Some criteria of chaos in non-autonomous discrete systems

TL;DR

This paper develops new criteria for chaos in non-autonomous discrete systems, extending existing results for autonomous systems and relaxing assumptions, with applications to Li-Yorke and distributional chaos.

Contribution

It introduces novel criteria for chaos in non-autonomous systems, broadening the theoretical understanding and applicability compared to prior autonomous system results.

Findings

Criteria for strong Li-Yorke chaos established

Distributional chaos criteria in sequences derived

Conditions for chaos induced by coupled-expansion identified

Abstract

This paper establishes some criteria of chaos in non-autonomous discrete systems. Several criteria of strong Li-Yorke chaos are given. Based on these results, some criteria of distributional chaos in a sequence are established. Moreover, several criteria of distributional chaos induced by coupled-expansion for an irreducible transition matrix are obtained. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts. One example is provided for illustration.

Equations

$$x_{n+1}=f_n(x_n),n\ge 0,$$

$$\underset{n\to \infty }{\mathrm{liminf}}d(f_0^n(x),f_0^n(y))=0\mathrm{and}\underset{n\to \infty }{\mathrm{limsup}}d(f_0^n(x),f_0^n(y))>\delta .\text{vspace}-0.2cm$$

$$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\chi_{[0,\epsilon)}\big{(}d(f_{0}^{p_{i}}(x),f_{0}^{p_{i}}(y))\big{)}=1\;{\rm and}\;\liminf_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\chi_{[0,\delta)}\big{(}d(f_{0}^{p_{i}}(x),f_{0}^{p_{i}}(y))\big{)}=0,\vspace{-0.2cm}$$

$${\Sigma }_N^{+}(A):=\{s=(s_0,s_1,\cdots ):1\le s_j\le N,a_{s_j{s}_{j+1}}=1,j\ge 0\}.\text{vspace}-0.2cm$$

$$f_n(V_{i,n})\supset \bigcup _{a_{ij}=1}{V}_{j,n+1},1\le i\le N,n\ge 0,\text{vspace}-0.2cm$$

$$V_{\alpha }^{m,n}:=\bigcap _{k=0}^m{f}_n^{-k}(V_{a_k,n+k}).$$

$${\omega }_0=(r_0,\cdots ,t_0),{\omega }_1=(b_0,\cdots ,t_0),{\omega }_2=(b_0,\cdots ,t_0),{\omega }_3=(s_0,\cdots ,b_0),\text{vspace}-0.2cm$$

$$\Omega :=\{\gamma =(B_1,B_2,\cdots ):B_i={\omega }_1\mathrm{or}{\omega }_2,i\ge 1\}.$$

$$\hat{\gamma }:=(b_0,\cdots ,b_{m_1-1},{\omega }_3,{\omega }_0,B_1,b_0,\cdots ,b_{m_2-1},{\omega }_3,{\omega }_0,B_1,B_2,\cdots ).$$

$$S:=\{x_{\hat{\gamma }}:\gamma \in \Omega \}.\text{vspace}-0.2cm$$

$$d(f_0^{h_k}(x_{\widehat{{\gamma }_1}}),f_0^{h_k}(x_{\widehat{{\gamma }_2}}))\ge \delta ,k\ge 1.\text{vspace}-0.2cm$$

$$\underset{n\to \infty }{\mathrm{limsup}}d(f_0^n(x_{\widehat{{\gamma }_1}}),f_0^n(x_{\widehat{{\gamma }_2}}))\ge \delta .$$

$$n_k=(k-1)l_1+\frac{k(k-1)}{2}{l}_2+(k-1)l_3+\sum _{t=1}^k{m}_t,k\ge 1.$$

$$d(f_0^{n_k}(x_{\widehat{{\gamma }_1}}),f_0^{n_k}(x_{\widehat{{\gamma }_2}}))\le d(V_{\beta }^{m_k,n_k}),k\ge 1.\text{vspace}-0.2cm$$

$$\underset{k\to \infty }{\lim }d(f_0^{n_k}(x_{\widehat{{\gamma }_1}}),f_0^{n_k}(x_{\widehat{{\gamma }_2}}))=0.\text{vspace}-0.2cm$$

$$\underset{n\to \infty }{\mathrm{liminf}}d(f_0^n(x_{\widehat{{\gamma }_1}}),f_0^n(x_{\widehat{{\gamma }_2}}))=0.$$

$${\hat{x}}_{n+1}={\hat{f}}_n({\hat{x}}_n),n\ge 0,$$

$${\hat{f}}_0:=f_{k_1-1}\circ f_{k_1-2}\circ \cdots f_0,{\hat{f}}_n:=f_{k_{n+1}-1}\circ f_{k_{n+1}-2}\circ \cdots f_{k_n},n\ge 1.\text{vspace}-0.2cm$$

$${\omega }_0:=(m_0,\cdots ,l_0),{\omega }_1:=(r_0,\cdots ,l_0),{\omega }_2:=(r_0,\cdots ,l_0),\text{vspace}-0.2cm$$

$$\hat{\beta }:=(r_0,{\omega }_0,{\omega }_{b_0}^{p_1},r_0,\cdots ,r_{m_2-2},R_2,{\omega }_0,{\omega }_{b_1}^{p_2},r_0,\cdots ,r_{m_3-2},R_3,{\omega }_0,{\omega }_{b_2}^{p_3},\cdots ),$$

