Devaney chaos, Li-Yorke chaos, and multi-dimensional Li-Yorke chaos for topological dynamics

Devaney chaos plus shadowing implies distributional chaos

We explore connections among the regional proximal relation, the asymptotic relation and the distal relation for a topological dynamical system with the shadowing property, and show that if a Devaney chaotic system has the shadowing property then it is distributionally chaotic.

两个系统Devaney混沌的区别

Difference between Devaney chaos associated with two systems

讨论了基系统的Devaney混沌与其诱导超空间系统的Devaney混沌之间的关系。我们表明，后者不一定意味着前者。我们还认为，即使在强化条件下，这一含义也不成立。此外，我们还给出了超空间系统周期密度的一个等价条件。

全正则连续统上的熵与精确Devaney混沌

Entropy and exact Devaney chaos on totally regular continua

研究了完全正则连续域上，即（拓扑上）可求长曲线上的恰Devaney混沌映射的拓扑熵。在引入所谓的P-Lipschitz映射（其中P是有限不变集）之后，我们给出了它们的拓扑熵的一个上界。证明了：如果非退化全正则连续统X包含一个不断开X的自由弧，或者X包含任意大的广义星，则X存在一个具有任意小熵的正Devaney混沌映射.指出了传递/Devaney混沌映射的拓扑熵的最佳下界的进一步研究的一个可能应用。

集值映射及其逆极限上的Devaney混沌

Devaney Chaos on a Set-valued Map and Its Inverse Limit

本文研究集值映射与其逆极限在周期点集、传递性、敏感性和Devaney混沌方面的关系，发现周期点集稠密性相互蕴含，敏感性不相互蕴含，而广义逆极限的传递性与Devaney混沌可推出集值映射的相应性质。

Devaney chaos in non-autonomous discrete systems

This paper is concerned with Devaney chaos in non-autonomous discrete systems. It is shown that in its definition, the two former conditions, i.e., transitivity and density of periodic points, in a set imply the last one, i.e., sensitivity, in the case that the set is unbounded, while a similar result holds under two additional conditions in the other case that the set is bounded. Furthermore, some chaotic behavior is studied for a class of non-autonomous systems, each of which is governed by a convergent sequence of continuous maps.

集值视角下多映射的Devaney混沌

Devaney chaos of multiple mappings from a set-valued view

该研究从集值视角定义多映射的Devaney混沌相关概念，厘清多映射与其连续自映射在周期点、传递性上的互不蕴含关系，给出多映射具备周期点、传递性、敏感性及Devaney混沌的充分条件。

Li-Yorke chaos for invertible mappings on compact metric spaces

In this paper, we construct a homeomorphism on the unit closed disk to show that an invertible mapping on a compact metric space is Li-Yorke chaotic does not imply its inverse being Li-Yorke chaotic.

Li-Yorke sensitivity does not imply Li-Yorke chaos

We construct an infinite-dimensional compact metric space $X$, which is a closed subset of $\mathbb{S}\times\mathbb{H}$, where $\mathbb{S}$ is the unit circle and $\mathbb{H}$ is the Hilbert cube, and a skew-product map $F$ acting on $X$ such that $(X,F)$ is Li-Yorke sensitive but possesses at most countable scrambled sets. This disproves the conjecture of Akin and Kolyada that Li-Yorke sensitivity implies Li-Yorke chaos from the article [Akin E., Kolyada S., Li-Yorke sensitivity, Nonlinearity 16, (2003), 1421-1433].

