Chaos for Discrete Dynamical System

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.

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. Let 𝑎,𝑏∈𝑋 with 𝑎≠𝑏 and 𝑝_𝑘→∞ be a sequence of positive integers. If for any sequence 𝐶=𝐶₁⋯𝐶_𝑘⋯ where 𝐶_𝐾=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,𝑓^𝑝𝑘(𝑥(𝐶))∈𝐶_𝑘 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 paper 𝑋 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 lim_𝑛→∞inf𝑑𝑓^𝑛(𝑥),𝑓^𝑛𝑦=0,lim_𝑛→∞sup𝑑𝑓^𝑛(𝑥),𝑓^𝑛𝑦>0. (1)

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 {𝑝_𝑖} be an increasing sequence of positive integers, 𝑥,𝑦∈𝑋,𝑡>0. Let 𝐹_𝑥𝑦𝑡,𝑝_𝑖=lim_𝑛→∞1inf𝑛^𝑛_𝑘=1𝜒_[0,𝑡)𝑑𝑓^𝑝𝑘(𝑥),𝑓^𝑝𝑘𝑦,𝐹^∗_𝑥𝑦𝑡,𝑝_𝑖=lim_𝑛→∞1sup𝑛^𝑛_𝑘=1𝜒_[0,𝑡)𝑑𝑓^𝑝𝑘(𝑥),𝑓^𝑝𝑘𝑦, (2) where 𝜒_𝐴(𝑦) is 1 if 𝑦∈𝐴 and 0 otherwise. Obviously, 𝐹_𝑥𝑦 and 𝐹^∗_𝑥𝑦 are both nondecreasing functions. If for 𝑡≤0 we define 𝐹_𝑥𝑦(𝑡)=𝐹^∗_𝑥𝑦(𝑡)=0, then 𝐹_𝑥𝑦 and 𝐹^∗_𝑥𝑦 are probability distributional functions.

Definition 5. Let 𝐷⊂𝑋, If ∀𝑥,𝑦∈𝐷,𝑥≠𝑦, we have (1)∃𝛿>0,𝐹_𝑥𝑦=𝛿,𝑝_𝑖(2)=0,∀𝑡>0,𝐹^∗_𝑥𝑦𝑡,𝑝_𝑖=1, (3)
then 𝐷 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. Let 𝑆⊂𝑋. If there exist two strictly increasing sequences of positive integers {𝑝_𝑖} and {𝑞_𝑖} such that for any 𝑥,𝑦∈𝑆, 𝑥≠𝑦, lim_𝑖→∞𝑑𝑓^𝑝𝑖(𝑥),𝑓^𝑝𝑖𝑦=0,lim_𝑖→∞𝑑𝑓^𝑞𝑖(𝑥),𝑓^𝑞𝑖𝑦>0, (4) 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 {𝑝_𝑖} be an increasing sequence of positive integers, then, PR𝑓,𝑝_𝑖=𝑓𝑥,𝑦∈𝑋×𝑋∣∀𝜀>0,∃𝑖∈𝑁suchthat𝑑^𝑝𝑖(𝑥),𝑓^𝑝𝑖𝑦<𝜀 (5) is called proximal relation with respect to {𝑝_𝑖}.
Thus AR𝑓,𝑝_𝑖=𝑥,𝑦∈𝑋×𝑋∣lim_𝑖→∞𝑑𝑓^𝑝𝑖(𝑥),𝑓^𝑝𝑖𝑦=0 (6) is called asymptotic relation with respect to {𝑝_𝑖}.
Thus DR𝑓,𝑝_𝑖=𝑋×𝑋−PR𝑓,𝑝_𝑖 (7) is called distal relation with respect to {𝑝_𝑖}.
So DCR𝑓,𝑝_𝑖=𝑥,𝑦∈𝑋×𝑋∣𝑥,𝑦isadistributivelychaoticpointpairof𝑓in𝑝_𝑖 (8) is called distributively chaotic respect to {𝑝_𝑖}.

Definition 8 (see [3]). 𝑓 is (topologically) transitive if for any two nonempty open sets 𝑈,𝑉⊂𝑋 there exists 𝑛>0 such that 𝑓^𝑛(𝑈)∩𝑉≠∅. 𝑓 is (topologically) weakly mixing if for any three nonempty open sets 𝑈,𝑉,𝑊⊂𝑋 there exists 𝑛>0 such that 𝑓^𝑛(𝑊)∩𝑉≠∅ and 𝑓^𝑛(𝑊)∩𝑈≠∅.

Definition 9 (see [4–6]). Let 𝑓 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(𝑓^𝑛(𝑥),𝑓^𝑛(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]). Let 𝑓 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(𝑓^𝑛(𝑥),𝑓^𝑛(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 that 𝑓 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. Let Σ 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,…,𝑠_𝑚≠𝑡_𝑚 for infinitely many 𝑚 and 𝑠_𝑛=𝑡_𝑛 for infinitely many 𝑛.