$$D:=\{x_{\hat{\beta }}:\beta \in E\}.\text{vspace}-0.2cm$$

$$n_i:=|(r_0,{\omega }_0,\cdots ,{\omega }_{b_{t_i}}^{p_{t_i+1}})|,i\ge 1.\text{vspace}-0.2cm$$

$$n_i=p_{t_i+1}l+p_{t_i+1}{2}^{-(t_i+1)},i\ge 1.$$

$${\Omega }_0:=\{\omega :\omega \mathrm{is}\mathrm{an}\mathrm{allowable}\mathrm{word}\mathrm{for}A\mathrm{with}|\omega |=l\}.\text{vspace}-0.2cm$$

$$V_{\mathcal{C}}^n:=\bigcap _{k=0}^{l-1}{f}_n^{-k}(V_{c_{k+1},n+k}).\text{vspace}-0.2cm$$

$$d_n:=\inf \{d(V_{\mathcal{C}}^n,V_{\mathcal{D}}^n):\mathcal{C}\ne \mathcal{D}\in \Omega \}\ge {\delta }_0,n\ge 0.$$

$$\sum _{j=0}^{n_i-1}{\chi }_{[0,{\delta }_0)}(d(f_0^j(x_{\widehat{{\beta }_1}}),f_0^j(x_{\widehat{{\beta }_2}})))\le n_i-(p_{t_i+1}l-l+1),i\ge 1.\text{vspace}-0.2cm$$

$$\lim_{i\to\infty}\frac{1}{n_{i}}\sum_{j=0}^{n_{i}-1}\chi_{[0,\delta_{0})}\big{(}d(f_{0}^{j}(x_{\hat{\beta_{1}}}),f_{0}^{j}(x_{\hat{\beta_{2}}}))\big{)}\leq 1-\lim_{i\to\infty}\frac{p_{t_{i}+1}l-l+1}{p_{{t_{i}}+1}l+p_{{t_{i}}+1}2^{-(t_{i}+1)}}=0,\vspace{-0.2cm}$$

$$\liminf_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\chi_{[0,\delta_{0})}\big{(}d(f_{0}^{j}(x_{\hat{\beta_{1}}}),f_{0}^{j}(x_{\hat{\beta_{2}}}))\big{)}=0.$$

$$k_i:=|(r_0,{\omega }_0,\cdots ,r_0,\cdots ,r_{m_i-1})|,i\ge 1.\text{vspace}-0.2cm$$

Full text

**Some criteria of chaos in non-autonomous discrete systems

**

Hua Shao*†, Guanrong Chen†, Yuming Shi‡*

† Department of Electronic Engineering, City University of Hong Kong,

Hong Kong SAR, P. R. China

‡ Department of Mathematics, Shandong University

Jinan, Shandong 250100, P. R. China

Abstract. This paper establishes some criteria of chaos in non-autonomous discrete systems. Several criteria of strong Li-Yorke chaos are given. Based on these results, some criteria of distributional chaos in a sequence are established. Moreover, several criteria of distributional chaos induced by coupled-expansion for an irreducible transition matrix are obtained. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts. One example is provided for illustration.

Keywords: non-autonomous discrete system; strong Li-Yorke chaos; distributional chaos; coupled-expansion; irreducible transition matrix.

2010 Mathematics Subject Classification: 37B55, 37D45, 37B10.

1. Introduction

Chaos of the non-autonomous discrete system (briefly, NDS)

[TABLE]

has attracted considerable attention recently [1, 2, 7, 10, 11, 12, 14, 16, 17, 19, 21, 25, 26], where $\{f_{n}\}_{n=0}^{\infty}$ is a sequence of maps from $X$ to $X$, with $(X,d)$ being a metric space. Many complex systems of real-world problems in the fields of biology, physics, chemistry and engineering, are indeed non-autonomous, putting the model (1.1) into focus. The positive orbit $\{x_{n}\}^{\infty}_{n=0}$ of system (1.1) starting from an initial point $x_{0}\in X$ is given by $x_{n}=f^{n}_{0}(x_{0})$, where $f^{n}_{0}:=f_{n-1}\circ\cdots\circ f_{0}$, $n\geq 1$.

Coupled-expansion is always associated with chaos [1, 3, 4, 5, 6, 8, 10, 13, 14, 15, 16, 20, 21, 22, 23]. In 2009, the concept of coupled-expansion for a transition matrix was extended to NDSs, for which a criterion of strong Li-Yorke chaos was established in [14]. Later, the assumptions in this result were weakened in [10], where some new criteria of strong Li-Yorke chaos were established via coupled-expansion for an irreducible transition matrix in bounded and closed subsets of a complete metric space, and in compact subsets of a metric space, respectively. Then, a new concept of weak coupled-expansion for a transition matrix was introduced for NDSs, and several criteria of chaos induced by weak coupled-expansion for an irreducible transition matrix were established in [21]. Recently, Li-Yorke $\delta$-chaos for some $\delta>0$ and distributional $\delta^{\prime}$-chaos in a sequence for some $\delta^{\prime}>0$ were proved to be equivalent for system (1.1) in the case that $(X,d)$ is a compact metric space and $f_{n}$ are continuous maps for all $n\geq 0$ [11]. Consequently, those criteria of strong Li-Yorke chaos can be regarded as criteria of distributional chaos in a sequence for system (1.1). It arouses our interest to investigate the relationship of coupled-expansion for a transition matrix and distributional chaos in a sequence for system (1.1). In the present paper, we establish some new criteria of strong Li-Yorke chaos for system (1.1), which not only relax the assumptions of the counterparts in the literature, but also can be regarded as criteria of distributional chaos in a sequence under certain conditions.

