# Chaos for Discrete Dynamical System

Authors: Lidong Wang; Heng Liu; Yuelin Gao
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We prove that a dynamical system is chaotic in the sense of Martelli and Wiggins, when it is a transitive distributively chaotic in a sequence. Then, we give a sufficient condition for the dynamical system to be chaotic in the strong sense of Li-Yorke. We also prove that a dynamical system is distributively chaotic in a sequence, when it is chaotic in the strong sense of Li-Yorke.


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## Body

## 1. Introduction

Since Li and Yorke first gave the definition of chaos by using strict mathematical language in 1975 [1], the research on chaos has greatly influenced modern science, not just natural sciences but also several social sciences, such as economics, sociology, and philosophy. The theory of chaos convinced scientists that a simple definite system can produce complicated features and a complex system instead possibly follows a simple law. However, scientists in different fields, finding different chaotic connotations, gave different definitions of chaos such as Li-Yorke chaos, distributional chaos, and Devaney chaos. In order to establish a satisfactory definitional and terminological framework for complex dynamical systems that are based on strict mathematical definitions, these concepts with less ambiguous are necessary, and their interdependence has to be clarified. There is no doubt that the mathematical definition of Li-Yorke chaos has a large influence than any other one, whereas distributional chaos possesses some statistical connotations besides the uncertainty of long-term behaviors. So, comparing distributional chaos with Li-Yorke chaos is a meaningful and significant problem.In order to reveal the inner relations between Li-Yorke chaos and distributional chaos, the author brought up the definition of distributional chaos in a sequence in [2]. In this paper, we mainly prove the relations between some different chaoses in discrete dynamical systems.The main theorems are stated as follows.Theorem 1.
If a dynamical system(X,f) exhibits transitive distributional chaos in a sequence, then,(1)
it is chaotic in the sense of Martelli;(2)
it is chaotic in the sense of Wiggins.Theorem 2.
Leta,b∈X with a≠b and pk→∞ be a sequence of positive integers. If for any sequence C=C1⋯Ck⋯ where CK=B(a,1/k)¯ or B(b,1/k)¯(B(a,1/k)={x∣d(a,x)<(1/k)}) there exists x(C)∈X, such that for each k≥1,fpk(x(C))∈Ck then system (X,f) is chaotic in the strong sense of Li-Yorke.Theorem 3.
If a dynamical system(X,f) exhibits chaotic in the strong sense of Li-Yorke, then it is distributively chaotic in a sequence.


## 2. Problem Statement and Preliminaries

Throughout this paperX will denote a compact metric space with metric d.


