Collective Sensitivity and Collective Accessibility of Non-Autonomous Discrete Dynamical Systems

Abstract

The concepts of collectively accessible, collectively sensitive, collectively infinitely sensitive, and collectively Li–Yorke sensitive are defined in non-autonomous discrete systems. It is proved that, if the mapping sequence $h_{1,\infty }=(h_1,h_2,\dots )$ is $\mathcal{W}$-chaotic, then $h_{n,\infty }=(h_n,h_{n+1},\dots )(\forall n\in \mathbb{N}=\{1,2,\dots \})$ would also be $\mathcal{W}$-chaotic. $\mathcal{W}$-chaos represents one of the following five properties: collectively accessible, sensitive, collectively sensitive, collectively infinitely sensitive, and collectively Li–Yorke sensitive. Then, the relationship of chaotic properties between the product system $(H_1\times H_2,f_{1,\infty }\times g_{1,\infty })$ and factor systems $(H_1,f_{1,\infty })$ and $(H_2,g_{1,\infty })$ was presented. Furthermore, in this paper, it is also proved that, if the autonomous discrete system $(X,\widehat{h})$ induced by the p-periodic discrete system $(H,h_{1,\infty })$ is $\mathcal{W}$-chaotic, then the p-periodic discrete system $(H,f_{1,\infty })$ would also be $\mathcal{W}$-chaotic.

1. Introduction

As a natural extension of autonomous discrete dynamical systems (ADDSs), non-autonomous discrete dynamical systems (NDDSs), are an important part of topological dynamical systems. Compared with classical dynamical systems (ADDSs), NDDSs can describe various dynamical behaviors more flexibly and conveniently.

NDDSs, firstly introduced by Kolyada [1] in 1996, are equivalent to some non-autonomous differential equations (see [2,3]). Chaos in NDDSs has become a hot research topic since the beginning of the 21st century. In 2011, Canovas [4] studied the limit behavior of sequences with the form $f_n\circ \dots \circ f_1(x)$$(x\in [0,1],n\in \mathbb{N})$, and discussed whether the simplicity (respectively chaoticity) of f implies the simplicity (respectively chaoticity) of $f_{1,\infty }$, where $f_{1,\infty }={\{f_n\}}_{n=1}^{\infty }$ converges uniformly to f. In 2012, Huang [5] studied the sensitivity of a special non-autonomous discrete system (named periodical discrete system). In 2015, Huang [6] extended the results of the sensitivity and strong sensitivity of ADDSs to NDDSs, and the conditions are weaker than those of ADDSs. In 2019, Vasisht [7] discussed the sensitivity of stronger forms via Furstenberg families in NDDSs, among which some examples are provided to illustrate the conclusions. In 2020, Kumar [8] studied the approximate controllability of specific non-autonomous second-order nonlinear differential problems with finite delay in the infinite dimensional space. Li [9] extended the general transitivity and sensitivity of non-autonomous systems to strong transitivity and strong sensitivity. Shao [10,11] established several criteria for strong Li–Yorke chaos and distributional chaos in non-autonomous discrete systems. Some other research about chaos or non-autonomous discrete systems includes [12,13,14,15,16,17,18], and others.

In this paper, collectively accessible, collectively sensitive, collectively infinitely sensitive, and collectively Li–Yorke sensitive classes of NDDSs are discussed, and some equivalence relations are presented. The main results are in Section 3, Section 4 and Section 5.

2. Preliminaries

Let $\mathbb{N}=\{1,2,\dots \}.$$h_n:H\to H$ ($n\in \mathbb{N}$, $(H,\rho )$ is a compact metric space) be a continuous mapping sequence denoted by $h_{1,\infty }=(h_1,h_2,\dots )={(h_n)}_{n=1}^{\infty }.$ This mapping sequence defines a non-autonomous discrete dynamical system $(H,h_{1,\infty })$. Under this sequence, the orbit of the point $x\in H$ is ${Orb}_{h_{1,\infty }}(x)=\{x,h_1(x),h_2\circ h_1(x),\dots ,h_1^n(x),\dots \}(n\in \mathbb{N})$, where $h_1^n=h_n\circ \dots \circ h_2\circ h_1$. Similarly, $h_n^m=h_m\circ \dots \circ h_{n+1}\circ h_n(m\ge n)$, and $h_1^0$ is the identity mapping.

In this section, the definitions related to sensitivity and accessibility will be given.

Definition 1.

The system$(H,h_{1,\infty })$is called collectively accessible, if for any$\epsilon >0$and any nonempty open subsets$A_1,A_2,\dots ,A_k,B_1,B_2,\dots ,B_k\subset H$, there exist$a_i\in A_i$for any$i\in \{1,2,\dots ,k\}$and$b_j\in B_j$for any$j\in \{1,2,\dots ,k\}$such that one of the following holds:

 (i) 

there is an$i_0\in \{1,2,\dots ,k\}$such that$\rho (h_1^n(a_{i_0}),h_1^n(b_j))<\epsilon$for any$j\in \{1,2,\dots ,k\}$and some integer$n>0$;

 (ii) 

there is a$j_0\in \{1,2,\dots ,k\}$such that$\rho (h_1^n(a_i),h_1^n(b_{j_0}))<\epsilon$for any$i\in \{1,2,\dots ,k\}$and some integer$n>0$.

Definition 2.

Let$(H,h_{1,\infty })$be a non-autonomous discrete dynamical system. Then$h_{1,\infty }$is said to be

 (1) 

sensitive if there is a$\eta >0$, for any$a\in H$and any$\epsilon >0$, there exist a$b\in B(a,\epsilon )$and$n\in \mathbb{N}$, such that$\rho (h_1^n(a),h_1^n(b))>\eta$.