Recall that distributional chaos is a special kind of distributional chaos in a sequence in the case that the sequence is the set of all nonnegative integers. However, generally it is stronger than distributional chaos in a sequence. In 2015, one criterion of distributional chaos induced by coupled-expansion for a transition matrix was established for autonomous discrete systems [6]. Motivated by this result, here we are interested in finding some criteria of distributional chaos induced by weak coupled-expansion for a transition matrix for the non-autonomous system (1.1).

The rest of the present paper is organized as follows. Section 2 presents some basic concepts and useful lemmas. In Section 3, two criteria of strong Li-Yorke chaos for system (1.1) are obtained, which relax the assumptions of the counterparts in the literature (see Remark 3.1). Moreover, other two criteria of strong Li-Yorke chaos are derived by applying the relationship of Li-Yorke chaos between system (1.1) and its induced system. In Section 4, some criteria of distributional chaos in a sequence are established for system (1.1) by the results obtained in Section 3. Furthermore, several criteria of distributional chaos induced by weak coupled-expansion for an irreducible transition matrix are obtained. Finally, an example is provided in Section 5 for illustration.

2. Preliminaries

In this section, some basic concepts and useful lemmas are presented.

For convenience, denote $f^{n}_{i}:=f_{i+n-1}\circ\cdots\circ f_{i}$ and $f^{-n}_{i}:=(f_{i}^{n})^{-1}$, $i\geq 0$, $n\geq 1$. Let $A,B$ be nonempty subsets of $X$. The boundary of $A$ is denoted by $\partial A$; the diameter of $A$ is denoted by $d(A):=\sup\{d(x,y):x,y\in A\}$; and the distance between $A$ and $B$ is denoted by $d(A,B):=\inf\{d(a,b):a\in A,b\in B\}$. The set of all nonnegative integers and positive integers are denoted by $\mathbf{N}$ and $\mathbf{Z^{+}}$, respectively.

Definition 2.1 ([14], Definition 2.7). System (1.1) is said to be Li-Yorke $\delta$-chaotic for some $\delta>0$ if it has an uncountable Li-Yorke $\delta$-scrambled set $S$ in $X$; that is, for any $x,y\in S\subset X$,

[TABLE]

Further, it is said to be chaotic in the strong sense of Li-Yorke if all the orbits starting from the points in $S$ are bounded.

Definition 2.2 ([11], Definitions 2.1 and 2.2). System (1.1) is said to be distributionally $\delta$-chaotic in a sequence $P=\{p_{n}\}_{n=1}^{\infty}$ for some $\delta>0$ if it has an uncountable distributionally $\delta$-scrambled set $D\subset X$ in $P$; that is, for any $x,y\in D\subset X$ and any $\epsilon>0$,

[TABLE]

where $\chi_{[0,\epsilon)}$ is the characteristic function defined on the set $[0,\epsilon)$. Further, if $P=\mathbf{N}$, then system (1.1) is said to be distributionally $\delta$-chaotic.

The relationship between Li-Yorke chaos and distributional chaos in a sequence for system (1.1) is shown below.

Lemma 2.1 ([11], Theorem 3.6). Let $(X,d)$ be compact and $f_{n}$ be continuous in $X$, $n\geq 0$. Then, system (1.1) is Li-Yorke $\delta$-chaotic for some $\delta>0$ if and only if it is distributionally $\delta^{\prime}$-chaotic in a sequence for some $\delta^{\prime}>0$.

A matrix $A=(a_{ij})_{N\times N}$ ($N\geq 2$) is said to be a transition matrix if $a_{ij}=0$ or $1$ for all $i,j$; $\sum_{j=1}^{N}a_{ij}\geq 1$ for all $i$; and $\sum_{i=1}^{N}a_{ij}\geq 1$ for all $j$, $1\leq i,j\leq N$. A transition matrix $A=(a_{ij})_{N\times N}$ is said to be irreducible if, for each pair $1\leq i,j\leq N$, there exists $k\in\mathbf{Z^{+}}$ such that $a^{(k)}_{ij}>0$, where $a^{(k)}_{ij}$ denotes the $(i,j)$ entry of matrix $A^{k}$. A finite sequence $\omega=(s_{1},s_{2},\cdots,s_{k})$ is said to be an allowable word of length $k$ for $A$ if $a_{s_{i}s_{i+1}}=1$, $1\leq i\leq k-1$, where $1\leq s_{i}\leq N$, $1\leq i\leq k$. For convenience, the length of $\omega$ is denoted by $|\omega|$ and $\omega^{n}:=\underbrace{(\omega,\cdots,\omega)}_{n}$, $n\geq 1$.