### 2.1. Several Definitions and Lemmas

D
⊂
X is said to be a chaotic set of f if for any pair (x,y)∈D×D, x≠y, we have
(1)limn→∞infd(fn(x),fn(y))=0,limn→∞supd(fn(x),fn(y))>0.Definition 4.
f is said to be chaotic in the sense of Li and Yorke (for short: Li-Yorke chaotic), if it has a chaotic set D which is uncountable.Let{pi} be an increasing sequence of positive integers, x,y∈X,t>0. Let
(2)Fxy(t,{pi})=limn→∞inf1n∑k=1n‍χ[0,t)(d(fpk(x),fpk(y))),Fxy*(t,{pi})=limn→∞sup1n∑k=1n‍χ[0,t)(d(fpk(x),fpk(y))),
where χA(y) is 1 if y∈A and 0 otherwise. Obviously, Fxy and Fxy* are both nondecreasing functions. If for t≤0 we define Fxy(t)=Fxy*(t)=0, then Fxy and Fxy* are probability distributional functions.Definition 5.
LetD⊂X, If ∀x,y∈D,x≠y, we have
(3)(1)∃δ>0,Fxy=(δ,{pi})=0,(2)∀t>0,Fxy*(t,{pi})=1,
thenD is said to be a distributively chaotic set in a sequence. The two points are said to be distributively chaotic point pair in a sequence. f is said to be distributively chaotic in a sequence, if f has a distributively chaotic set in a sequence which is uncountable.Definition 6.
LetS⊂X. If there exist two strictly increasing sequences of positive integers {pi} and {qi}  such that for any x,y∈S, x≠y,
(4)limi→∞d(fpi(x),fpi(y))=0,limi→∞d(fqi(x),fqi(y))>0,
then S is said to be a strong scrambled set. f is said to be chaotic in the strong sense of Li-Yorke, if f has an uncountable strong scrambled set.Definition 7.
Let{pi} be an increasing sequence of positive integers, then,
(5)PR(f,{pi})={(x,y)∈X×X∣∀ε>0,∃i∈Nsuchthatd(fpi(x),fpi(y))<εX×X}
is called proximal relation with respect to {pi}.
Thus(6)AR(f,{pi})={(x,y)∈X×X∣limi→∞d(fpi(x),fpi(y))=0}
is called asymptotic relation with respect to {pi}.
Thus(7)DR(f,{pi})=X×X-PR(f,{pi})
is called distal relation with respect to {pi}.
So(8)DCR(f,{pi})={(x,y)∈X×X∣(x,y)isadistributivelychaoticpointpairoffinpiX×X}
is called distributively chaotic respect to {pi}.Definition 8 (see [3]).
f is (topologically) transitive if for any two nonempty open sets U,V⊂X there exists n>0 such that fn(U)∩V≠∅. f is (topologically) weakly mixing if for any three nonempty open sets U,V,W⊂X there exists n>0 such that fn(W)∩V≠∅ and fn(W)∩U≠∅.Definition 9 (see [4–6]).
Letf be a continuous map from a compact metric space (X,d) into itself. The orbit of a point x∈X is said to be unstable if there exists r>0 such that for every ϵ>0 there are y∈X and n≥1 satisfying inequalities d(x,y)<ϵ and d(fn(x),fn(y))>r. The map f is said to be chaotic in the sense of Martelli if there exists x0∈X such that x0 has dense orbit which is unstable.Definition 10 (see [7]).
Letf be a continuous map from a compact metric space (X,d) into itself. We say f has sensitive dependence on initial conditions if there exists r>0 such that for any x∈X and ϵ>0, there is some y∈X and a nonnegative integer n satisfying d(x,y)<ϵ and d(fn(x),fn(y))>r. f is said to be chaotic in the sense of Wiggins, if f is transitive and has sensitive dependence on initial conditions.Definition 11.
Let(X,d) be a compact metric space, f:X→X be a continuous map, and D be an uncountable distributively scrambled set in a sequence.We say thatf exhibits dense distributional chaos in a sequence if the set D may be chosen to be dense. If D is not only dense but additionally consists of points with dense orbits, then we say that f exhibits transitive distributional chaos in a sequence.Lemma 12.
LetΣ be an infinite sequence set of {0,1}. Then, there exists an uncountable subset E⊂Σ such that for any different points s=s1s2,…,t=t1t2,…,sm≠tm for infinitely many m and sn=tn for infinitely many n.Proof.
For a proof, see [8].Lemma 13.
If{pi} and {qi} are both infinite increasing subsequences of {mi} which is a sequence of positive integers, then there exists an infinite increasing subsequence {ti}⊂{mi} such that
(9)AR(f,{pi})∩DR(f,{qi})⊂DCR(f,{ti}).Proof.
For a proof, see [9].Lemma 14.
f is weakly mixing if and only if for any m≥2,fm is transitive.Proof.
For a proof, see [10].