 (2) 

collectively sensitive if there is an$\eta >0$, for any$\epsilon >0$and any finite number of different points$a_1,a_2,\dots ,a_k\in H$, the existence of k distinct points$b_1,b_2,\dots ,b_k\in H$makes the following two conditions true:

 (i) 

$\rho (a_i,b_i)<\epsilon$for any$1\le i\le k$;

 (ii) 

there exist$1\le i_0,j_0\le k$, and positive integer$n\in \mathbb{N}$such that

$$\rho (h_1^n(a_i),h_1^n(b_{j_0}))>\eta (1\le i\le k)or\rho (h_1^n(a_{i_0}),h_1^n(b_j))>\eta (1\le j\le k).$$

 (3) 

collectively infinitely sensitive if there is an$\eta >0$, for any$\epsilon >0$and any finite number of different points$a_1,a_2,\dots ,a_k\in H$, the existence of k distinct points$b_1,b_2,\dots ,b_k\in H$makes the following two conditions true:

 (i) 

$\rho (a_i,b_i)<\epsilon$for any$1\le i\le k$;

 (ii) 

there exist$1\le i_0,j_0\le k$such that

$$\underset{n\to \infty }{lim\; sup}\rho (h_1^n(a_i),h_1^n(b_{j_0}))>\eta (1\le i\le k)or\underset{n\to \infty }{lim\; sup}\rho (h_1^n(a_{i_0}),h_1^n(b_j))>\eta (1\le j\le k).$$

 (4) 

collectively Li–Yorke sensitive if there is an$\eta >0$, for any $\epsilon >0$and any finite number of different points$a_1,a_2,\dots ,a_k\in H$, the existence of k distinct points$b_1,b_2,\dots ,b_k\in H$makes the following two conditions true:

 (i) 

$\rho (a_i,b_i)<\epsilon$for any$1\le i\le k$;

 (ii) 

there exist$1\le i_0,j_0\le k$such that

$$\underset{n\to \infty }{lim\; sup}\rho (h_1^n(a_i),h_1^n(b_{j_0}))>\eta and\underset{n\to \infty }{lim\; inf}\rho (h_1^n(a_i),h_1^n(b_{j_0}))=0(1\le i\le k)$$

or

$$\underset{n\to \infty }{lim\; sup}\rho (h_1^n(a_{i_0}),h_1^n(b_j))>\eta and\underset{n\to \infty }{lim\; inf}\rho (h_1^n(a_{i_0}),h_1^n(b_j))=0(1\le j\le k).$$

Remark 1.

Obviously, when$k=1$, collectively sensitive is equivalent to sensitive. If$h_{1,\infty }$is collectively sensitive, then$h_{1,\infty }$must be sensitive.

3. Chaos of Mapping Sequences ${\mathit{h}}_{\mathbf{1},\infty }$ and ${\mathit{h}}_{\mathit{n},\infty }$

In this section, it is always assumed that $h_n(n\in \mathbb{N})$ are surjections.

Theorem 1.

If the mapping sequence$h_{1,\infty }=(h_1,h_2,\dots )$ is $\mathcal{W}$-chaotic, then the mapping sequence $h_{n,\infty }=(h_n,h_{n+1},\dots )$ is $\mathcal{W}$-chaotic for any $n\in \mathbb{N}$, where $\mathcal{W}$-chaos represents one of the following five properties: collectively accessible, sensitive, collectively sensitive, collectively infinitely sensitive, and collectively Li–Yorke sensitive.

Proof. 

In this case, it only needs to prove that $n=2$.

 (1) 

collectively accessible

Let any finite nonempty open subsets $A_1,A_2,\dots ,A_k,B_1,B_2,\dots ,B_k\subset H$. Since $h_1$ is surjective, taking an inverse image of each element in $A_i(i\in \{1,2,\dots ,k\})$ and $B_j(j\in \{1,2,\dots ,k\})$ under $h_1$, one can obtain $A_i^{*}(i\in \{1,2,\dots ,k\})$ and $B_j^{*}(j\in \{1,2,\dots ,k\})$ separately. Since $h_{1,\infty }$ is collectively accessible, then for any $\epsilon >0$ and finite nonempty open subsets $A_1^{*},A_2^{*},\dots ,A_k^{*},B_1^{*},B_2^{*},\dots ,B_k^{*}\subset H$, there exist $a_i^{*}\in A_i^{*}$ for any $i\in \{1,2,\dots ,k\}$ and $b_j^{*}\in B_j^{*}$ for any $j\in \{1,2,\dots ,k\}$ such that one of the following holds:

(i)

there is an $i_0\in \{1,2,\dots ,k\}$ such that $\rho (h_1^n(a_{i_0}^{*}),h_1^n(b_j^{*}))<\epsilon$ for any $j\in \{1,2,\dots ,k\}$ and some integer $n>0$;

(ii)

there is a $j_0\in \{1,2,\dots ,k\}$ such that $\rho (h_1^n(a_i^{*}),h_1^n(b_{j_0}^{*}))<\epsilon$ for any $i\in \{1,2,\dots ,k\}$ and some integer $n>0$.

Assume that condition (i) is held (the second condition is similar). Then there exist $a_{i_0}\in A_i$ and $b_j\in B_j$ such that $h_1(a_{i_0}^{*})=a_{i_0}$ and $h_1(b_j^{*})=b_j$ for any $j(j\in \{1,2,\dots ,k\})$. So, for any $\epsilon >0$ and any nonempty open subsets $A_1,A_2,\dots ,A_k,B_1,B_2,\dots ,B_k\subset H$, there exist $a_i\in A_i$ for any $i\in \{1,2,\dots ,k\}$, $b_j\in B_j$ for any $j\in \{1,2,\dots ,k\}$, and $i_0\in \{1,2,\dots ,k\}$ such that

$$\rho (h_2^n(a_{i_0}),h_2^n(b_j))=\rho (h_1^n(a_{i_0}^{*}),h_1^n(b_j^{*}))<\epsilon$$

for any $j\in \{1,2,\dots ,k\}$ and some integer $n>0$.

Thus, $h_{2,\infty }$ is collectively accessible.

 (2) 

sensitive

Similar to the proof of collectively sensitive.

 (3) 

collectively sensitive

Let any $k(k\in \mathbb{N})$ different points $a_1,a_2,\dots ,a_k\in H$. Since $h_1$ is a surjective, then there exists $a_i^{*}\in H$, such that $h_1(a_i^{*})=a_i$, for any $1\le i\le k$. Since $h_{1,\infty }$ is collectively sensitive, then there is an $\eta >0$, for any $\epsilon >0$, and the finite number of points $a_1^{*},a_2^{*},\dots ,a_k^{*}\in H$, there exist k points $b_1^{*},b_2^{*},\dots ,b_k^{*}\in H$ such that:

(i)

$\rho (a_i^{*},b_i^{*})<\epsilon$ for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$, and positive integer $n\in \mathbb{N}$ such that

$$\rho (h_1^n(a_i^{*}),h_1^n(b_{j_0}^{*}))>\eta (1\le i\le k)\mathrm{or}\rho (h_1^n(a_{i_0}^{*}),h_1^n(b_j^{*}))>\eta (1\le j\le k).$$

Without loss of generality, assume that there exist $1\le i_0,j_0\le k$, and positive integer $n\in \mathbb{N}$ such that $\rho (h_1^n(a_i^{*}),h_1^n(b_{j_0}^{*}))>\eta (1\le i\le k)$.

