## 非自治复合系统的集态敏感性和集态可达性

1 四川轻化工大学数学与统计学院 四川自贡 643000

2 四川师范大学数学科学学院 成都 610068

3 企业信息化与物联网测控技术四川省高校重点实验室 四川自贡 643000

## The Collectively Sensitivity and Accessible in Non-Autonomous Composite Systems

Yang Xiaofang,1, Tang Xiao,2, Lu Tianxiu,1,3

1 College of Mathematics and Statistics, Sichuan University of Science and Engineering, Sichuan Zigong 643000

2 School of Mathematical Science, Sichuan Normal University, Chengdu 610068

3 Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things, Sichuan Zigong 643000

 基金资助: 四川省科技计划.  19YYJC2845企业信息化与物联网测控技术四川省高校重点实验室开放基金.  2020WZJ01四川轻化工大学人才引进项目.  2020RC24研究生创新基金项目.  Y2020077

 Fund supported: the Science and Technology Plan of Sichuan Province.  19YYJC2845the Key Laboratory of Colleges and Universities Open Fund for Enterprise Information and Internet of Measurement and Control Technology in Sichuan Province.  2020WZJ01the Talent Introduction Program.  2020RC24the Graduate Student Innovation Fund.  Y2020077

Abstract

In this paper, collectively sensitivity, collectively infinity sensitivity, collectively Li-Yorke sensitivity and collectively accessible are defined in the non-autonomous discrete system. First of all, it is showed that, on compact metric spaces, mapping sequence $(f_k)^\infty_{k=1}$ is ${\cal P}$-chaos if and only if $\forall n\in {\Bbb N}$ ($N$ is the set of natural numbers and does not contain 0). Then, under the condition that $f_{1, \infty}$ is uniformly convergence, it is proved that $f_{1, \infty}$ is ${\cal CP}$-chaos if and only if for any $m\in {\Bbb N}$, $f_{1, \infty}^{[m]}$ is ${\cal CP}$-chaos. Where ${\cal P}$-chaos denote one of the five properties: transitivity, sensitivity, infinitely sensitivity, accessibility and exact, ${\cal CP}$-chaos denote one of the four properties: collectively sensitivity, collectively infinity sensitivity, collectively Li-Yorke sensitivity and collectively accessible.

Keywords： Non-autonomous discrete system ; Composite mapping ; Transitivity ; Sensitivity ; Accessibility

Yang Xiaofang, Tang Xiao, Lu Tianxiu. The Collectively Sensitivity and Accessible in Non-Autonomous Composite Systems. Acta Mathematica Scientia[J], 2021, 41(5): 1545-1554 doi:

## 1 引言

$f_{1, \infty} = (f_{n})^{\infty}_{n = 1}$是紧度量空间$(I, d)$上的一个连续自映射序列. 对于任意正整数$i, n\in{\Bbb N}$, 记$f^{n}_{i} = f_{i+(n-1)}\circ\cdots\circ f_{i}$, $f^{0}_{1} = {\rm id}_{I}$ (恒等映射). 称系统$(I, f_{1, \infty})$为非自治离散动力系统. $\forall x\in I$, 点$x$在映射序列$f_{1, \infty}$下的轨道为$\{f^{n}_{1}(x): n \in {\Bbb N}\} = orb(x, f_{1, \infty})$. 换句话说, 这个轨道也就是非自治微分方程

21世纪初以来, 关于非自治离散系统的混沌性问题一直倍受学者们关注. Canovas[4]研究了映射序列$f_{1, \infty}$的极限行为, 考虑当$f_{1, \infty}$一致收敛于$f$时, 是否有极限映射$f$的混沌性意味着原映射序列$f_{1, \infty}$的混沌性. Kumar[5]在无限维空间中讨论了具有有限延迟的非自治二阶非线性微分方程的近似可控性. 2020年, 邵华[6-7]建立了非自治离散系统中强Li-Yorke混沌和分布混沌的判断准据, 并讨论了非自治集值系统与有限子移位系统之间的拓扑等度半共轭性和等度共轭性. 同年, 黎日松[8]则进一步将非自治系统的一般传递性和敏感性扩展到了更强形式的传递性和敏感性. 此外, 我们也曾经得到过关于$f_{1, \infty}$的分布混沌性, 敏感性和${\cal F}$ -混沌性的一些结果[9-11]. 其它有关非自治离散系统混沌性的研究, 参见文献[12-24].