The one-sided sequence space $\Sigma_{N}^{+}:=\{\alpha=(a_{0},a_{1},\cdots):1\leq a_{i}\leq N,\;i\geq 0\}$ is a metric space with the metric $\rho(\alpha,\beta):=\sum_{i=0}^{\infty}d^{\prime}(a_{i},b_{i})/2^{i}$, where $\alpha=(a_{0},a_{1},\cdots),\beta=(b_{0},b_{1},\cdots)\in\Sigma_{N}^{+}$, $d^{\prime}(a_{i},b_{i})=0$ if $a_{i}=b_{i}$, and $d^{\prime}(a_{i},b_{i})=1$ if $a_{i}\neq b_{i}$, $i\geq 0$. Note that $(\Sigma_{N}^{+},\rho)$ is a compact metric space. Define the shift map $\sigma:\Sigma_{N}^{+}\to\Sigma_{N}^{+}$ by $\sigma((s_{0},s_{1},s_{2},\cdots)):=(s_{1},s_{2},\cdots)$. This map is continuous and $(\Sigma_{N}^{+},\sigma)$ is called the one-sided symbolic dynamical system on $N$ symbols. For a given transition matrix $A=(a_{ij})_{N\times N}$, denote

[TABLE]

$\Sigma_{N}^{+}(A)$ is a compact subset of $\Sigma_{N}^{+}$ and invariant under $\sigma$. The map $\sigma_{A}:=\sigma|_{\Sigma_{N}^{+}(A)}:\Sigma_{N}^{+}(A)\to\Sigma_{N}^{+}(A)$ is said to be a subshift of finite type for matrix $A$. For more details about symbolic dynamical systems and subshifts of finite type, see [9, 24].

The following two lemmas will also be useful in the sequel.

Lemma 2.2 ([18], Lemma 2.2). $\Sigma_{2}^{+}$* has an uncountable subset $E$ such that for any different points $\alpha=(a_{0},a_{1},\cdots),\beta=(b_{0},b_{1},\cdots)\in E$, $a_{n}=b_{n}$ for infinitely many $n$ and $a_{m}\neq b_{m}$ for infinitely many $m$.*

Lemma 2.3 ([22], Theorem 2.2). Let $A=(a_{ij})_{N\times N}$ be an irreducible transition matrix with $\sum_{j=1}^{N}a_{i_{0}j}\geq 2$ for some $1\leq i_{0}\leq N$. Then

• (i)

for any $1\leq i,j\leq N$ and any $M\in\mathbf{Z^{+}}$, there exists at least one allowable word $\omega=(i,\cdots,j)$ for $A$ such that $|\omega|>M$;

• (ii)

for any given allowable word $\omega=(b_{1},b_{2},\cdots,b_{k})$ for $A$, if $k>2N(N^{2}-2N+2)$, then there exists another different allowable word $\omega^{\prime}=(c_{1},c_{2},\cdots,c_{k})$ for $A$ with $c_{1}=b_{1}$ and $c_{k}=b_{k}$.

Next, the definition of weak coupled-expansion for a transition matrix is introduced.

Definition 2.3. Let $A=(a_{ij})_{N\times N}$ be a transition matrix. If there exists a sequence $\{V_{i,n}\}_{n=0}^{\infty}$ of nonempty subsets of $X$ with $V_{i,n}\cap V_{j,n}=\partial V_{i,n}\cap\partial V_{j,n}$ ($d(V_{i,n},V_{j,n})>0$) for any $1\leq i\neq j\leq N$ and $n\geq 0$ such that

[TABLE]

then system (1.1) is said to be (strictly) weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$. In the special case that $V_{i,n}=V_{i}$ for all $n\geq 0$ and $1\leq i\leq N$, it is said to be (strictly) $A$-coupled-expanding in $V_{i}$, $1\leq i\leq N$.

Remark 2.1. Definition 2.3 is a slight revision of Definition 2.4 in [21].

3. Some criteria of strong Li-Yorke chaos

In this section, two criteria of strong Li-Yorke chaos for system (1.1) are established. Further, other two criteria of strong Li-Yorke chaos are given by applying the relationship of Li-Yorke chaos between system (1.1) and its induced system.

The following result can be easily verified.

Lemma 3.1. Let $A=(a_{ij})_{N\times N}$ be a transition matrix and $V_{i}$, $1\leq i\leq N$, be disjoint nonempty compact subsets of $X$. Assume that $f_{n}$ is continuous in $\bigcup_{i=1}^{N}V_{i}$, $n\geq 0$, and $\{V_{i,n}\}_{n=0}^{\infty}$ is a sequence of nonempty closed subsets of $V_{i}$, $1\leq i\leq N$. Then, $V_{\alpha}^{m,n}$ is compact and satisfies that $V_{\alpha}^{m+1,n}\subset V_{\alpha}^{m,n}\subset\bigcup_{i=1}^{N}V_{i,n}$ for all $m,n\geq 0$ and all $\alpha=(a_{0},a_{1},\cdots)$ $\in\Sigma_{N}^{+}(A)$, where

[TABLE]

Further, $(\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n})\cap(\bigcap_{m=0}^{\infty}V_{\beta}^{m,n})=\emptyset$ for any $\alpha\neq\beta\in\Sigma_{N}^{+}(A)$ and $m,n\geq 0$.

