=max⁡{ρ(A,B),ρ(B,A)}.Defineℱ(X) as the class of all upper semicontinuous fuzzy sets u:X→[0,1] such that [u]α∈𝒦(X), where α-cuts and thesupport of u are defined by
(4)ii[u]α={x∈X∣u(x)≥α},α∈[0,1],supp⁡(u)={x∈X∣u(x)>0}¯,
respectively.Moreover, letℱ0(X) denote the space of all nonempty fuzzy sets on X and let ∅X denote theempty fuzzy set (∅X(x)=0 for all x∈X).Alevel-wise metricd∞ on ℱ(X) is defined by
(5)d∞(u,v)=sup⁡α∈[0,1]H([u]α,[v]α)
for all u,v∈ℱ(X). It is well known that if (X,d) is complete, then (ℱ(X),d∞) is also complete but is not compact and is not separable (see [19, 28, 29]).Lemma 2 (see [9, 14]).
LetA be an open subset of X. Define e(A)={u∈ℱ(X):[u]0⊆A}, and then e(A) is an open subset of ℱ(X).Letf:X→X be continuous. A usual fuzzification (often called Zadeh’s extension) f^:𝔽(X)→𝔽(X) is defined by
(6)f^(u)(x)=sup⁡y∈f-1(x)u(y)
for any u∈𝔽(X) and x∈X.Now let us introduceg-fuzzification. Denote Dm(I) as the set of all nondecreasing right continuous functions g:I→I for which g(x)=x if x=0 and x=1. Let Cm(I) be the set of all continuous maps from Dm(I). For any g∈Dm(I), a g-fuzzification f^g:𝔽(X)→𝔽(X) is defined by
(7)f^g(u)(x)=sup⁡y∈f-1(x){g(u(y))}sforanyu∈𝔽(X),x∈X.
An α-cut [u]αg of a fuzzy set u∈𝔽(X) with respect to g∈Dm(I) is
(8)[u]αg={x∈supp⁡(u)∣g(u(x))≥α}forα∈(0,1].Lemma 3 (see [19]).
Letf:X→X be continuous and let f^g be g-fuzzification. Then,
(9)f([u]αg)=[f^g(u)]α
holds for any u∈𝔽0(X), g∈Dm(I), and α∈(0,1].Lemma 4 (see [19]).
Letg∈Dm(I), u∈𝔽0(X), and α∈(0,1]. If [u]αg≠∞. then there is c∈(0,1] such that [u]αg=[u]c.


## 3.M-Chaotic Relations between f^g and f

In this section, we study the relations between Martelli’s chaotic properties of the original system(X,f) and g-fuzzified system (𝔽(X),f^g), where 𝔽(X) is equipped with the level-wise topology, that is, the metric topology induced by d∞.On the one hand, some conditions are discussed, under whichf^gM-chaotic implies fM-chaotic. On the other hand, several examples are presented to illustrate that, in general, fM-chaotic does not imply f^gM-chaotic.Proposition 5.
Define[u]0g={x∈[u]0∣g(u(x))>0}. Then
(10)[u]0g=[u]0
holds for every u∈𝔽0(X) and g∈Dm(I).Proof.
The inclusion[u]0g⊆[u]0 follows directly from the definition of [u]0g. If x∈[u]0, then, because g is nondecreasing, we have g(u(x))>0, which implies x∈[u]0g, and, consequently, [u]0g⊇[u]0 holds.Proposition 6.
LetU be subset of X and let f:X→X be continuous. Then, f^g(e(U))⊆e(f(U)).Proof.
Ifu∈f^g(e(U)), then there exists u*∈e(U) such that u=f^g(u*). Hence, due to Lemma 3 and Proposition 5, we have that [u]0=[f^g(u*)]0=f([u*]0g)=f([u*]0), since [u*]0⊆U and [u]0=f([u*]0)⊆f(U); thus u∈e(f(U)), and the inclusion follows.Proposition 7.
Letu∈𝔽0(X), g∈Dm(I), and αi∈(0,1], i=1,…,n,…. Then, there exists an αn∈(0,1] such that [f^gn(u)]αn=[fgn^(u)]α2n-1=fn([u]α2n).Proof.
We do the proof by mathematical induction.
Whenn=1, by Lemmas 3 and 4, the formula gives us [f^g(u)]α1=f([u]α1g)=f([u]α2); therefore, the statement holds for n=1.
Assume that the statement is true forn=k; that is,
(11)[f^gk(u)]αk=[fgk^(u)]α2k-1=fk([u]α2k).
Note that [f^gk(u)]αk+2=fk([u]α2k+2).
Whenn=k+1,
(12)[f^gk+1(u)]αk+1=[f^gf^gk(u)]αk+1=f([f^gk(u)]αk+1g)=f([f^gk(u)]αk+2)=f(fk([u]α2k+2))=fk+1([u]α2(k+1)).