Claim. $Q=\{n\in \mathbb{N}\mid \rho (h_1^n(a_i^{*}),h_1^n(b_{j_0}^{*}))>\eta (1\le i\le k)\}$ is an infinite set.

Hypothesis Q is a finite set, then there must be an $m\in \mathbb{N}$ such that $m\ge max\{Q\}$. Since $h_n(n\in \mathbb{N})$ are continuous maps on a compact metric space, then $h_n(n\in \mathbb{N})$ is uniformly continuous on H. So $h_1^i(i\in \{1,2,\dots ,m\})$ is uniformly continuous on H. Then, for any $\epsilon >0$, there is a $\delta >0$ such that, for any $x,y\in H$: $\rho (x,y)<\delta$, $\rho (h_1^i(x),h_1^i(y))<\epsilon$ for any $i\in \{1,2,\dots ,m\}$. Thus,

when $i=1$, there is a ${\delta }_1>0$ such that $\rho (h_1(x),h_1(y))<\epsilon$ for any $x,y\in H:\rho (x,y)<{\delta }_1$;

when $i=2$, there is a ${\delta }_2>0$ such that $\rho (h_1^2(x),h_1^2(y))<\epsilon$ for any $x,y\in H:\rho (x,y)<{\delta }_2$;

…

when $i=m$, there is ${\delta }_m>0$ such that $\rho (h_1^m(x),h_1^m(y))<\epsilon$ for any $x,y\in H:\rho (x,y)<{\delta }_m$.

These contradict that $h_{1,\infty }$ is collectively sensitive. So, Q is an infinite set.

Thus, there exist $b_j\in H$ such that $h_1(b_j^{*})=b_j$ for any $1\le j\le k$. Since $h_1$ is a continuous mapping on a compact metric space H, then $h_1$ is uniformly continuous on H. By the definition of continuous, $\rho (h_1(a_i^{*}),h_1(b_i^{*}))=\rho (a_i,b_i)<\epsilon$ for any $1\le i\le k$ and, because $\rho (h_2^n(a_i),h_2^n(b_j))=\rho (h_1^n(a_i^{*}),h_1^n(b_j^{*})$ for any $1\le i,j\le k$, then there is an $\eta >0$, for any $\epsilon >0$ and any finite number of different points $a_1,a_2,\dots ,a_k\in H$, there exist k distinct points $b_1,b_2,\dots ,b_k\in H$ such that:

(i)

$\rho (a_i,b_i)<\epsilon$ for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$, and positive integer $n\in \mathbb{N}$ such that

$$\rho (h_2^n(a_i),h_2^n(b_{j_0}))>\eta (1\le i\le k)\mathrm{or}\rho (h_2^n(a_{i_0}),h_2^n(b_j))>\eta (1\le j\le k).$$

Therefore, $h_{2,\infty }$ is collectively sensitive.

 (4) 

collectively infinitely sensitive

Similar to the proof below.

 (5) 

collectively Li–Yorke sensitive

Let any finite number of different points $a_1,a_2,\dots ,a_k\in H$. Since $h_1$ is surjective, then there exist $a_i^{*}\in H$ such that $h_1(a_i^{*})=a_i$ for any $1\le i\le k$. And because $h_{1,\infty }$ is collectively Li–Yorke sensitive, then there is an $\eta >0$, for any $\epsilon >0$ and the finite number of points $a_1^{*},a_2^{*},\dots ,a_k^{*}\in H$, the existence of k points $b_1^{*},b_2^{*},\dots ,b_k^{*}\in H$ makes the following two conditions true:

(i)

$\rho (a_i^{*},b_i^{*})<\epsilon$ for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$ such that

$$\underset{n\to \infty }{lim\; sup}\rho (h_1^n(a_i^{*}),h_1^n(b_{j_0}^{*}))>\eta \mathrm{and}\underset{n\to \infty }{lim\; inf}\rho (h_1^n(a_i^{*}),h_1^n(b_{j_0}^{*}))=0(1\le i\le k)$$

or

$$\underset{n\to \infty }{lim\; sup}\rho (h_1^n(a_{i_0}^{*}),h_1^n(b_j^{*}))>\eta \mathrm{and}\underset{n\to \infty }{lim\; inf}\rho (h_1^n(a_{i_0}^{*}),h_1^n(b_j^{*}))=0(1\le j\le k).$$

Then, there exist $b_j\in H$ such that $h_1(b_j^{*})=b_j$ for any $1\le j\le k$. Due to $h_1$ is uniformly continuous on H, then $\rho (h_1(a_i^{*}),h_1(b_i^{*}))=\rho (a_i,b_i)<\epsilon$ for any $1\le i\le k$. And because $\rho (h_2^n(a_i),h_2^n(b_j)=\rho (h_1^n(a_i^{*}),h_1^n(b_j^{*})$ for any $1\le i,j\le k$, then there is an $\eta >0$, for any $\epsilon >0$ and any finite number of different points $a_1,a_2,\dots ,a_k\in H$, the existence of k distinct points $b_1,b_2,\dots ,b_k\in H$ makes the following two conditions true:

(i)

$\rho (a_i,b_i)<\epsilon$ for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$ such that

$$\underset{n\to \infty }{lim\; sup}\rho (h_2^n(a_i),h_2^n(b_{j_0}))>\eta \mathrm{and}\underset{n\to \infty }{lim\; inf}\rho (h_2^n(a_i),h_2^n(b_{j_0}))=0(1\le i\le k)$$

or

$$\underset{n\to \infty }{lim\; sup}\rho (h_2^n(a_{i_0}),h_2^n(b_j))>\eta \mathrm{and}\underset{n\to \infty }{lim\; inf}\rho (h_2^n(a_{i_0}),h_2^n(b_j))=0(1\le j\le k).$$

Thus, $h_{2,\infty }$ is collectively Li–Yorke sensitive.