## 2.1. Several Definitions and Lemmas

D
⊂
X is said to be a chaotic set of f if for any pair (x,y)∈D×D, x≠y, we have
(1)limn→∞infd(fn(x),fn(y))=0,limn→∞supd(fn(x),fn(y))>0.Definition 4.
f is said to be chaotic in the sense of Li and Yorke (for short: Li-Yorke chaotic), if it has a chaotic set D which is uncountable.Let{pi} be an increasing sequence of positive integers, x,y∈X,t>0. Let
(2)Fxy(t,{pi})=limn→∞inf1n∑k=1n‍χ[0,t)(d(fpk(x),fpk(y))),Fxy*(t,{pi})=limn→∞sup1n∑k=1n‍χ[0,t)(d(fpk(x),fpk(y))),
where χA(y) is 1 if y∈A and 0 otherwise. Obviously, Fxy and Fxy* are both nondecreasing functions. If for t≤0 we define Fxy(t)=Fxy*(t)=0, then Fxy and Fxy* are probability distributional functions.Definition 5.
LetD⊂X, If ∀x,y∈D,x≠y, we have
(3)(1)∃δ>0,Fxy=(δ,{pi})=0,(2)∀t>0,Fxy*(t,{pi})=1,
thenD is said to be a distributively chaotic set in a sequence. The two points are said to be distributively chaotic point pair in a sequence. f is said to be distributively chaotic in a sequence, if f has a distributively chaotic set in a sequence which is uncountable.Definition 6.
LetS⊂X. If there exist two strictly increasing sequences of positive integers {pi} and {qi}  such that for any x,y∈S, x≠y,
(4)limi→∞d(fpi(x),fpi(y))=0,limi→∞d(fqi(x),fqi(y))>0,
then S is said to be a strong scrambled set. f is said to be chaotic in the strong sense of Li-Yorke, if f has an uncountable strong scrambled set.Definition 7.
Let{pi} be an increasing sequence of positive integers, then,
(5)PR(f,{pi})={(x,y)∈X×X∣∀ε>0,∃i∈Nsuchthatd(fpi(x),fpi(y))<εX×X}
is called proximal relation with respect to {pi}.
Thus(6)AR(f,{pi})={(x,y)∈X×X∣limi→∞d(fpi(x),fpi(y))=0}
is called asymptotic relation with respect to {pi}.
Thus(7)DR(f,{pi})=X×X-PR(f,{pi})
is called distal relation with respect to {pi}.
So(8)DCR(f,{pi})={(x,y)∈X×X∣(x,y)isadistributivelychaoticpointpairoffinpiX×X}
is called distributively chaotic respect to {pi}.Definition 8 (see [3]).
f is (topologically) transitive if for any two nonempty open sets U,V⊂X there exists n>0 such that fn(U)∩V≠∅. f is (topologically) weakly mixing if for any three nonempty open sets U,V,W⊂X there exists n>0 such that fn(W)∩V≠∅ and fn(W)∩U≠∅.Definition 9 (see [4–6]).
Letf be a continuous map from a compact metric space (X,d) into itself. The orbit of a point x∈X is said to be unstable if there exists r>0 such that for every ϵ>0 there are y∈X and n≥1 satisfying inequalities d(x,y)<ϵ and d(fn(x),fn(y))>r. The map f is said to be chaotic in the sense of Martelli if there exists x0∈X such that x0 has dense orbit which is unstable.Definition 10 (see [7]).
Letf be a continuous map from a compact metric space (X,d) into itself. We say f has sensitive dependence on initial conditions if there exists r>0 such that for any x∈X and ϵ>0, there is some y∈X and a nonnegative integer n satisfying d(x,y)<ϵ and d(fn(x),fn(y))>r. f is said to be chaotic in the sense of Wiggins, if f is transitive and has sensitive dependence on initial conditions.Definition 11.
Let(X,d) be a compact metric space, f:X→X be a continuous map, and D be an uncountable distributively scrambled set in a sequence.We say thatf exhibits dense distributional chaos in a sequence if the set D may be chosen to be dense. If D is not only dense but additionally consists of points with dense orbits, then we say that f exhibits transitive distributional chaos in a sequence.Lemma 12.
LetΣ be an infinite sequence set of {0,1}. Then, there exists an uncountable subset E⊂Σ such that for any different points s=s1s2,…,t=t1t2,…,sm≠tm for infinitely many m and sn=tn for infinitely many n.Proof.
For a proof, see [8].Lemma 13.
If{pi} and {qi} are both infinite increasing subsequences of {mi} which is a sequence of positive integers, then there exists an infinite increasing subsequence {ti}⊂{mi} such that
(9)AR(f,{pi})∩DR(f,{qi})⊂DCR(f,{ti}).Proof.
For a proof, see [9].Lemma 14.
f is weakly mixing if and only if for any m≥2,fm is transitive.Proof.
For a proof, see [10].