On the other hand, fk+1([u]α2(k+1))=fk+1([u]α2(k+1)-1g)=[fgk+1^(u)]α2k+1.
This completes the proof.Remark 8.
The proof of Proposition7 can also be done as follows.
Due to Lemmas3 and 4, we have that
(13)[f^gn(u)]αn=[f^gf^gn-1(u)]αn=f([f^gn-1(u)]αng)=f([f^gn-1(u)]αn+1)=f([f^gf^gn-2(u)]αn+1)=f2([f^gn-2(u)]αn+1g)=f2([f^gn-2(u)]αn+2)i⋮=fn([u]α2n-1g)=fn([u]α2n).
On the other hand, by Lemma 3 again, we obtain fn([u]α2n-1g)=[fgn^(u)]α2n-1, and, consequently, [f^gn(u)]αn=[fgn^(u)]α2n-1=fn([u]α2n) holds.Theorem 9.
Letu0 be a transitive point of (𝔽(X),f^g). Then, every x∈[u0]αg is a transitive point of (X,f) for α∈(0,1].Proof.
Sinceu0 is a transitive point of (𝔽(X),f^g), there exists k∈ℕ such that d∞(f^gk(u0),v)<ɛ for any ν∈𝔽(X) and ɛ>0. By using Proposition 7 and Lemma 3, we obtain
(14)d∞(f^gk(u0),v)=sup⁡β∈[0,1]H([f^gk(u0)]β,[ν]β)=sup⁡α,β∈(0,1]H([fgk^(u0)]α,[ν]β)=sup⁡α,β∈(0,1]H(fk[u0]αg,[ν]β)<ɛ,
for some α,β∈[0,1]. Hence, for each y∈[ν]β, there exists x∈[u0]αg such that d(x,y)<ɛ, which means that every x∈[u0]αg is a transitive point of (X,f).Theorem 10.
IfL(u0)=𝔽(X), then there exists x0∈[u0]αg such that L(x0)=X.Proof.
It follows directly from Theorem9.The following example shows that, in general, the converse of Theorem10 is not true.Example 11 (irrational rotation of circle).
Letλ be an irrational number and Rλ:S1→S1 is defined by Rλ(eiθ)=ei(θ+2πλ). It is well known that, for each z∈S1, the orbit of z is dense in S1 and, consequently, L(z)=S1. Nevertheless, it is not necessary for some ν∈𝔽(S1) to exist such that L(ν)=𝔽(S1). In fact, assume that u∈𝔽(S1) and diam⁡([u]0g)=1. Given that 0<ɛ<1/2, let U=B(1^,ɛ/2) and V=B(u,ɛ/2), and by Proposition 5, we obtain
(15)ω∈U=B(1^,ɛ2)⟹diam⁡([ω]0g)=diam⁡([ω]0)≤ɛ2,ν∈V=B(u,ɛ2)⟹diam⁡([ν]0g)=diam⁡([ν]0)≥1-ɛ,
since
(16)diam⁡([R^λgn(v)]0)=diam⁡(Rλn[v]0)≥1-ε
for n∈ℕ. Hence, U∩R^λgn(V)=∅, which means that there exists no ν∈V such that R^λgn(ν)=ω for some ω∈𝔽(X), and, consequently, L(ν)≠𝔽(X).Theorem 12.
Letf:X→X be continuous, let f^g be the g-extension of f, and let u0∈𝔽(X). If the orbit of u0 is unstable in 𝔽(X), then there exists x0∈[u0]βg such that the orbit of x0 is unstable in X, where β∈[0,1].Proof.
Let the assumptions be satisfied. Then, there existsδu0 such that for every ϵ>0 we can find ν∈𝔽(X) and k∈ℕ satisfying ν∈B(u0,ɛ) and
(17)d∞(f^gk(u0),f^gk(ν))=sup⁡α∈[0,1]H([f^gk(u0)]α,[f^gk(ν)]α)=sup⁡β,γ∈[0,1]H([fgk^(u0)]β,[fgk^(ν)]γ)=sup⁡β,γ∈[0,1]H(fk[u0]βg,fk[ν]γg)>δu0.
Thus, there exist x0∈[u0]βg and y0∈[ν]γg such that d(x0,y0)>δu0. Since ν∈B(u0,ɛ), we have d(x0,y0)<ɛ. This proves that there exists x0∈[u0]βg such that the orbit of x0 is unstable in X with instable constant δu0.Example 13.
Consider the foregoing example (Example11); because Rλ is isometric, it does not exhibit sensitive dependence on initial conditions and hence the orbit of each z∈S1 is stable, which implies that, by Theorem 12, there exists no orbit of u∈𝔽(S1) that is unstable.By combining Theorems9, 10, and 12, we obtain the following theorem.Theorem 14.