The proof is completed. □

The inverse of Theorem 1 should be considered. It is found that only the collective accessibility is valid, while the other four properties are not. An example is used to illustrate this conclusion.

Example 1.

Let $H=[0,1]$,

$$h_1(x)=\left\{\begin{array}{cc}4x& forx\in \left[0,\frac{1}{4}\right]\\ 1& forx\in \left[\frac{1}{4},\frac{1}{2}\right]\\ -2x+2& forx\in \left[\frac{1}{2},1\right]\end{array}\right.$$

$$h(x)=\left\{\begin{array}{cc}3x& forx\in \left[0,\frac{1}{3}\right]\\ -3x+2& forx\in \left[\frac{1}{3},\frac{2}{3}\right]\\ 3x-2& forx\in \left[\frac{2}{3},1\right]\end{array}\right.$$

for any $n\in \mathbb{N}$. $h_{1,\infty }=(h_1,h,h,h,\dots )$.

Obviously, h is a triangle-tent map, the images of $h^n$ oscillate between 0 and 1. The larger the $n\in \mathbb{N}$, the denser the oscillation interval. Then for any nonempty open set $U\subset H$, there is a large enough positive integer n such that $h_1^n(U)$ covers $[0,1]$. Especially, for any $a\in H$ and $\epsilon >0$, there is an $n\in \mathbb{N}$ such that $[0,1]$ is covered by $h_1^n(B(a,\epsilon ))$. That is to say, there exist an $\eta >0$ and a $b\in B(a,\epsilon )$ such that $\rho (h_1^n(a),h_1^n(b))>\eta$. Then, h is sensitive. Thus, $h_{2,\infty }$ is sensitive.

Take $a=\frac{1}{3}$ and $\epsilon =\frac{1}{16}$. For any $b\in B(a,\epsilon )$, and $n\in \mathbb{N}$, one can deduce that $\rho (h_1^n(a),h_1^n(b))=0$. So, $h_{1,\infty }$ is not sensitive.

Theorem 2.

If the mapping sequence$h_{n,\infty }=(h_n,h_{n+1},\dots )$is collectively accessible, then the mapping sequence$h_{1,\infty }=(h_1,h_2,\dots )$is collectively accessible for any$n\in \mathbb{N}$.

Proof. 

Take any finite nonempty open subsets $A_1,A_2,\dots ,A_k,B_1,B_2,\dots ,B_k$ of H. Since $h_1$ is surjective, taking an image of each element in $A_i(i\in \{1,2,\dots ,k\})$ and $B_j(j\in \{1,2,\dots ,k\})$ under $f_1$, one can obtain $A_i^{*}(i\in \{1,2,\dots ,k\})$ and $B_j^{*}(j\in \{1,2,\dots ,k\})$ separately. Since $h_{2,\infty }$ is collective accessibility, then for any $\epsilon >0$ and for finite nonempty open subsets $A_1^{*},A_2^{*},\dots ,A_s^{*},B_1^{*},B_2^{*},\dots ,B_k^{*}\subset H$, there exist $a_i^{*}\in A_i^{*}$ for any $i\in \{1,2,\dots ,k\}$ and $b_j^{*}\in B_j^{*}$ for any $j\in \{1,2,\dots ,k\}$ such that one of the following holds:

(i)

there is an $i_0\in \{1,2,\dots ,k\}$ such that $\rho (h_2^n(a_{i_0}^{*}),h_2^n(b_j^{*}))<\epsilon$ for any $j\in \{1,2,\dots ,k\}$ and some integer $n>0$;

(ii)

there is a $j_0\in \{1,2,\dots ,k\}$ such that $\rho (h_2^n(a_i^{*}),h_2^n(b_{j_0}^{*}))<\epsilon$ for any $i\in \{1,2,\dots ,k\}$ and some integer $n>0$.

The case that condition (i) was held is going to be proved.

In fact, there exist $a_{i_0}\in A_i$ and $b_j\in B_j$ such that $h_1(a_{i_0})=a_{i_0}^{*}$ and $h_1(b_j)=b_j^{*}$ for each $j(j\in \{1,2,\dots ,k\})$. So, for any $\epsilon >0$ and any nonempty open subsets $A_1,A_2,\dots ,A_k,B_1,B_2,\dots ,B_k\subset H$, there exist $a_i\in A_i$ for any $i\in \{1,2,\dots ,k\}$, $b_j\in B_j$ for any $j\in \{1,2,\dots ,k\}$, and $i_0\in \{1,2,\dots ,k\}$ such that

$$\rho (h_1^n(a_{i_0}),h_1^n(b_j))=\rho (h_2^n(a_{i_0}^{*}),h_2^n(b_j^{*}))<\epsilon$$

for any $j\in \{1,2,\dots ,k\}$ and some integer $n>0$.

Thus, $h_{1,\infty }$ is collectively accessible. □

According to the proof of Theorems 1 and 2, the following corollary can be obtained.

Corollary 1.

Let${(h_n)}_{n=1}^{\infty }=(h_1,h_2,\dots ,h_n,\dots )$be a mapping sequence on metric space$(H,\rho )$. If $h_{1,\infty }$ is $\mathcal{W}$-chaotic, then there exists an $n\in \mathbb{N}$ such that $h_{n,\infty }$ is $\mathcal{W}$-chaotic.

Corollary 2.

Let${(h_n)}_{n=1}^{\infty }=(h_1,h_2,\dots ,h_n,\dots )$be a mapping sequence on metric space$(H,\rho )$. If$h_{n,\infty }$is collectively accessible, then there exists an$n\in \mathbb{N}$such that$h_{1,\infty }$is collectively accessible.