Proof. For a proof, see [8].

Lemma 13. If {𝑝_𝑖} and {𝑞_𝑖} are both infinite increasing subsequences of {𝑚_𝑖} which is a sequence of positive integers, then there exists an infinite increasing subsequence {𝑡_𝑖}⊂{𝑚_𝑖} such that 𝐴𝑅𝑓,𝑝_𝑖∩𝐷𝑅𝑓,𝑞_𝑖⊂𝐷𝐶𝑅𝑓,𝑡_𝑖. (9)

Proof. For a proof, see [9].

Lemma 14. 𝑓 is weakly mixing if and only if for any 𝑚≥2,𝑓_𝑚 is transitive.

Proof. For a proof, see [10].

3. Proof of Main Theorem

Proof of Theorem 1. (1) There is no isolated points in 𝑋 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.
Let 𝐷 be a dense scrambled set in the sequence {𝑝_𝑘} consisting of transitive points, and let 𝑟>0 be such that 𝐹_𝑥𝑦(𝑟,{𝑝_𝑘})=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 𝐹_𝑥0𝑦(𝑟,{𝑝_𝑘})=0. This implies that 𝑑(𝑓^𝑝𝑘(𝑥₀),𝑓^𝑝𝑘(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 any 𝜖>0. In 𝜖-neighborhood of any point 𝑥, we can find points 𝑦,𝑧∈𝐷 such that 𝑑(𝑓^𝑝𝑘(𝑦),𝑓^𝑝𝑘(𝑧))>𝑟. Then, 𝑑(𝑓^𝑝𝑘(𝑥),𝑓^𝑝𝑘(𝑦))>𝑟/2 or 𝑑(𝑓^𝑝𝑘(𝑥),𝑓^𝑝𝑘(𝑧))>𝑟/2.

Proof of Theorem 2. Let 𝐸 be an uncountable subset of Σ, as in Lemma 12. For each 𝑠=𝑠₀𝑠₁⋯𝑠_𝑛⋯∈𝐸, by the hypotheses, we can choose a point 𝑥(𝑠)∈𝑋 such that for any 𝑘, if 𝑛!<𝑘≤(𝑛+1)! then, 𝑥^𝑠_𝑝𝑘∈𝐵1𝑎,𝑘if𝑠_𝑛=0,𝐵1𝑏,𝑘if𝑠_𝑛=1. (10) Put 𝐷={𝑥^𝑠∣𝑠∈𝐸}. Clearly, if 𝑠≠𝑡 then 𝑥^𝑠≠𝑥^𝑡. It follows that 𝐸 being uncountable implies so is 𝐷.
Let 𝑥^𝑠,𝑥^𝑡∈𝐷 be any different points, where 𝑠=𝑠₀𝑠₁⋯𝑠_𝑖⋯,𝑡=𝑡₀𝑡₁⋯𝑡_𝑖⋯∈𝐸. By the property of 𝐸, we know that there exist sequences of positive integers 𝑚_𝑖→∞,𝑛_𝑖→∞ such that 𝑠_𝑚𝑖≠𝑡_𝑚𝑖,𝑠_𝑛𝑖=𝑡_𝑛𝑖 for all 𝑖, and for 𝑚_𝑖 large enough 1/𝑚<𝑑(𝑎,𝑏)/4=𝛿, we have 𝑑(𝑥^𝑠_𝑚𝑖,𝑥^𝑡_𝑚𝑖)>𝛿. Thus, lim_𝑖→∞𝑑𝑥^𝑠_𝑚𝑖,𝑥^𝑡_𝑚𝑖≥𝛿, (11) this shows lim_𝑖→∞sup𝑑𝑥^𝑠_𝑚𝑖,𝑥^𝑡_𝑚𝑖≥𝛿. (12) Meanwhile, for 𝑛_𝑖 large enough, 𝑥^𝑠_𝑛𝑖 and 𝑥^𝑡_𝑛𝑖 lie in the same ball of diameter less than 1/𝑛_𝑖. Thus, 𝑑(𝑥^𝑠_𝑛𝑖,𝑥^𝑡_𝑛𝑖)<1/𝑛, so lim_𝑖→∞𝑑𝑥^𝑠_𝑛𝑖,𝑥^𝑡_𝑛𝑖=0. (13) This shows lim_𝑖→∞inf𝑑𝑥^𝑠_𝑛𝑖,𝑥^𝑡_𝑛𝑖=0. (14)
Above all, (𝑋,𝑓) is chaotic in the strong sense of Li-Yorke.