G-空间上映射的Li-Yorke混沌

Li-Yorke chaos for maps on G-Spaces

本文在G-空间上定义并研究G-Li-Yorke混沌，证明其可迭代性及与Li-Yorke混沌的关系，并给出G-传递且存在公共不动点条件下映射为G-Li-Yorke混沌的充分条件。

非紧空间上可逆映射的Li-Yorke混沌

Li-Yorke chaos for invertible mappings on non-compact spaces

本文给出了两个例子，证明了一个可逆映射是Li-Yorke混沌的并不意味着它的逆是Li-Yorke混沌的，其中一个是无限维希尔伯特空间上的可逆有界线性算子，另一个是单位开圆盘上的同胚.此外，我们还利用最后一个例子证明了Li-Yorke混沌在拓扑共轭下是不保持的。

Robust Devaney chaos in the two-dimensional border-collision normal form

The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on $\mathbb{R}^2$ can be reduced to a four-parameter family known as the two-dimensional border-collision normal form. We prove that throughout an open region of parameter space this family has an attractor satisfying Devaney's definition of chaos. This strengthens existing results on the robustness of chaos in piecewise-linear maps. We further show that the stable manifold of a saddle fixed point, despite being a one-dimensional object, densely fills an open region containing the attractor. Finally we identify …

超空间上Devaney混沌的等价条件

Equivalent conditions of Devaney chaos on the hyperspace

设$T$是紧度量空间$X$的连续自映射.变换$T$在$X$的所有非空闭子集的超空间$K(X)$上自然地导出连续自映射$T_K$。本文证明了超空间上的系统$(K(X),T_K)$是Devaney混沌的当且仅当$(K(X),T_K)$是HY-系统当且仅当$(X,T)$是HY-系统，其中一个系统$(Y,S)$称为HY-系统，如果它是全传递的并且具有稠密的小周期集.

Banach空间上线性算子的稠密一致Li-Yorke混沌

Dense uniform Li-Yorke chaos for linear operators on a Banach space

该研究针对Banach空间上线性算子的稠密一致Li-Yorke混沌问题，建立相关充分与等价条件，证明此类算子大量存在，并证实ℓ^p上两类后向加权移位的Li-Yorke混沌与稠密一致Li-Yorke混沌等价。

Devaney的混沌

Devaney's chaos revisited

本文给出了Devaney混沌的几种等价定义

序列中Li-Yorke $δ$ -混沌与分布$δ$ -混沌的等价关系

The equivalence relationship between Li-Yorke $δ$-chaos and distributional $δ$-chaos in a sequence

本文讨论了序列的Li-Yorke混沌与分布混沌的关系。指出了序列$Q$中所有分布$δ$ -扰码对的集合是一个$G_δ$集合，证明了Li-Yorke $δ$ -混沌等价于序列中的分布$δ$ -混沌，一致混沌集合是序列中的分布扰码集合，一类传递系统蕴涵着序列中的分布混沌.

Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems

We study chaotic properties of uniformly convergent nonautonomous dynamical systems. We show that, contrary to the autonomous systems on the compact interval, positivity of topological sequence entropy and occurrence of Li-Yorke chaos are not equivalent, more precisely, neither of the two possible implications is true.

关于模一个理想的Li-Yorke混沌变换群

On Li--Yorke chaotic transformation groups modulo an ideal

本文在拓扑变换半群中引入模理想的Li-Yorke混沌概念，刻画一类阿贝尔无限Li-Yorke混沌Fort变换群，并证明其可余分解为非Li-Yorke混沌变换群。

双序amenable groups正熵作用下的稳定集与平均Li-Yorke混沌

Stable sets and mean Li-Yorke chaos in positive entropy actions of bi-orderable amenable groups

证明了对于可数无限离散双序顺从群G-系统，正熵蕴含着平均Li-Yorke混沌.给出了整数格群和整数幂幺上三角形矩阵群的例子。

$L^p$空间上复合算子的Li-Yorke混沌

Li-Yorke Chaos for Composition Operators on $L^p$-Spaces

该研究在线性动力学已有Li-Yorke混沌刻画的基础上，针对$L^p$空间上的复合算子给出更简洁实用的Li-Yorke混沌刻画，推导出加权移位算子混沌的简洁判据，并以实例验证结果的紧性。

具有A-耦合展开的时变离散动力系统的强Li-Yorke混沌

Strong Li-Yorke chaos for time-varying discrete dynamical systems with A-coupled-expansion