4. Chaos of the Product System $({\mathit{H}}_{\mathbf{1}}\times {\mathit{H}}_{\mathbf{2}},{\mathit{f}}_{\mathbf{1},\infty }\times {\mathit{g}}_{\mathbf{1},\infty })$

Let $f_{1,\infty }={(f_n)}_{n=1}^{\infty }$, $g_{1,\infty }={(g_n)}_{n=1}^{\infty }$ be the continuous mapping sequences on compact metric spaces $(H_1,{\rho }_1)$ and $(H_2,{\rho }_2)$, respectively, where $(H_1,{\rho }_1)$ and $(H_2,{\rho }_2)$ are compact metric spaces. $(H_1\times H_2,f_{1,\infty }\times g_{1,\infty })$ is called the product of $(H_1,f_{1,\infty })$ and $(H_2,g_{1,\infty })$. For any $(x_1,y_1),(x_2,y_2)\in H_1\times H_2$, define

$$\rho ((x_1,y_1),(x_2,y_2))=max\{{\rho }_1(x_1,x_2),{\rho }_2(y_1,y_2)\}$$

as the metric on $H_1\times H_2.$

Theorem 3.

If$f_{1,\infty }\times g_{1,\infty }$is collectively accessible, then$f_{1,\infty }$ and $g_{1,\infty }$ are collectively accessible.

Proof. 

Assume that $f_{1,\infty }\times g_{1,\infty }$ is collectively accessible, then for any $\epsilon >0$ and any nonempty open subsets $A_1\times B_1,A_2\times B_2,\dots ,A_k\times B_k,A_1^{*}\times B_1^{*},A_2^{*}\times B_2^{*},\dots ,A_k^{*}\times B_k^{*}\subset H_1\times H_2$, there exist $(a_i,b_i)\in A_i\times B_i$ for any $i\in \{1,2,\dots ,k\}$ and $(a_j^{*},b_j^{*})\in A_j^{*}\times B_j^{*}$ for any $j\in \{1,2,\dots ,k\}$ such that one of the following holds:

(i) there is an $i_0\in \{1,2,\dots ,k\}$ such that $\rho ((f_1^n\times g_1^n)(a_{i_0},b_{i_0}),(f_1^n\times g_1^n)(a_j^{*},b_j^{*}))<\epsilon$ for any $j\in \{1,2,\dots ,k\}$ and some integer $n>0$.

(ii) there is a $j_0\in \{1,2,\dots ,k\}$ such that $\rho ((f_1^n\times g_1^n)(a_i,b_i),(f_1^n\times g_1^n)(a_{j_0}^{*},b_{j_0}^{*}))<\epsilon$ for any $i\in \{1,2,\dots ,k\}$ and some integer $n>0$.

Without loss of generality, it is only necessary to deal with the case that condition (i) is held (the second case is similar). Since

$$\begin{array}{cc}& \rho ((f_1^n\times g_1^n)(a_{i_0},b_{i_0}),(f_1^n\times g_1^n)(a_j^{*},b_j^{*}))\hfill \\ & =max\{{\rho }_1(f_1^n(a_{i_0}),f_1^n(a_j^{*})),{\rho }_2(g_1^n(b_{i_0}),g_1^n(b_j^{*}))\}\hfill \\ & <\epsilon \hfill \end{array}$$

for any $j\in \{1,2,\dots ,k\}$ and some integer $n>0$, then,

$${\rho }_1(f_1^n(a_{i_0}),f_1^n(a_j^{*}))<\epsilon and{\rho }_2(g_1^n(b_{i_0}),g_1^n(b_j^{*}))<\epsilon .$$

Thus, for any $\epsilon >0$ and any nonempty open subsets $A_1,A_2,\dots ,A_k,A_1^{*},A_2^{*},\dots ,A_k^{*}\subset H_1$, there exist $a_i\in A_i$ for any $i\in \{1,2,\dots ,k\}$, $a_j^{*}\in A_j^{*}$ for any $j\in \{1,2,\dots ,k\}$ and an $i_0\in \{1,2,\dots ,k\}$ such that ${\rho }_1(f_1^n(a_{i_0}),f_1^n(a_j^{*})<\epsilon$ for any $j\in \{1,2,\dots ,k\}$ and some integer $n>0$. Therefore $f_{1,\infty }$ is collectively accessible. And by the same discussion, $g_{1,\infty }$ is collectively accessible. So, $f_{1,\infty }$ and $g_{1,\infty }$ is collectively accessible.

The proof is completed. □

Theorem 4.

If$f_{1,\infty }\times g_{1,\infty }$is sensitive (resp., collectively sensitive, collectively infinitely sensitive, collectively Li–Yorke sensitive), then$f_{1,\infty }$or$g_{1,\infty }$is sensitive (resp., collectively sensitive, collectively infinitely sensitive, collectively Li–Yorke sensitive).

Proof. 

(1) sensitive

Similar to the proof of collectively sensitive.

 (2) 

collectively sensitive

Assume that $f_{1,\infty }\times g_{1,\infty }$ is collectively sensitive, then there is an $\eta >0$, for any $\epsilon >0$ and any finite number of different points $(a_1,b_1),(a_2,b_2),\dots ,(a_k,b_k)\in H_1\times H_2$, there exist k distinct points $(a_1^{*},b_1^{*}),(a_2^{*},b_2^{*}),\dots ,(a_k^{*},b_k^{*})\in H_1\times H_2$ such that:

(i)

$\rho ((a_i,b_i),(a_i^{*},b_i^{*}))<\epsilon$ for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$, and positive integer $n\in \mathbb{N}$ such that

$$\rho (f_1^n\times g_1^n(a_i,b_i),f_1^n\times g_1^n(a_{j_0}^{*},b_{j_0}^{*}))>\eta (1\le i\le k)$$

or

$$\rho (f_1^n\times g_1^n(a_{i_0},b_{i_0}),f_1^n\times g_1^n(a_j^{*},b_j^{*}))>\eta (1\le j\le k).$$

Then

$$\begin{array}{cc}& \rho ((a_i,b_i),(a_i^{*},b_i^{*}))\hfill \\ & =max\{{\rho }_1(a_i,a_i^{*}),{\rho }_2(b_i,b_i^{*})\}\hfill \\ & <\epsilon .\hfill \end{array}$$