$\forall m\in {\Bbb N}$, 定义

$(I, g_{1, \infty}) $$m 次迭代的或者称其为系统 (I, f_{1, \infty}) 的一个复合系统. 记 g_{1, \infty} = f_{1, \infty}^{[m]} . 这篇文章将讨论非自治复合系统的集态敏感性、集态无限敏感性、集态可达性和集态Li-Yorke敏感性等. ## 2 相关定义 (I, d) 是一个紧度量空间, f_n:I\mapsto I (n\geq1) 是连续自映射序列. 定义2.1 (1) f_{1, \infty} 是敏感依赖的, 如果存在 \eta>0 , 使得对任意 a\in I$$ \varepsilon>0$, 存在$b\in B(a, \varepsilon) $$n\in {\Bbb N} , 满足 d(f^n_1(a), f^n_1(b))>\eta ; \rm(2)$$ f_{1, \infty}$称为无限敏感的, 如果存在$\eta>0$, 使得对任意$a\in I $$\varepsilon>0 , 存在 b\in B(a, \varepsilon)$$ n\in {\Bbb N}$, 满足$\limsup\limits_{n\rightarrow \infty}d(f^n_1(a), f^n_1(b))\geq \eta$;

$\rm(3) $$f_{1, \infty} 称为传递的, 如果对任意非空开子集 U_{1}, U_{2}\subset I , 存在 n\in {\Bbb N} , 满足 f_1^{n}(U_{1})\cap U_{2}\neq\emptyset ; \rm(4)$$ f_{1, \infty}$称为可达的, 如果对任意的$\varepsilon>0$和任意两个非空开子集$U_1, U_2\subset I$, 存在$a\in U_1$, $b\in U_2 $$n\in {\Bbb N} 使得 d(f^n_1(a), f^n_1(b))<\varepsilon ; \rm(5)$$ f_{1, \infty}$称为正合的, 如果对任意非空开子集$U\subset I$, 存在$n\in {\Bbb N}$使得$f_1^n(U) = I$.

$\rm(1)$对一切$1\leq i, j\leq s$, 满足$d(a_i, b_j)<\varepsilon$;

$\rm(2)$存在$1\leq i_0, j_0\leq s$使得

因为$(I, f_{1, \infty})$是集态无限敏感的, 则存在$\lambda>0$, 对任意$\varepsilon>0$和任意有限个不同的点$a_1, a_2, \cdots, a_s\in I$, 存在$s$个不同的点$b_1, b_2, \cdots, b_s\in I$使得下列两个条件成立:

(1) 对一切$1\leq i, j\leq s$, 满足$d(a_i, b_j)<\varepsilon$;

(2) 存在$1\leq i_0, j_0\leq s$使得

$\rm(1) $$\exists i_0\in \{1, 2, \cdots, s\} , 对 \forall j\in \{1, 2, \cdots, s\} , \exists n\in {\Bbb N} 满足 d(f_1^n(a_{i_0}), f_1^n(b_{j}))<\varepsilon ; \rm(2)$$ \exists j_0\in \{1, 2, \cdots, s\}$, 对$\forall i\in \{1, 2, \cdots, s\}$, $\exists n\in {\Bbb N}$满足$d(f_1^n(a_{i}), f_1^n(b_{j_0}))<\varepsilon.$

$\rm(1)$对一切$1\leq i, j\leq s$, 满足$d(a_i, b_j)<\varepsilon$;

$\rm(2)$存在$1\leq i_0, j_0\leq s$使得

## 3 映射序列$f_{1, \infty}$和$f_{n, \infty}$的混沌性

$a $$\varepsilon 的任意性, f_{2, \infty} 是无限敏感的. (ⅳ) (可达性) 充分性. 因为 f_{2, \infty} 是可达的, 则对 \forall \varepsilon>0 和任意非空开子集 U_1, U_2\subset I , 存在两点 a\in U_1 , b\in U_2$$ n\in {\Bbb N}$使得$d(f_2^n(a), f_2^n(b)))<\varepsilon$. 因为$f_1$是一个满射, 取$U_1 $$U_2 中每个元素在 f_1 之下的逆象分别构成集合 U_1^{*}$$ U_2^{*}$. 则存在$a^{*}\in U_1^{*}$, $b^{*}\in U_2^{*}$使得$f_1(a^{*}) = a $$f_1(b^{*}) = b . 因此, 可得 U_1$$ U_2$的任意性, $f_{1, \infty}$是可达的.