Proof of Theorem 3. Because 𝑓 is chaotic in the strong sense of Li-Yorke, there exists an infinite increasing sequence {𝑝_𝑖}⊂ℕ and uncountable set 𝑆⊂𝑋, such that for any 𝑥,𝑦∈𝑆 with 𝑥≠𝑦, we have lim_𝑖→∞𝑑𝑓^𝑝𝑖(𝑥),𝑓^𝑝𝑖𝑦=0, (15) so that (𝑥,𝑦)∈AR(𝑓,{𝑝_𝑖}).
Again, by the definition of chaos in the strong sense of Li-Yorke, there exists {𝑞_𝑖}⊂ℕ, such that lim_𝑖→∞𝑑𝑓^𝑞𝑖(𝑥),𝑓^𝑞𝑖𝑦>0, (16) so that (𝑥,𝑦)∈DR(𝑓,{𝑞_𝑖}).
Hence, 𝑆×𝑆−Δ⊂AR(𝑓,{𝑝_𝑖})∩DR(𝑓,{𝑞_𝑖}), where Δ={(𝑥,𝑥);𝑥∈𝑋}. Then by Lemma 13, there exists a subsequence {𝑡_𝑖}⊂ℕ such that 𝑆×𝑆⊂DCR(𝑓,{𝑡_𝑖}). This shows that 𝑆 is a distributively chaotic set in the sequence {𝑡_𝑖} of 𝑓.

Corollary 15. If system (𝑋,𝑓) satisfies conditions of Theorem 2. then it is distributively chaotic in the sequence.

Proof. By Theorems 2 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. Let 𝑓 be weakly mixing, 𝑎,𝑏∈𝑋 with 𝑎≠𝑏. Take arbitrarily a nonempty open set 𝑉₀⊂𝑋 such that 𝑉₀ is compact. Since 𝑓 is weakly mixing, there exists 𝑝₁>0 such that 𝑓^𝑝1(𝑉₀)∩𝐵(𝑎,1/𝑘)≠∅ and 𝑓^𝑝1(𝑉₀)∩𝐵(𝑏,1/𝑘)≠∅. Thus, we find points 𝑥₁,𝑥₂ such that 𝑓^𝑝1(𝑥₁)∈𝐵(𝑎,1/𝑘),𝑓^𝑝1(𝑥₂)∈𝐵(𝑏,1/𝑘). Assume that there exist positive integers 𝑝₁<𝑝₂<⋯<𝑝_𝑘 such that for each finite sequence 𝐴₁𝐴₂⋯𝐴_𝑘, where 𝐴_𝑖∈{𝐵(𝑎,1/𝑖),𝐵(𝑏,1/𝑖)}, there is a point 𝑥∈𝑉₀ satisfying 𝑓^𝑝𝑖(𝑥)∈𝐴_𝑖 for 𝑖=1,2,…,𝑘, the set of all such points will denoted by 𝑆_𝑘. By continuity of 𝑓, each 𝑥∈𝑆_𝑘 has an open nonempty neighborhood 𝑊_𝑥⊂𝑉₀ such that 𝑓^𝑝𝑖(𝑊_𝑥)⊂𝐴_𝑖, if 𝑓^𝑝𝑖(𝑥)∈𝐴_𝑖, it follows from Lemma 14 that there exists 𝑝_𝑘+1>𝑝_𝑘 such that for each 𝑥∈𝑆_𝑘,𝑓^𝑝𝑘+1(𝑊_𝑥)∩𝐵(𝑎,1/(𝑘+1))≠∅ and 𝑓^𝑝𝑘+1(𝑊_𝑥)∩𝐵(𝑏,1/(𝑘+1))≠∅. Thus by induction, we know that there exists a sequence 𝑝_𝑘→∞ of positive integers such that for any finite sequence 𝐴₁⋯𝐴_𝑘, there is a point 𝑥∈𝑉₀ satisfying 𝑓^𝑝𝑖(𝑥)∈𝐴_𝑖,1≤𝑖≤𝑘.
Let 𝐶=𝐶₁𝐶₂⋯ be an infinite sequence, where 𝐶_𝑘∈𝐵1𝑎,𝑘,𝐵1𝑏,𝑘. (17) For each 𝑘, we can take a point 𝑥_𝑘∈𝐶_𝑘 such that 𝑓^𝑝𝑖𝑥_𝑘∈𝐶_𝑖,1≤𝑖≤𝑘. (18) Since 𝑉₀ is compact, the infinite sequence {𝑥_𝑖} has a limit point in 𝑉₀, say 𝑥_𝐶, it is not difficult to show 𝑓^𝑝𝑘(𝑥_𝐶)⊂𝐶_𝑘.
Thus by Theorem 2, 𝑓 is a strong chaos in the sense of Li-Yorke.

Acknowledgment

This work is supported by the NSFC no. 11271061, the NSFC no. 11001038, the NSFC no. 61153001, and the Independent Research Foundation of the Central Universities no. DC 12010111.