That is to say, ${\rho }_1((a_i,a_i^{*})<\epsilon$ and ${\rho }_2(b_i,b_i^{*}))<\epsilon$ for any $1\le i\le k$. And by (ii), one has

$$\begin{array}{cc}& \rho (f_1^n\times g_1^n(a_i,b_i),f_1^n\times g_1^n(a_{j_0}^{*},b_{j_0}^{*}))\hfill \\ & =max\{{\rho }_1(f_1^n(a_i),f_1^n(a_{j_0}^{*})),{\rho }_2(g_1^n(b_i),g_1^n(b_{j_0}^{*}))\}\hfill \\ & >\eta \hfill \end{array}$$

or

$$\begin{array}{cc}& \rho (f_1^n\times g_1^n(a_{i_0},b_{i_0}),f_1^n\times g_1^n(a_j^{*},b_j^{*}))\hfill \\ & =max\{{\rho }_1(f_1^n(a_{i_0}),f_1^n(a_j^{*})),{\rho }_2(g_1^n(b_{i_0}),g_1^n(b_j^{*}))\}\hfill \\ & >\eta .\hfill \end{array}$$

So, there exist $1\le i_0,j_0\le k$, and positive integer $n\in \mathbb{N}$ such that

$${\rho }_1(f_1^n(a_i),f_1^n(a_{j_0}^{*}))>\eta or{\rho }_2(g_1^n(b_i),g_1^n(b_{j_0}^{*}))>\eta (1\le i\le k);$$

$${\rho }_1(f_1^n(a_{i_0}),f_1^n(a_j^{*}))>\eta or{\rho }_2(g_1^n(b_{i_0}),g_1^n(b_j^{*}))>\eta (1\le j\le k).$$

Thus, $f_{1,\infty }$ or $g_{1,\infty }$ is collectively sensitive.

 (3) 

collectively infinitely sensitive

Similar to the proof below.

 (4) 

collectively Li–Yorke sensitive

Assume that $f_{1,\infty }\times g_{1,\infty }$ is collectively Li–Yorke sensitive, then there is an $\eta >0$, for any $\epsilon >0$ and any finite number of different points $(a_1,b_1),(a_2,b_2),\dots ,(a_k,b_k)\in H_1\times H_2$, the existence of k distinct points $(a_1^{*},b_1^{*}),(a_2^{*},b_2^{*}),\dots ,(a_k^{*},b_k^{*})\in H_1\times H_2$ makes the following two conditions true:

(i)

$\rho ((a_i,b_i),(a_i^{*},b_i^{*}))<\epsilon$ for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$ such that

$$\underset{n\to \infty }{lim\; sup}\rho ((f_1^n\times g_1^n)(a_i,b_i),(f_1^n\times g_1^n)(a_{j_0}^{*},b_{j_0}^{*}))>\eta$$

and

$$\underset{n\to \infty }{lim\; inf}\rho ((f_1^n\times g_1^n)(a_i,b_i),(f_1^n\times g_1^n)(a_{j_0}^{*},b_{j_0}^{*}))=0(1\le i\le k),$$

or

$$\underset{n\to \infty }{lim\; sup}\rho ((f_1^n\times g_1^n)(a_{i_0},b_{i_0}),(f_1^n\times g_1^n)(a_j^{*},b_j^{*}))>\eta$$

and

$$\underset{n\to \infty }{lim\; inf}\rho ((f_1^n\times g_1^n)(a_{i_0},b_{i_0}),(f_1^n\times g_1^n)(a_j^{*},b_j^{*}))=0(1\le j\le k).$$

Then,

$$\begin{array}{cc}& \rho ((a_i,b_i),(a_i^{*},b_i^{*}))\hfill \\ & =max\{{\rho }_1(a_i,a_i^{*}),{\rho }_2(b_i,b_i^{*})\}\hfill \\ & <\epsilon .\hfill \end{array}$$

So, ${\rho }_1((a_i,a_i^{*})<\epsilon$ and ${\rho }_2(b_i,b_i^{*}))<\epsilon$ for any $1\le i\le k$.

According to (ii),

$$\begin{array}{cc}& \underset{n\to \infty }{lim\; sup}\rho ((f_1^n\times g_1^n)(a_i,b_i),(f_1^n\times g_1^n)(a_{j_0}^{*},b_{j_0}^{*}))\hfill \\ & =\underset{n\to \infty }{lim\; sup}max\{{\rho }_1(f_1^n(a_i),f_1^n(a_{j_0}^{*})),{\rho }_2(g_1^n(b_i),g_1^n(b_{j_0}^{*}))\}\hfill \\ & >\eta \hfill \end{array}$$

and

$$\begin{array}{cc}& \underset{n\to \infty }{lim\; inf}\rho (f_1^n\times g_1^n(a_i,b_i),f_1^n\times g_1^n(a_{j_0}^{*},b_{j_0}^{*}))\hfill \\ & =\underset{n\to \infty }{lim\; inf}max\{{\rho }_1(f_1^n(a_i),f_1^n(a_{j_0}^{*})),{\rho }_2(g_1^n(b_i),g_1^n(b_{j_0}^{*}))\}\hfill \\ & =0.\hfill \end{array}$$

So, there exist $1\le i_0,j_0\le k$, and positive integer $n\in \mathbb{N}$ such that

$$\underset{n\to \infty }{lim\; sup}{\rho }_1(f_1^n(a_i),f_1^n(a_{j_0}^{*}))>\eta and\underset{n\to \infty }{lim\; inf}{\rho }_1(f_1^n(a_i),f_1^n(a_{j_0}^{*}))=0,$$

or

$$\underset{n\to \infty }{lim\; sup}{\rho }_2(g_1^n(b_i),g_1^n(b_{j_0}^{*}))>\eta and\underset{n\to \infty }{lim\; sup}{\rho }_2(g_1^n(b_i),g_1^n(b_{j_0}^{*}))=0.$$

Thus, $f_{1,\infty }$ or $g_{1,\infty }$ is collectively Li–Yorke sensitive.

The proof is completed.

□

The inverse of Theorem 4 should be considered.

Theorem 5.

If$f_{1,\infty }$ or $g_{1,\infty }$ is sensitive (resp., collectively sensitive, collectively infinitely sensitive), then $f_{1,\infty }\times g_{1,\infty }$ is sensitive (resp., collectively sensitive, collectively infinitely sensitive).

Proof. 

 (1) sensitive

Similar to the proof of collectively sensitive.