本文研究了由A-耦合展开引起的时变（即，非自治）度量空间中的离散系统。通过严格耦合展开，分别建立了完全度量空间的有界子集和闭子集以及度量空间的紧子集中不可约转移阵的强Li-Yorke意义下的混沌判据，其条件比现有文献中的弱.提供了一个示例用于说明。

Recurrence determinism and Li-Yorke chaos for interval maps

Recurrence determinism, one of the fundamental characteristics of recurrence quantification analysis, measures predictability of a trajectory of a dynamical system. It is tightly connected with the conditional probability that, given a recurrence, following states of the trajectory will be recurrences. In this paper we study recurrence determinism of interval dynamical systems. We show that recurrence determinism distinguishes three main types of $ω$-limit sets of zero entropy maps: finite, solenoidal without non-separable points, and solenoidal with non-separable points. As a corollary we obt…

Mean Li-Yorke chaos along polynomials of several variables and prime numbers

In this paper, for any given polynomial, by analyzing the limiting behavior of ergodic averages along polynomials of several variables and prime numbers, we prove that for a topology dynamical system, positive entropy implies mean Li-Yoke chaos along non-constant polynomials of several variables and prime numbers.

Mean proximality and mean Li-Yorke chaos

We prove that if a topological dynamical system is mean sensitive and contains a mean proximal pair consisting of a transitive point and a periodic point, then it is mean Li-Yorke chaotic (DC2 chaotic). On the other hand we show that a system is mean proximal if and only if it is uniquely ergodic and the unique measure is supported on one point.

S-极限跟踪与Li-Yorke型混沌的全局描述

S-limit shadowing and a global description of Li-Yorke type chaos

该研究将紧度量空间连续自映射的链分支链邻近划分推广为相空间$G_δ$划分，在s-极限跟踪假设下实现了对应多类Furstenberg族的Li-Yorke型混沌的全局刻画。

Li-Yorke chaos in nonautonomous Hopf bifurcation patterns - I

We analyze the characteristics of the global attractor of a type of dissipative nonautonomous dynamical systems in terms of the Sacker and Sell spectrum of its linear part. The model gives rise to a pattern of nonautonomous Hopf bifurcation which can be understood as a generalization of the classical autonomous one. We pay special attention to the dynamics at the bifurcation point, showing the possibility of occurrence of Li-Yorke chaos in the corresponding attractor and hence of a high degree of unpredictability.

一致空间中的灵敏度与Devaney混沌

Sensitivity and Devaney's chaos in uniform spaces

给出了一致空间上连续群作用灵敏性的充分条件。

正熵系统中的稳定集与平均Li-Yorke混沌

Stable sets and mean Li-Yorke chaos in positive entropy systems

证明了在具有正熵的拓扑动力系统中，存在一个测度理论上“相当大”的集合，使得在该集合中任意点的稳定或不稳定集合的闭包上，均发生多变量形式的平均Li-Yorke混沌.证明了渐近元组集和平均Li-Yorke元组集与拓扑熵元组集的交在拓扑熵元组集中是稠密的.

弱拓扑下二维空间中的Devaney混沌

Devaney chaos in a Two-Dimensional Space with Weak Topology

本文证明弱拓扑下二维线性映射可产生Devaney混沌，给出相关动力学性质的等价条件，揭示弱拓扑与范数拓扑的动力学性质存在显著差异。

混沌理论在拓扑动力学中的新进展

Recent development of chaos theory in topological dynamics

综述了拓扑动力学中混沌理论的最新进展，重点介绍了Li-Yorke混沌、Devaney混沌、分布混沌、正拓扑熵、弱混合集等混沌理论及其相互关系。

多重映射的混合性、Li-Yorke混沌、分布混沌与Kato混沌

Mixing, Li-Yorke chaos, distributional chaos and Kato's chaos to multiple mappings

本文从集值视角定义多重映射的传递性与混合性等概念，证明多重映射的混合性蕴含三类混沌，给出Kato混沌的迭代等价性，并构造符号动力系统验证仅两个强非游荡点即可产生分布混沌。

区间映射Li-Yorke混沌的不变性

On the Invariance of Li-Yorke Chaos of Interval Maps

在他们著名的“周期三蕴含混沌”的论文中，李和约克证明了，如果一个连续区间映射f有一个周期为3的点，那么存在一个不可数的乱集S，在这个乱集上f有非常复杂的动力学。一个问题自然地产生了：这个集合S在f下能被选择为不变的吗？对于湍流映射，答案是肯定的，否则是否定的。在本说明中，我们将使用符号动力学来实现我们的目标。特别地，我们得到帐篷映射T（x）= 1 -| 2x-1| [0，1]上有一个稠密的不可数不变1-乱集，它由传递点组成.