## 3. Proof of Main Theorem

Proof of Theorem1.
(1) There is no isolated points inX as otherwise the set of points with dense orbit is at most countable. But in the case of compact set without isolated points, the existence of dense orbit implies transitivity.
LetD be a dense scrambled set in the sequence {pk} consisting of transitive points, and let r>0 be such that Fxy(r,{pk})=0 for all distinct x,y∈D. Let us fix any x0∈D. Because orbit of x0 is dense, for any ϵ>0, there exists y∈D and k≥1 satisfying the inequalities d(x0,y)<ϵ and Fx0y(r,{pk})=0. This implies that d(fpk(x0),fpk(y))>r for some k≥1. This shows that the orbit of x0 is unstable. So, (X,f) is chaotic in the sense of Martelli.
(2) Fix anyϵ>0. In ϵ-neighborhood of any point x, we can find points y,z∈D such that d(fpk(y),fpk(z))>r. Then, d(fpk(x),fpk(y))>r/2 or d(fpk(x),fpk(z))>r/2.Proof of Theorem2.
LetE be an uncountable subset of Σ, as in Lemma 12. For each s=s0s1⋯sn⋯∈E, by the hypotheses, we can choose a point x(s)∈X such that for any k, if n!<k≤(n+1)! then,
(10)xpks∈{B(a,1k)¯ifsn=0,B(b,1k)¯ifsn=1.
Put D={xs∣s∈E}. Clearly, if s≠t then xs≠xt. It follows that E being uncountable implies so is D.
Letxs,xt∈D be any different points, where s=s0s1⋯si⋯,t=t0t1⋯ti⋯∈E. By the property of E, we know that there exist sequences of positive integers mi→∞,ni→∞ such that smi≠tmi,sni=tni for all i, and for mi large enough 1/m<d(a,b)/4=δ, we have d(xmis,xmit)>δ. Thus,
(11)limi→∞d(xmis,xmit)≥δ,
this shows
(12)limi→∞supd(xmis,xmit)≥δ.
Meanwhile, for ni large enough, xnis and xnit lie in the same ball of diameter less than 1/ni. Thus, d(xnis,xnit)<1/n, so
(13)limi→∞d(xnis,xnit)=0.
This shows
(14)limi→∞infd(xnis,xnit)=0.
Above all,(X,f) is chaotic in the strong sense of Li-Yorke.Proof of Theorem3.
Becausef is chaotic in the strong sense of Li-Yorke, there exists an infinite increasing sequence {pi}⊂ℕ and uncountable set S⊂X, such that for any x,y∈S with x≠y, we have
(15)limi→∞d(fpi(x),fpi(y))=0,
so that (x,y)∈AR(f,{pi}).
Again, by the definition of chaos in the strong sense of Li-Yorke, there exists{qi}⊂ℕ, such that
(16)limi→∞d(fqi(x),fqi(y))>0,
so that (x,y)∈DR(f,{qi}).
Hence,S×S-Δ⊂AR(f,{pi})∩DR(f,{qi}), where Δ={(x,x);x∈X}. Then by Lemma 13, there exists a subsequence {ti}⊂ℕ such that S×S⊂DCR(f,{ti}). This shows that S is a distributively chaotic set in the sequence {ti} of f.Corollary 15.
If system(X,f) satisfies conditions of Theorem 2. then it is distributively chaotic in the sequence.Proof.
By Theorems2 and 3, we can easily prove it.Corollary 16.
Let(X,d) be a locally compact metric space containing at least two points. If system (X,f) is weakly mixing, then it must be chaotic in the strong sense of Li-Yorke.Proof.
Letf be weakly mixing, a,b∈X with a≠b. Take arbitrarily a nonempty open set V0⊂X such that V-0 is compact. Since f is weakly mixing, there exists p1>0 such that fp1(V0)∩B(a,1/k)≠∅ and fp1(V0)∩B(b,1/k)≠∅. Thus, we find points x1,x2 such that fp1(x1)∈B(a,1/k),fp1(x2)∈B(b,1/k). Assume that there exist positive integers p1<p2<⋯<pk such that for each finite sequence A1A2⋯Ak, where Ai∈{B(a,1/i),B(b,1/i)}, there is a point x∈V0 satisfying fpi(x)∈Ai for i=1,2,…,k, the set of all such points will denoted by Sk. By continuity of f, each x∈Sk has an open nonempty neighborhood Wx⊂V0 such that fpi(Wx)⊂Ai, if fpi(x)∈Ai, it follows from Lemma 14 that there exists pk+1>pk such that for each x∈Sk,fpk+1(Wx)∩B(a,1/(k+1))≠∅ and fpk+1(Wx)∩B(b,1/(k+1))≠∅. Thus by induction, we know that there exists a sequence pk→∞ of positive integers such that for any finite sequence A1⋯Ak, there is a point x∈V0 satisfying fpi(x)∈Ai,1≤i≤k.
LetC=C1C2⋯ be an infinite sequence, where
(17)Ck∈{B(a,1k)¯,B(b,1k)¯}.
For each k, we can take a point xk∈Ck such that
(18)fpi(xk)∈Ci,1≤i≤k.
Since V-0 is compact, the infinite sequence {xi} has a limit point in V-0, say xC, it is not difficult to show fpk(xC)⊂Ck.
Thus by Theorem2, f is a strong chaos in the sense of Li-Yorke.



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*Source: 212036-2013-03-28.xml*