Theorem 3.1. Let all the assumptions of Lemma 3.1 hold and suppose that $A$ is irreducible with $\sum_{j=1}^{N}a_{i_{0}j}\geq 2$ for some $1\leq i_{0}\leq N$. If

• (i)

$V_{\alpha}^{m,n}\neq\emptyset$* for all $m,n\geq 0$ and all $\alpha=(a_{0},a_{1},\cdots)\in\Sigma_{N}^{+}(A)$, where $V_{\alpha}^{m,n}$ is specified in (3.1);*

• (ii)

there exist $\beta=(b_{0},b_{1},\cdots)\in\Sigma_{N}^{+}(A)$, an increasing subsequence $\{m_{k}\}_{k=1}^{\infty}\subset\mathbf{Z^{+}}$, and $1\leq s_{0}\leq N$, such that $b_{m_{k}}=s_{0}$, $k\geq 1$, and $d(V_{\beta}^{m_{k},n_{k}})$ converges to [math] as $k\to\infty$ for all increasing subsequence $\{n_{k}\}_{k=1}^{\infty}\subset\mathbf{N}$;

then system (1.1) is chaotic in the strong sense of Li-Yorke.

Proof. Since $A$ is a transition matrix, there exist $1\leq t_{0},r_{0}\leq N$ such that $a_{t_{0}b_{0}}=a_{b_{0}r_{0}}=1$. By Lemma 2.3, there exist four allowable words for matrix $A$:

[TABLE]

where $|\omega_{0}|=l_{1}$, $|\omega_{1}|=|\omega_{2}|=l_{2}$ with $\omega_{1}\neq\omega_{2}$, and $|\omega_{3}|=l_{3}$ (if $s_{0}=b_{0}$, then set $\omega_{3}=(b_{0})$ with length $1$). Denote

[TABLE]

Then $\Omega\subset\Sigma_{N}^{+}(A)$ is uncountable. Note that $a_{b_{m_{k}-1}s_{0}}=1$, $k\geq 1$, since $b_{m_{k}}=s_{0}$. For any $\gamma=(B_{1},B_{2},\cdots)\in\Omega$, set

[TABLE]

Clearly, $\hat{\gamma}\in\Sigma_{N}^{+}(A)$ and $\hat{\gamma_{1}}\neq\hat{\gamma_{2}}$ if and only if $\gamma_{1}\neq\gamma_{1}$. By assumption (i) and Lemma 3.1, one has that $V_{\hat{\gamma}}^{m,0}$ is nonempty, compact, and satisfies that $V_{\hat{\gamma}}^{m+1,0}\subset V_{\hat{\gamma}}^{m,0}$, $m\geq 0$. Thus $\bigcap_{m=0}^{\infty}V_{\hat{\gamma}}^{m,0}\neq\emptyset$. Fix one point $x_{\hat{\gamma}}\in\bigcap_{m=0}^{\infty}V_{\hat{\gamma}}^{m,0}$, and denote

[TABLE]

It follows from Lemma 3.1 that $x_{\hat{\gamma_{1}}}\neq x_{\hat{\gamma_{2}}}$ if and only if $\hat{\gamma_{1}}\neq\hat{\gamma_{2}}$, which holds if and only if $\gamma_{1}\neq\gamma_{2}$. So, $S$ is uncountable.

Next, it will be shown that $S$ is a Li-Yorke $\delta$-scrambled set for system (1.1), where $\delta:=\min_{1\leq i\neq j\leq N}d(V_{i},V_{j})>0$. For any $x_{\hat{\gamma_{1}}}\neq x_{\hat{\gamma_{2}}}\in S$, one has that $\hat{\gamma_{1}}\neq\hat{\gamma_{2}}$. By (3.1)-(3.3), there exist an infinite sequence $\{h_{k}\}_{k=1}^{\infty}$ and $1\leq s_{1}\neq s_{2}\leq N$ such that $f_{0}^{h_{k}}(x_{\hat{\gamma_{1}}})\in V_{s_{1},h_{k}}\subset V_{s_{1}}$ and $f_{0}^{h_{k}}(x_{\hat{\gamma_{2}}})\in V_{s_{2},h_{k}}\subset V_{s_{2}}$, $k\geq 1$. So,

[TABLE]

Thus,

[TABLE]

On the other hand, by (3.3), one has that $\sigma_{A}^{n_{k}}(\hat{\gamma})=(b_{0},\cdots,b_{m_{k}},\cdots)$ for any $\gamma\in\Omega$, where

[TABLE]

Then, $f_{0}^{n_{k}+i}(x_{\hat{\gamma_{1}}}),f_{0}^{n_{k}+i}(x_{\hat{\gamma_{2}}})\in V_{b_{i},n_{k}+i}$, $0\leq i\leq m_{k}$. Thus, $f_{0}^{n_{k}}(x_{\hat{\gamma_{1}}}),f_{0}^{n_{k}}(x_{\hat{\gamma_{2}}})\in V_{\beta}^{m_{k},n_{k}}$, $k\geq 1$, implying that

[TABLE]

This, together with assumption (ii), yields that

[TABLE]

So,

[TABLE]

Hence, $S$ is an uncountable $\delta$-scrambled set of system (1.1) by (3.4) and (3.5). Moreover, for any $x_{\hat{\gamma}}\in S$, its positive orbit $\{f_{0}^{n}(x_{\hat{\gamma}})\}_{n=0}^{\infty}\subset\bigcup_{i=1}^{N}V_{i}$ by (3.1) and (3.3). Thus, all the orbits starting from the points in $S$ are bounded. Therefore, system (1.1) is chaotic in the strong sense of Li-Yorke. The proof is complete.