 (2) 

collectively sensitive

Assume that $f_{1,\infty }$ is collectively sensitive. Then there is an $\eta >0$, for any $\epsilon >0$ and any finite number of different points $a_1,a_2,\dots ,a_k\in H_1$, there exist k distinct points $a_1^{*},a_2^{*},\dots ,a_k^{*}\in H_1$ such that:

(i)

${\rho }_1(a_i,a_i^{*})<\epsilon$ for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$, and positive integer $m\in \mathbb{N}$ such that

$${\rho }_1(f_1^m(a_i),f_1^m(a_{j_0}^{*}))>\eta (1\le i\le k)\mathrm{or}{\rho }_1(f_1^m(a_{i_0}),f_1^m(a_j^{*}))>\eta (1\le j\le k).$$

For any finite number of different points $b_1,b_2,\dots ,b_k\in H_2$, there exist k points $b_1^{*},b_2^{*},\dots ,b_k^{*}\in H_2$ such that ${\rho }_2(b_i,b_i^{*})<\epsilon$ for any $1\le i\le k$. Then,

$$\begin{array}{cc}& \rho ((a_i,b_i),(a_i^{*},b_i^{*}))\hfill \\ & =max\{{\rho }_1(a_i,a_i^{*}),{\rho }_2(b_i,b_i^{*})\}\hfill \\ & <\epsilon \hfill \end{array}$$

for any $1\le i\le k$.

Take $n=m$, then

$$\begin{array}{cc}& \rho ((f_1^n\times g_1^n)(a_i,b_i),(f_1^n\times g_1^n)(a_{j_0}^{*},b_{j_0}^{*}))\hfill \\ & =max\{{\rho }_1(f_1^n(a_i),f_1^n(a_{j_0}^{*})),{\rho }_2(g_1^n(b_i),g_1^n(b_{j_0}^{*}))\}\hfill \\ & >\eta .\hfill \end{array}$$

Similarly, $\rho ((f_1^n\times g_1^n)(a_{i_0},b_{i_0}),(f_1^n\times g_1^n)(a_j^{*},b_j^{*}))\ge \eta (1\le j\le k)$ can be obtained.

Therefore, for any $\epsilon >0$ and any finite number of different points $(a_1,b_1),(a_2,b_2),\dots ,(a_k,b_k)\in H_1\times H_2$, there exist $(a_1^{*},b_1^{*}),(a_2^{*},b_2^{*}),\dots ,(a_k^{*},b_k^{*})\in H_1\times H_2$ such that:

(i)

$\rho ((a_i,b_i),(a_i^{*},b_i^{*}))<\epsilon$ for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$, and positive integer $n\in \mathbb{N}$ such that

$$\rho (f_1^n\times g_1^n(a_i,b_i),f_1^n\times g_1^n(a_{j_0}^{*},b_{j_0}^{*}))>\eta (1\le i\le k)$$

or

$$\rho (f_1^n\times g_1^n(a_{i_0},b_{i_0}),f_1^n\times g_1^n(a_j^{*},b_j^{*}))>\eta (1\le j\le k).$$

So, $f_{1,\infty }\times g_{1,\infty }$ is collectively sensitive.

 (3) 

collectively infinitely sensitive

Assume that $f_{1,\infty }$ is collectively infinitely sensitive. Then there is an $\eta >0$, for any $\epsilon >0$ and any finite number of different points $a_1,a_2,\dots ,a_k\in H_1$, the existence of k distinct points $a_1^{*},a_2^{*},\dots ,a_k^{*}\in H_1$ makes the following two conditions true:

(i)

${\rho }_1(a_i,a_i^{*})<\epsilon$ for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$ such that

$$\underset{n\to \infty }{lim\; sup}{\rho }_1(f_1^m(a_i),f_1^m(a_{j_0}^{*}))>\eta \mathrm{or}\underset{n\to \infty }{lim\; sup}{\rho }_1(f_1^m(a_{i_0}),f_1^m(a_j^{*}))>\eta .$$

For any finite number of different points $b_1,b_2,\dots ,b_k\in H_2$, there exist k points $b_1^{*},b_2^{*},\dots ,b_k^{*}\in H_2$ such that ${\rho }_2(b_i,b_i^{*})<\epsilon$ for any $1\le i\le k$. Then,

$$\begin{array}{cc}& \rho ((a_i,b_i),(a_i^{*},b_i^{*}))\hfill \\ & =max\{{\rho }_1(a_i,a_i^{*}),{\rho }_2(b_i,b_i^{*})\}\hfill \\ & <\epsilon \hfill \end{array}$$

for any $1\le j\le k$.

Take $n=m$, then

$$\begin{array}{cc}& \underset{n\to \infty }{lim\; sup}\rho ((f_1^n\times g_1^n)(a_{i_0},b_{i_0}),(f_1^n\times g_1^n)(a_j^{*},b_j^{*}))\hfill \\ & =\underset{n\to \infty }{lim\; sup}max\{{\rho }_1(f_1^n(a_{i_0}),f_1^n(a_j^{*})),{\rho }_2(g_1^n(b_{i_0}),g_1^n(b_j^{*}))\}\hfill \\ & >\eta .\hfill \end{array}$$

Therefore, for any $\epsilon >0$ and any finite number of different points $(a_1,b_1),(a_2,b_2),\dots ,(a_k,b_k)\in H_1\times H_2$, the existence of k points $(a_1^{*},b_1^{*}),(a_2^{*},b_2^{*}),\dots ,(a_k^{*},b_k^{*})\in H_1\times H_2$ makes the following two conditions true:

(i)

$\rho ((a_i,b_i),(a_i^{*},b_i^{*}))<\epsilon$ for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$ such that

$$\underset{n\to \infty }{lim\; sup}\rho ((f_1^n\times g_1^n)(a_i,b_i),(f_1^n\times g_1^n)(a_{j_0}^{*},b_{j_0}^{*}))>\eta$$

or

$$\underset{n\to \infty }{lim\; sup}\rho ((f_1^n\times g_1^n)(a_{i_0},b_{i_0}),(f_1^n\times g_1^n)(a_j^{*},b_j^{*}))>\eta .$$

So, $f_{1,\infty }\times g_{1,\infty }$ is collectively sensitive.

The proof is completed. □

5. Chaos of the $p$-Periodic Discrete System $(H,f_{\mathbf{1},\infty })$

If an NDDS satisfies:

$$h_{n+p}(x)=h_n(x),\forall x\in H,n\ge 0,p\in \mathbb{N},$$

then $(H,h_{1,\infty })$ is called a p-periodic discrete system. Particularly, if $p=1,(H,h_{1,\infty })$ is an autonomous discrete system. Let $\widehat{h}=h_p\circ \dots \circ h_1$, then $(H,\widehat{h})$ is said to be an ADDS induced by the p-periodic discrete system $(H,h_{1,\infty })$ (see [5]).