On definition of Devaney chaos for a continuous group action on a Hausdorff uniform space

We show that the existence of a dense set of periodic points for a topologically transitive non-minimal continuous group action on a Hausdorff uniform space with an infinite acting group does not necessarily imply a sensitive dependence to the initial conditions in such a system. This leads to define the chaos in the sense of Devaney for a continuous group action on a Hausdorff uniform spaces with an infinite acting group in the original way, i.e. a non-minimal topologically transitive and sensitive system with a dense set of periodic points is a chaotic system in the sense of Devaney.

拓扑熵为零且具有$ω$ -极限集的枝晶映射的Li-Yorke混沌

Li-Yorke chaos for dendrite maps with zero topological entropy and $ω$-limit sets

设$X$是一个端点集$E(X)$闭的树状映射，$f:~X \to X$是一个拓扑熵为零的连续映射.令$P(f)$是$f$的周期点的集合。证明了若$L$是$f$的无限$ω$ -极限集，则有$L\cap P(f)\subset E(X)^{\prime}$，其中$E(X)^{\prime}$是$E(X)$的所有聚集点的集合.此外，如果$E(X)$是可数的，而$L$是不可数的，则$L\cap P(f)=\emptyset$。我们还表明，如果$E(X)^{\prime}$是有限的，那么$f$的任何不可数的$ω$ -极限集有一个分解，因此，如果$f$有一个李约克对$(x,y)$与$ω\_f(x)$或$ω\_f(y)$是不可数的，那么$f$是李约克混沌。

对于图映射，一个置乱对意味着Li-Yorke混沌

For graph maps, one scrambled pair implies Li-Yorke chaos

对于动力系统$(X,f)$，$X$是具有度量$d$的紧致度量空间，并且$f$是连续映射$X\to X$，如果$S$中的每对$(x,y)$都被置乱，则集合$S\subseteq X$被置乱，即，$\liminf_{n\to+\infty}d(f^n(x),f^n(y))=0$和$\limsup_{n\to+\infty}d(f^n(x),f^n(y))>0$。如果系统$(X,f)$具有不可数的置乱集，则系统是Li-Yorke混沌的.众所周知，对于区间和圆映射，扰码对的存在意味着Li-Yorke混沌，实际上是Cantor扰码集的存在。我们证明了同样的结果适用于图映射。我们进一步证明了在紧可数度量空间上，一个加扰对蕴涵着一个无限加扰集的存在。

叶状空间与伪群的混沌

Chaos for foliated spaces and pseudogroups

本文将 Devaney 混沌推广至叶状空间与伪群，证明敏感性仅在紧情形下由其余条件推出，并给出非紧反例及 Auslander-Yorke 二分法的对应结论。

关于 s-极限跟踪与 Li-Yorke 型混沌的若干结果

Some results on s-limit shadowing and Li-Yorke type chaos

本文研究紧度量空间连续自映射的 s-极限跟踪性质，通过相空间的 $G_δ$-划分将 Li-Yorke 型混沌的描述推广到一般 Furstenberg 族，补充了已有结果并给出例证。

从描述性视角看Devaney的混沌及一些关于拓扑共轭系统的结果

A Descriptive Perspective on Devaney's Chaos and Some Results on Topologically Conjugate Systems

本文从描述性接近视角重新审视Devaney混沌条件，定义了描述性遍历性、描述性周期对象的稠密性及描述性敏感性，发现Banks定理在描述性视角下不成立，部分概念在特定条件下保持拓扑共轭不变性。