Remark 3.1. It can be easily verified that the weak $A$-coupled-expansion in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$, implies the condition in assumption (i) of Theorem 3.1. However, the converse is not true in general, even in the special case that $f_{n}=f$ and $V_{i,n}=V_{i}$, $n\geq 0$, $1\leq i\leq N$ (see Example 3.1.1 in [4]). Hence, assumption (i) of Theorem 3.1 here is strictly weaker than assumption ${\rm(ii_{1})}$ of Theorems 3.2 and 3.3 in [21]. Moreover, both assumption ${\rm(iii_{1})}$ of Theorem 3.2 and assumption ${\rm(iii_{4})}$ of Theorem 3.3 in [21] imply assumption (ii) of Theorem 3.1 above (see, for example, the proof of Corollary 4.1 below). Hence, Theorems 3.2 and 3.3 in [21] are corollaries of Theorem 3.1 here.

The following result is a direct consequence of Theorem 3.1.

Corollary 3.1. Let all the assumptions of Theorem 3.1 hold except that assumption (i) is replaced by that system (1.1) is weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$. Then, system (1.1) is chaotic in the strong sense of Li-Yorke.

Let $\{k_{n}\}_{n=1}^{\infty}\subset\mathbf{Z^{+}}$ be an increasing subsequence. The following system:

[TABLE]

is called the induced system by system (1.1) through $\{k_{n}\}_{n=1}^{\infty}$ [16], where

[TABLE]

Let $\{x_{n}\}_{n=0}^{\infty}$ be the orbit of system (1.1) starting from $x_{0}$ and $\{\hat{x}_{n}\}_{n=0}^{\infty}$ be the orbit of the induced system (3.6) starting from $\hat{x}_{0}:=x_{0}$. Then, $\hat{x}_{n}=f_{0}^{k_{n}}(x_{0})$, $n\geq 0$, and thus the orbit $\{\hat{x}_{n}\}_{n=0}^{\infty}$ of system (3.6) is a part of the orbit of system (1.1) starting from the same initial point $x_{0}$.

Next, recall the relationship of (strong) Li-Yorke chaos between systems (1.1) and (3.6).

Lemma 3.2 ([16], Theorem 3.1). If system (3.6) is Li-Yorke $\delta$-chaotic for some $\delta>0$ through $\{k_{n}\}_{n=1}^{\infty}$, so is system (1.1). Further, if system (3.6) is chaotic in the strong sense of Li-Yorke through $\{k_{n}\}_{n=1}^{\infty}$, so is system (1.1) in the case that $X$ is bounded.

The following result follows directly from Theorem 3.1 and Lemma 3.2.

Theorem 3.2. Assume that there exists an increasing subsequence $\{k_{n}\}_{n=1}^{\infty}\subset\mathbf{Z^{+}}$ such that all the assumptions of Theorem 3.1 hold for system (3.6). Then system (1.1) is Li-Yorke $\delta$-chaotic for some $\delta>0$. Further, system (1.1) is chaotic in the strong sense of Li-Yorke in the case that $X$ is bounded.

Applying Lemma 3.2 to Corollary 3.1, one obtains the following result.

Corollary 3.2. Assume that there exists an increasing subsequence $\{k_{n}\}_{n=1}^{\infty}\subset\mathbf{Z^{+}}$ such that all the assumptions of Corollary 3.1 hold for system (3.6). Then, system (1.1) is Li-Yorke $\delta$-chaotic for some $\delta>0$. Further, system (1.1) is chaotic in the strong sense of Li-Yorke in the case that $X$ is bounded.

4. Some criteria of distributional chaos in a sequence and distributional chaos

In this section, several criteria of distributional chaos in a sequence and of distributional chaos are established for system (1.1), respectively.

Applying Lemma 2.1 to Theorems 3.1 and 3.2 and Corollaries 3.1 and 3.2, respectively, the following result can be obtained.

Theorem 4.1. Let all the assumptions of Theorem 3.1 or Corollary 3.1 or Theorem 3.2 or Corollary 3.2 hold. If $X$ is compact and $f_{n}$ is continuous in $X$ for all $n\geq 0$, then system (1.1) is distributionally $\delta$-chaotic in a sequence for some $\delta>0$.

The next result gives a criterion of distributional chaos for system (1.1), which is induced by weak coupled-expansion for an irreducible transition matrix.

Theorem 4.2. Let all the assumptions of Lemma 3.1 hold, $\{f_{n}\}_{n=0}^{\infty}$ be equi-continuous in $\bigcup_{i=1}^{N}V_{i}$, and $A$ be irreducible with $\sum_{j=1}^{N}a_{i_{0}j}\geq 2$ for some $1\leq i_{0}\leq N$. If

• ${\rm(i)}$

system (1.1) is weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$;

• ${\rm(ii)}$

there exists a periodic point $\gamma=(r_{0},r_{1},\cdots)\in\Sigma_{N}^{+}(A)$ such that $d(V_{\gamma}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$;

then system (1.1) is distributionally $\delta$-chaotic for some $\delta>0$.