$U_1 $$U_2 的任意性, f_{2, \infty} 是可达的. (ⅴ) (正合性) 充分性. 因为 f_{2, \infty} 是正合的, 则对任意非空开子集 U\subset I , 存在 n\in{\Bbb N} 使得 f_2^n(U) = X . 也就是说 f^n_2(U)\subseteq I$$ I\subseteq f^n_2(U)$都成立. 所以, 对任意$a\in U$, 可以得到$f^n_2(a)\in I$. 因为$f_1$是一个满射, 取$U$中每个元素在$f_1$之下的逆象构成集合$U^{*}$. 那么, 存在一个$a^{*}\in U^{*}$, $f_1(a^{*}) = a$满足$f_1^n(a^{*}) = f_2^n(a)\in I$. 再根据点$a$的任意性, 可知$f_1^n(U^{*})\subseteq I$. 又因为$I\subseteq f_2^n(U) = f_1^n(U^{*})$, 则$I\subseteq f_1^n(U^{*})$.$U$的任意性, 可得$f_1^n(U^{*}) = I$. 因此, $f_{1, \infty}$是正合的.

$\rm(1) $$\forall1\leq i, j \leq s , 有 d(a_i, b_j)<\delta ; \rm(2)$$ \forall n>N$, $\exists 1\leq i_0, j_0\leq s$, 有

根据引理4.1和不等式

必要性. 因为$f_{1, \infty}$是集态敏感的, 设其集态敏感常数为$\delta$, 则对$\forall \varepsilon>0$和任意有限个不同的点$a_1, a_2, \cdots, a_s\in I$, 存在$s$个不同的点$b_1, b_2, \cdots, b_s\in I$使得

(1) 对一切$1\leq i, j\leq s$, 满足$d(a_i, b_j)<\varepsilon$;

(2) 存在$1\leq i_0, j_0\leq s $$n\in {\Bbb N} 使得 又因为 (f_n)_{n = 1}^{\infty} 是一致收敛的, 根据引理4.4, 对 \forall m\in{\Bbb N} , 存在 \xi>0$$ N\in {\Bbb N}$使得对上述$s$个不同的点$a_1, a_2, \cdots, a_s\in I $$b_1, b_2, \cdots, b_s\in I , 满足: (a) 对 \forall 1\leq i, j\leq s , d(a_i, b_j)<\xi ; (b) 对 \forall n>N , \exists 1\leq i'_0, j'_0\leq s 使得 所以, \exists N_0>2m , 对任意有限个不同的点 a_1, a_2, \cdots, a_s\in I$$ b_1, b_2, \cdots, b_s\in I$, 满足:

(a$'$)$\forall 1\leq i, j\leq s$, 有$d(a_i, b_j)<\xi$;

(b$'$)$\forall n> N_0$, $\exists1\leq i'_0, j'_0\leq s$使得

必要性. 因为$f_{1, \infty}$集态无限敏感, 所以存在一个常数$\lambda>0$, 对$\forall \varepsilon>0$和任意有限个不同的点$a_1, a_2, \cdots, a_s\in I$, 存在$s$个不同的点$b_1, b_2, \cdots, b_s \in I$使得下列两个条件成立:

(1) 对$\forall 1\leq i, j\leq s$, 有$d(a_i, b_j)<\varepsilon$;

(2) 存在$i_0 $$j_0 满足 1\leq i_0, j_0\leq s 使得 不失一般性, 我们考虑条件(2)中的 \limsup\limits_{n\rightarrow \infty}d(f_1^n(a_i), f_1^n(b_{j_0}))>\lambda(1\leq i\leq s) . 因为 \forall p:0\leq p\leq m-1 , \forall k\in{\Bbb N} , f_k^p 是集态一致连续的, 对上述常数 \lambda >0 , \exists \beta>0 , 对任意有限个不同的点 a_1, a_2, \cdots, a_s\in I$$ b_1, b_2, \cdots, b_s\in I$, 有

(a) 对$\forall 1\leq i, j\leq s$, $d(a_i, b_j)<\beta$;

(b) $\exists 1\leq i_0, j_0\leq s$, 满足$d(f_{k}^{p}(a_i), d(f_{k}^{p}(b_{j_0}))<\frac{\lambda}{2}, (0\leq i\leq s).$