Iff^g is M-chaotic, then f is M-chaotic.We will need some notions from Denjoy map [13]. Recall that the circle S1 can be considered as the quotient space ℝ/ℤ, where ℝ and ℤ are the sets of real numbers and integers, respectively. The irrational rotation of the circle Rλ:S1→S1 is then given by
(18)Rλ(x)=x+λ(mod⁡1),
where λ is irrational. Recall that a Denjoy map can be constructed as follows. Take any point x0∈S1. We cut out each point Rλn(x0) on the orbit of x0 and replace it with a small interval In. For n∈ℕ,(a)
ℒ(I0)=1/4, ℒ(In+1)<ℒ(In), ℒ(In)=ℒ(I-n), and ∑n∈ℤ‍ℒ(In)=1, where ℒ(In) denotes the length of the interval In;(b)
lim⁡n→∞(ℒ(In+1)/ℒ(In))=1.Consequently, a new circleS* has been constructed. The Denjoy homeomorphism Dλ:S*→S* is an orientation preserving homeomorphism of S*. There exists a Cantor set Cλ⊂S* on which Dλ acts minimally. It is known that there exists a continuous surjection hλ:S*→S1 that semiconjugates Dλ with Rλ. In [30], the authors show that the system (𝕂(Cλ),Dλ) is not sensitive.Proposition 15.
Letx∈Cλ; then the orbit of x is unstable in (Cλ,Dλ) with the constant 1/4.Proof.
Suppose thaty∈B(x,ɛ) and hλ(x)≠hλ(y) for any ɛ>0. Since the orbit of x0 is dense in S1, there exist some k∈ℕ such that R-λk(x0)∈[hλ(x),hλ(y)], where [hλ(x),hλ(y)] is the closed arc in S1. Thus, we have x0∈Rλk([hλ(x),hλ(y)]). Consequently, due to the construction of Denjoy map, we obtain I0⊂[Dλk(x),Dλk(y)], which means that d(Dλk(x),Dλk(y))>1/4.The following proposition shows that the instability of the orbit in(Cλ,Dλ) cannot be inherited by its g-fuzzification. More specifically, there exist points arbitrarily close to u∈𝔽(Cλ) which eventually also close to u under iteration of Dλ^g, although there exist some x∈[u]0 such that the orbits of these points are unstable in (Cλ,Dλ). It should be mentioned that our approach was inspired by the idea in [8] where a continuous map iλ was defined.Defineiλ:𝕂(Cλ)→𝔽(Cλ) by iλ(K)=λχK for any K∈𝕂(Cλ) and any λ∈(0,1], where χK is the characteristic function of K (that is to say, χK(x)=1 if x∈K and χK(x)=0 if x∉K). Hence, iλ∘D¯λ=Dλ^g∘iλ. Note that iλ is continuous.Proposition 16.
Letu∈𝔽(Cλ); then there exist some ν∈𝔽(Cλ) and n>0 such that d∞(Dλ^gn(u),Dλ^gn(v))<ɛ.Proof.
Since(𝕂(Cλ),D¯λ) is not sensitive, for ɛ>0 and δ>0, there exist M∈𝕂(Cλ) and B(M,δ) such that, for all N∈B(M,δ),
(19)H(D¯λn(M),D¯λn(N))<ɛ.
Suppose that u∈e(M) (recall that e(M)={u∈𝔽(Cλ)∣[u]0⊆M}), and by continuity of iλ and (19), we have
(20)H(D¯λn([u]0),D¯λn(N))<ɛ⟹H(iλ∘D¯λn([u]0),iλ∘D¯λn(N))<ɛ⟹H(Dλ^gn∘iλ([u]0),Dλ^gn∘iλ(N))=d∞(Dλ^gn(u),Dλ^gn(ν))<ɛ.
Without loss of generality, assume thatν=iλ(N)∈𝔽(Cλ). This completes the proof.Remark 17.
Theorem9 together with Theorem 10 shows that f^gM-chaotic implies fM-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 andg-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)
IfL(u0)=𝔽(X), then there exists x0∈[u0]αg such that L(x0)=X (Theorem 10).(b)
The instability oforb(u,f^g) implies the instability of orb(x,f), where u∈𝔽(X), x∈[u]βg, and β∈[0,1] (Theorem 12).(c)
f^gM-chaotic implies fM-chaotic (Theorem 14).(d)
fM-chaotic does not imply f^gM-chaotic (Example 11 and Proposition 16).It is worth noting that anyg-fuzzification is connected to a crisp discrete dynamical system in two different ways [19]. One way is to connect two systems via α-cut, and another approach is to consider g-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 g-fuzzified dynamical systems, and this will be one aspect of our future works.



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