Proof. Since $A$ is a transition matrix, there exist $1\leq l_{0},m_{0}\leq N$ such that $a_{l_{0}r_{0}}=a_{r_{0}m_{0}}=1$. By Lemma 2.3, one has that there exist three allowable words for matrix $A$:

[TABLE]

where $|\omega_{1}|=|\omega_{2}|=l$ and $\omega_{1}\neq\omega_{2}$. Let $E\subset\Sigma_{2}^{+}$ be the set satisfying the property in Lemma 2.2. For any $\beta=(b_{0},b_{1},b_{2},\cdots)\in E$, set

[TABLE]

where $p_{1}:=2|(r_{0},\omega_{0})|$, $p_{n}=2^{n}|(r_{0},\omega_{0},\cdots,R_{n},$ $\omega_{0})|$ for $n\geq 2$; $m_{n}:=2^{n}|(r_{0},\omega_{0},\cdots,\omega_{b_{n-2}}^{p_{n-1}})|$ for $n\geq 2$; and for any $n\geq 2$, $R_{n}:=(r_{0})$ if $r_{m_{n}-1}=r_{0}$, and otherwise, $R_{n}:=(r_{m_{n}-1},\cdots,r_{0})$, while the fact that there exists an allowable word $(r_{m_{n}-1},\cdots,r_{0})$ for the matrix $A$ since $A$ is irreducible has been used in the case that $r_{m_{n}-1}\neq r_{0}$. Clearly, $\hat{\beta}\in\Sigma_{N}^{+}(A)$ and $\hat{\beta_{1}}\neq\hat{\beta_{2}}$ if and only if $\beta_{1}\neq\beta_{2}$. By assumption (i) and Lemma 3.1, $V_{\hat{\beta}}^{m,0}$ is nonempty, compact, and satisfies that $V_{\hat{\beta}}^{m+1,0}\subset V_{\hat{\beta}}^{m,0}$, $m\geq 0$. Thus, $\bigcap_{m=0}^{\infty}V_{\hat{\beta}}^{m,0}\neq\emptyset$. Fix one point $x_{\hat{\beta}}\in\bigcap_{m=0}^{\infty}V_{\hat{\beta}}^{m,0}$, and denote

[TABLE]

Using Lemma 3.1 again, one has that $x_{\hat{\beta_{1}}}\neq x_{\hat{\beta_{2}}}$ if and only if $\hat{\beta_{1}}\neq\hat{\beta_{2}}$, which holds if and only if $\beta_{1}\neq\beta_{2}$. Hence, $D$ is uncountable since $E$ is uncountable.

In the following, it will be shown that $D$ is a distributionally $\delta_{0}$-scrambled subset for system (1.1), where $\delta_{0}:=\min_{1\leq i\neq j\leq N}d(V_{i},V_{j})$. Fix any $x_{\hat{\beta_{1}}},x_{\hat{\beta_{2}}}\in D$ with $x_{\hat{\beta_{1}}}\neq x_{\hat{\beta_{2}}}$. Then, $\beta_{1}\neq\beta_{2}\in E$. Suppose that $\beta_{1}=(b_{0}^{(1)},b_{1}^{(1)},b_{2}^{(1)},\cdots)$ and $\beta_{2}=(b_{0}^{(2)},b_{1}^{(2)},b_{2}^{(2)},\cdots)$. Thus, there exists an increasing subsequence $\{t_{i}\}_{i=1}^{\infty}\subset\mathbf{Z^{+}}$ such that $b_{t_{i}}^{(1)}\neq b_{t_{i}}^{(2)}$, $i\geq 1$, by Lemma 2.2. Set

[TABLE]

Then,

[TABLE]

Denote

[TABLE]

For any $\mathcal{C}=(c_{1},\cdots,c_{l})\in\Omega_{0}$ and any $n\geq 0$, set

[TABLE]

It is evident that

[TABLE]

By (4.1) and (4.3), one has that

[TABLE]

This, together with (4.2), implies that

[TABLE]

which yields that

[TABLE]

On the other hand, denote

[TABLE]

Thus,

[TABLE]

Fix any $\epsilon>0$. It is claimed that there exists $M>0$ such that, for all $m\geq M$,

[TABLE]

where $P$ is the period of $\gamma$. For simplicity, only consider the case of $P=2$. The general cases can be proved in a similar way. Since $\{f_{n}\}_{n=0}^{\infty}$ is equi-continuous in $\bigcup_{i=1}^{N}V_{i}$, there exists $0<\delta_{1}<\epsilon$ such that, for any $x,y\in X$ with $d(x,y)<\delta_{1}$,

[TABLE]

By assumption (ii), there exists $M>0$ such that, for any $m\geq M$,

[TABLE]

Fix any $n\geq 0$ and $m\geq M$. By assumption (i), one has that

[TABLE]