Theorem 6.

The autonomous discrete system$(X,\widehat{h})$induced by the p-periodic discrete system$(H,h_{1,\infty })$is$\mathcal{W}$-chaotic, then the p-periodic discrete system$(H,f_{1,\infty })$is$\mathcal{W}$-chaotic, where$\mathcal{W}$-chaos represents one of the following five properties: collectively accessible, sensitive, collectively sensitive, collectively infinitely sensitive, and collectively Li–Yorke sensitive.

Proof. 

In fact, for a p-periodic discrete system, using $h_1^{np}={(h_1^p)}^n$, these conclusions can be easily obtained. The proof of collectively Li–Yorke sensitivity is given below, and the proofs of other chaotic properties are similar.

Assume that $(H,\widehat{h})$ is collectively Li–Yorke sensitive, then there is an $\eta >0$, for any $\epsilon >0$ and any finite number of different points $a_1,a_2,\dots ,a_k\in H$, the existence of k distinct points $b_1,b_2,\dots ,b_k\in H$ makes the following two conditions true:

(i)

such that $\rho (a_i,b_i)<\epsilon$, for any $1\le i\le k$;

(ii)

there exist $1\le i_0,j_0\le k$ such that

$$\underset{n\to \infty }{lim\; sup}\rho ({\widehat{h}}_1^n(a_i),{\widehat{h}}_1^n(b_{j_0}))>\eta \mathrm{and}\underset{n\to \infty }{lim\; inf}\rho ({\widehat{h}}_1^n(a_i),{\widehat{h}}_1^n(b_{j_0}))=0(1\le i\le k),$$

or

$$\underset{n\to \infty }{lim\; sup}\rho ({\widehat{h}}_1^n(a_{i_0}),{\widehat{h}}_1^n(b_j))>\eta \mathrm{and}\underset{n\to \infty }{lim\; inf}\rho ({\widehat{h}}_1^n(a_{i_0}),{\widehat{h}}_1^n(b_j))=0(1\le j\le k).$$

Since

$$\underset{n\to \infty }{lim\; sup}\rho ({\widehat{h}}_1^n(a_i),h_1^n(b_j))=\underset{n\to \infty }{lim\; sup}\rho (h_1^{np}(a_i),h_1^{np}(b_j))\le \underset{n\to \infty }{lim\; sup}\rho (h_1^n(a_i),h_1^n(b_j))$$

and

$$\underset{n\to \infty }{lim\; inf}\rho ({\widehat{h}}_1^n(a_i),h_1^n(b_j))=\underset{n\to \infty }{lim\; inf}\rho (h_1^{np}(a_i),h_1^{np}(b_j))\ge \underset{n\to \infty }{lim\; inf}\rho (h_1^n(a_i),h_1^n(b_j))$$

for any $i,j\in \{1,2,\dots ,k\}$, then

$$\underset{n\to \infty }{lim\; sup}\rho (h_1^n(a_i),h_1^n(b_{j_0}))>\eta \mathrm{and}\underset{n\to \infty }{lim\; inf}\rho (h_1^n(a_i),h_1^n(b_{j_0}))=0(1\le i\le k),$$

or

$$\underset{n\to \infty }{lim\; sup}\rho (h_1^n(a_{i_0}),h_1^n(b_j))>\eta \mathrm{and}\underset{n\to \infty }{lim\; inf}\rho (h_1^n(a_{i_0}),h_1^n(b_j))=0(1\le j\le k).$$

Thus $(H,h_{1,\infty })$ is collectively Li–Yorke sensitive.

The proof is completed. □

Example 2.

Let$H=[0,1]$. Two mappings$\varphi (x)$, h(x) be defined by$\varphi (x)=4x(1-x)$for$x\in [0,1]$and h(x)$=\left\{\begin{array}{cc}3x& forx\in \left[0,\frac{1}{3}\right]\\ -3x+2& forx\in \left[\frac{1}{3},\frac{2}{3}\right]\\ 3x-2& forx\in \left[\frac{2}{3},1\right]\end{array}\right.$

Let ${(g_n)}_{n=1}^{\infty }=\{\varphi ,h,\varphi ,h,\varphi ,h,\dots \}$ then $\widehat{g}=g_2\circ g_1$. The function images of $\widehat{g}$ and ${\widehat{g}}^2$ are in Figure 1. Then the following conclusion (Proposition 1) can be obtained.

Proposition 1.

The mapping$\widehat{g}$is sensitive.

Proof. 

From Figure 1 (The function image of ${\widehat{g}}^n$ can be inferred), it can be seen that, for any $a\in H$ and $\epsilon >0$, there is an $n\in \mathbb{N}$ such that $[0,1]$ is covered by ${\widehat{g}}^n(B(a,\epsilon ))$. So, there exists an $\eta >0$ and a $b\in B(a,\epsilon )$ such that $\rho ({\widehat{g}}^n(a),{\widehat{g}}^n(b))>\eta$ (similar to the proof of Example 1). Thus, $\widehat{g}$ is sensitive. □

The numerical simulation diagram of $\widehat{g}$ is shown in Figure 2. The green dots and red dots represent the trajectories of 5000 iterations under the initial values $x_1=0.50001$ and $x_2=0.50002$, respectively.

After iteration, it can be seen from Figure 2 that the trajectory of $x_1$ (or $x_2$) is ergodic and disordered. $\rho (x_1,x_2)=0.00001$, but $\rho (\widehat{g}(x_1),\widehat{g}(x_2))=0.7393$. This means that the initial distance between $x_1$ and $x_2$ is very minimal, but after 3423 iterations, the distance of iteration points is large (the distance has increased more than 70,000 times). This phenomenon is a characterization of sensitivity.

Now, the function images of $g_1^2$, $g_1^3$, and $g_1^4$ are given in Figure 3, and the image of $g_1^n(n>4)$ can be inferred. Then, the following result is true (see Proposition 2).

Proposition 2.

The mapping sequence$g_{1,\infty }$is sensitive.

Proof. 

Similar to the proof of Proposition 1, this conclusion can also be obtained. □