$\alpha = min(\frac{\lambda}{2}, \frac{\beta}{2})$, 可以证明

必要性. 因为$f_{1, \infty}$是集态Li-Yorke敏感的, 设其敏感常数为$\delta>0$, 则对$\forall \varepsilon>0$和任意有限个不同的点$a_1, a_2, \cdots, a_s\in I$, 存在$s$个不同的点$b_1, b_2, \cdots, b_s\in I$使得:

(1) 对$\forall 1\leq i, j\leq s$, 有$d(a_i, b_j)<\varepsilon$;

(2) 存在$1\leq i_0, j_0\leq s$使得

(ⅰ) 由$\liminf\limits_{n\rightarrow \infty}d(f_{1}^{n}(a_i), f_1^n(b_{j_0})) = 0(1\leq i \leq s)$知, 存在一个递增序列$\{n_s\}_{s\in {\Bbb N}}$使得$\lim\limits_{n\rightarrow \infty}d(f_1^{n_s}(a_i), f_1^{n_s}(b_{j_0}) ) = 0(1\leq i \leq s)$. 对任意的$p\in {\Bbb N}\; (0\leq p\leq m-1)$, 注意到有$N_i = \{mj+i: j\in {\Bbb N}^{+}\} \cap \{n_s: s\in {\Bbb N}\}$, 那么$\cup_{i = 0}^{s-1}N_i = \{n_s:s\in{\Bbb N}\}$. 所以, 存在一个$i^{*}\in\{0, 1, \cdots, s-1\}$使得$N_{i^{*}}$是一个无限集. 令$N_{i^{*}} = \{n_s^{i^{*}}\}_{s = 0}^{\infty}$, 显然$N_{i^{*}} $$\{n_s\}_{s\in{\Bbb N}} 的一个子序列. 因此, \lim\limits_{s\rightarrow \infty}d(f_1^{n_s^ {i^{*}}}(a_i), f_1^{n_s^{i^{*}}}(b_{j_0})) = 0 . 由引理4.3, 对 \forall k\in{\Bbb N} , f_k^{m-i^{*}} 是集态一致连续的, 所以有 (ⅱ) 因为 \forall p:0\leq p\leq m-1 , \forall k\in{\Bbb N} , f_k^p 是集态一致连续的, 对上述常数 \delta >0 , 存在 \beta>0 , 对任意有限个不同的点 a_1, a_2, \cdots, a_s\in I$$ b_1, b_2, \cdots, b_s\in I$, 有

(a) 对$\forall 1\leq i, j\leq s$, $d(a_i, b_j)<\beta$;

(b) $\exists 1\leq i_0, j_0\leq s$, 满足$d(f_{k}^{p}(a_i), d(f_{k}^{p}(b_{j_0}))<\frac{\delta}{2}, (0\leq i\leq s)$.$\alpha = \min(\frac{\delta}{2}, \frac{\beta}{2})$, 可以证明

根据$(f_n)_{n = 1}^{\infty}$的一致收敛性和引理4.3可知, 对$\forall\varepsilon>0$, $\forall m\in{\Bbb N}$, 存在$\delta>0 $$N\in {\Bbb N} 使得对任意有限个不同的点 a_1, a_2, \cdots, a_s\in I 满足 b_1, b_2, \cdots, b_s\in I , d(a_i, b_j)<\delta$$ (\forall1\leq i, j \leq s)$, 且$\forall n>N$, $\exists 1\leq i_0, j_0\leq s$

(2) 存在$j_0\in \{1, 2, \cdots, s\} $$n\in{\Bbb N} , 对 \forall i\in \{1, 2, \cdots, s\} , 有 d(f_1^n(a_{i}), f_1^n(b_{j_0}))<\delta(\varepsilon) . 不失一般性, 我假设上述条件(1)成立, 结合 (f_n)_{n = 1}^{\infty} 的一致收敛性可得 因此, f_{1, \infty}^{[m]} 是集态可达的. 充分性. 令 m = 1 , 即得结论. 定理4.1–定理4.4可以综合描述为: 如果映射序列 (f_n)_{n = 1}^{\infty} 是一致收敛的, 则 f_{1, \infty}$$ {\cal CP}$ -混沌的当且仅当对$\forall m\in {\Bbb N}$, $f_{1, \infty}^{[m]}$${\cal CP}$ -混沌的.

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