Mean Li--Yorke chaos and multifractal analysis on subshifts

In the present paper, we use the generalized multifractal framework introduced by Olsen to study the Bowen entropy and packing entropy of historic sets with typical weights over aperiodic and irreducible shifts of finite type. Following those results and a transfer from almost everywhere to everywhere, we show that for each point $ω$ in a irreducible shift of finite type $Σ_A$, the Bowen entropy of the set consisting of all the points that are mean Li-Yorke pairs with $ω$ is $0$, and its packing entropy is full. This result is beyond the ergodic theory. Also, by the transfer from almost everyw…

Pinsker $\sigma$-代数特征与均值 Li-Yorke 混沌

Pinsker $σ$-algebra Character and mean Li-Yorke chaos

本文证明了在无限可数离散顺从群作用下Pinsker $\sigma$-代数为特征因子，并由此推导出一类拓扑动力系统中正拓扑熵蕴含均值Li-Yorke混沌。

模糊动力系统中的李 - 约克混沌

Li-Yorke chaos on fuzzy dynamical systems

研究李 - 约克混沌在模糊动力系统扩展系统中的表现，证明其主要变体的单向转移性质，通过结合近性和敏感性引入康托密集李 -约克混沌，并证明其在自然假设下的反向转移。

Mean Li-Yorke chaos along some good sequences

If a topological dynamical system $(X,T)$ has positive topological entropy, then it is multivariant mean Li-Yorke chaotic along a sequence $\{a_k\}_{k=1}^\infty$ of positive integers which is "good" for pointwise ergodic convergence with a mild condition; more specifically, there exists a Cantor subset $K$ of $X$ such that for every $n\ge2$ and pairwise distinct points $x_1,x_2,\dotsc,x_n$ in $K$ we have \[\liminf_{N\to\infty}\frac{1}{N}\sum_{k=1}^N\max_{1\leq i<j\leq n} d(T^{a_k}x_i,T^{a_k}x_j)=0\] and \[\limsup_{N\to\infty}\frac{1}{N}\sum_{k=1}^N\min_{1\leq i<j\leq n} d(T^{a_k}x_i,T^{a_k}x_j…

Replication of Li-Yorke Chaos Near a Homoclinic Orbit

We prove the presence of chaos near a homoclinic orbit in the modified Li-Yorke sense [10] by implementing chaotic perturbations. A Duffing oscillator is considered to show the effectiveness of our technique, and simulations that support the theoretical results are depicted. Ott-Grebogi-Yorke and Pyragas control methods are used to stabilize almost periodic motions.

时标上混杂系统的Li-Yorke混沌

Li-Yorke chaos in hybrid systems on a time scale

利用脉冲微分方程的约化技术，严格证明了时标动力方程（DETS）中混沌的存在性。本研究的结果基于Li-Yorke对混沌的定义。这是第一次在文献中，混沌得到的DETS。通过时间尺度上的Duffing方程给出了一个说明性的例子。

Persistence of Li-Yorke chaos in systems with relay

It is rigorously proved that the chaotic dynamics of the non-smooth system with relay function is persistent even if a chaotic perturbation is applied. We consider chaos in a modified Li-Yorke sense such that infinitely many almost periodic motions take place in its basis. It is demonstrated that the system under investigation possesses countable infinity of chaotic sets of solutions. Coupled Duffing oscillators are used to show the effectiveness of our technique, and simulations that support the theoretical results are represented. Moreover, a chaos control procedure based on the Ott-Grebogi-…

Ω极限集与一维不变混沌

Omega-limit sets and invariant chaos in dimension one

Ω极限集在一维动力学中起着重要的作用。在过去的50年里，至少出现了三种基本集合的定义。作者经常使用不同定义的结果。在这里，我们填补了缺少这些定义的等价性证明的空白。利用基集上的结果，推广了文[P.~ Oprocha，不变的混乱集和分布混沌，Dyn。24（2009），no. 1，31- 43.]有限图的连续映射。Li-Yorke混沌弱于正拓扑熵。当我们在Li-Yorke置乱集上加上不变性条件时，等价性就产生了。本文证明了连续图映射的正拓扑熵、马蹄形、不变Li-Yorke混沌集、一致不变分布混沌集和分布混沌对是等价的。