It follows from (4.9) that, for any $z_{1},z_{2}\in V_{\sigma_{A}(\gamma)}^{m,n+1}$, there exist $x,y\in V_{\gamma}^{m+1,n}$ such that $z_{1}=f_{n}(x)$ and $z_{2}=f_{n}(y)$. By (4.8), one has that

[TABLE]

This, together with (4.7), implies that

[TABLE]

Thus,

[TABLE]

Hence, (4.6) follows from (4.8) and (4.10) in the case of $P=2$. By (4.1) and (4.6), one has that

[TABLE]

This, together with (4.5), implies that

[TABLE]

So,

[TABLE]

Hence, $D$ is an uncountable distributionally $\delta_{0}$-scrambled subset for system (1.1) by (4.4) and (4.11). Therefore, system (1.1) is distributionally $\delta_{0}$-chaotic. The proof is complete.

Remark 4.1. Theorem 4.2 extends Theorem 3.2 in [6] from autonomous to non-autonomous systems.

Corollary 4.1. Let all the assumptions of Theorem 4.2 hold, except that assumption (ii) is replaced by

• ${\rm(ii_{a})}$

there exists a constant $\lambda>1$ such that

[TABLE]

then system (1.1) is distributionally $\delta$-chaotic for some $\delta>0$.

Proof. It can be shown that $d(V_{\gamma}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$ for all $\gamma=(r_{0},r_{1},\cdots)\in\Sigma_{N}^{+}(A)$. Indeed, it follows from (4.11) that, for any $x,y\in V_{\gamma}^{m,n}$,

[TABLE]

implying that

[TABLE]

Thus,

[TABLE]

This, together with the assumption that $\lambda>1$, yields that $d(V_{\gamma}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$. Hence, all the assumptions of Theorem 4.2 hold. Therefore, system (1.1) is distributionally $\delta$-chaotic for some $\delta>0$. This completes the proof.

The following result is somewhat better since it only requires $f_{n}$ be expanding in distance in one subset for all $n\geq 0$.

Corollary 4.2. Let all the assumptions of Theorem 4.2 hold, except that assumption (ii) is replaced by

• ${\rm(ii_{b})}$

there exist an integer $1\leq j_{0}\leq N$ and a constant $\lambda>1$ such that $a_{j_{0}j_{0}}=1$ and

[TABLE]

then system (1.1) is distributionally $\delta$-chaotic for some $\delta>0$.

Proof. With a similar proof to that of Corollary 4.1, one can show that for $\gamma=(j_{0},j_{0},j_{0},\cdots)$, $d(V_{\gamma}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$. Hence, all the assumptions of Theorem 4.2 hold. Therefore, system (1.1) is distributionally $\delta$-chaotic for some $\delta>0$. The proof is complete.

5. An example

In this section, an example is provided to illustrate the theoretical results.

Example 5.1. Consider the following non-autonomous logistic system:

[TABLE]

governed by the maps $f_{n}(x)=r_{n}x(1-x)$, $x\in[0,1]$, where $9/2\leq r_{n}\leq\mu$, $n\geq 0$, and $\mu\geq 9/2$ is a constant. It can be easily verified that

[TABLE]

Thus,

[TABLE]

which yields that $\{f_{n}\}_{n=0}^{\infty}$ is equi-continuous in $[0,1]$. Set

[TABLE]

It is evident that $V_{1}$ and $V_{2}$ are disjoint nonempty compact subsets of $[0,1]$, and

[TABLE]

Since $r_{n}\geq 9/2$, $n\geq 0$, one can see that

[TABLE]

which implies that system (5.1) is strictly $A$-coupled-expanding in $V_{1}$ and $V_{2}$, where $A=(a_{ij})_{2\times 2}$ with $a_{ij}=1$, $1\leq i,j\leq 2$, and thus $A$ is irreducible with $\sum_{j=1}^{2}a_{i_{0}j}=2$, $i_{0}=1,2$. On the other hand, one has that

[TABLE]

which yields that

[TABLE]

Therefore, all the assumptions in Theorem 3.1 hold for system (1.1) with $\beta=(1,1,\cdots)\in\Sigma_{2}^{+}(A)$, satisfying by (5.3) that $d(V_{\beta}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$. By Theorem 3.1, system (1.1) is chaotic in the strong sense of Li-Yorke.

Moreover, all the assumptions in Corollary 4.2 hold for system (1.1) with $j_{0}=1$, satisfying assumption ${\rm(ii_{b})}$ by (5.2) and (5.3). By Corollary 4.2, system (1.1) is distributionally $\delta$-chaotic for some $\delta>0$.

Remark 5.1. System (5.1) is an important model in biology, which describes the population growth under certain conditions. Comparing to Example 5.1 in [14], here it not only proves that system (5.1) is chaotic in the strong sense of Li-Yorke, but also proves that system (5.1) is distributionally $\delta$-chaotic for some $\delta>0$.

Finally, it is worth noting here that Theorem 4.2 and its corollaries still hold true when $f_{n}:X\to X$ is replaced by that $f_{n}:X_{n}\to X_{n+1}$ with compact subsets $X_{n}\subset X$ for any $n\geq 0$.

Acknowledgments

This research was supported by the Hong Kong Research Grants Council (GRF Grant CityU11200317) and the NNSF of China (Grant 11571202).