Mean Li-Yorke chaos along any infinite sequence for infinite-dimensional random dynamical systems

In this paper, we study the mean Li-Yorke chaotic phenomenon along any infinite positive integer sequence for infinite-dimensional random dynamical systems. To be precise, we prove that if an injective continuous infinite-dimensional random dynamical system $(X,ϕ)$ over an invertible ergodic Polish system $(Ω,\mathcal{F},\mathbb{P},θ)$ admits a $ϕ$-invariant random compact subset $K$ with $h_{top}(K,ϕ)>0$, then given a positive integer sequence $\mathbf{a}=\{a_i\}_{i\in\mathbb{N}}$ with $\lim_{i\to+\infty}a_i=+\infty$, for $\mathbb{P}$-a.s. $ω\inΩ$ there exists an uncountable subset $S(ω)\subs…

分布性与均值Li-Yorke混沌性在加权移位上的弗雷歇序列空间中

Distributional and mean Li-Yorke chaos for weighted shifts on Fréchet sequence spaces

本文研究了加权后向移位在一般弗雷歇序列空间中分布性混沌和均值Li-Yorke混沌的特征，并在Köthe序列空间λ_p(A,J)中推导了这两种混沌类型的判据。

Li-Yorke 混沌加权复合算子

Li-Yorke Chaotic Weighted Composition Operators

本文完整刻画了 $C_0(X)$ 与 $L^p(μ)$ 空间上加权复合算子的 Li-Yorke 混沌性，并由此给出 $c_0$、$\ell^p$ 及 Köthe 序列空间上加权移位算子 Li-Yorke 混沌的简化判别条件。

Peano连续统上的混沌

Chaos on Peano continua

本研究推广Agronsky和Ceder的结果，证明Peano连续统存在精确Devaney混沌的连续变换，界定含一维连续统、绝对邻域收缩核的Peano连续统类，通过局部小扰动证明该类中混沌系统的稠密性与相关通有性结论。

动力系统的商与 Cantor 扇上的混沌

Quotients of dynamical systems and chaos on the Cantor fan

本文定义动力系统在等价关系下的商，证明其敏感性、传递性和周期点性质，并据此分析 Cantor 扇与 Lelek 扇上多种混沌定义下的函数。

非自治动力系统的混沌继承

Inheriting of chaos in nonautonomous dynamical systems

考虑非自治离散动力系统$\{ f_n\}_{n\ge 1}$，其中每个$f_n$都是满射连续映射$[0,1]\to [0,1]$，使得$f_n$一致收敛到一个映射$f$ .我们表明，除其他外，如果$f$是混沌的意义上的李和约克然后非自治系统$ \{ f_n\}_{n\ge 1}$是李约克混沌，以及，同样是真实的分布混沌。如果$f$的拓扑熵为零，则非自治系统继承其无穷$ω$ -极限集.

Uniform stability and chaotic dynamics in nonhomogeneous linear dissipative scalar ordinary differential equations

The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor.

Li-Yorke chaos for ultragraph shift spaces

Recently, in connection with C*-algebra theory, the first author and Danilo Royer introduced ultragraph shift spaces. In this paper we define a family of metrics for the topology in such spaces, and use these metrics to study the existence of chaos in the shift. In particular we characterize all ultragraph shift spaces that have Li-Yorke chaos (an uncountable scrambled set), and prove that such property implies the existence of a perfect and scrambled set in the ultragraph shift space. Furthermore, this scrambled set can be chosen compact, what is not the case for a labelled edge shift (with t…

超越拓扑观念的 $C_0$- 半群相对混沌

Relative Chaos for $C_0$-Semigroups Beyond Topological Notions

本文提出了一种基于轨迹的相对混沌概念，用于描述 $C_0$- 半群在无限维空间中的不稳定性，克服了传统拓扑方法的局限性。