Bingzhe Hou, Xianfeng Ma, Gongfu Liao
We discuss the relation between Devaney chaos in the base system and Devaney chaos in its induced hyperspace system. We show that the latter need not imply the former. We also argue that this implication is not true even in the strengthened condition. Additionally we give an equivalent condition for the periodically density in the hyperspace system.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we study various chaos of topological group or semigroup actions.
On stronger forms of Devaney chaos
该研究定义并探究了以自然数集子集族为参数的✂-Devaney混沌这一Devaney混沌更强形式，构造不同族下的映射示例，证明无孤立点无限度量空间上该混沌定义中敏感性冗余，还给出Devaney混沌蕴含特定族下该混沌及相关P-混沌蕴含该混沌的条件。
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.
We give an equivalent definition of Devaney chaotic semiflow in terms of eventual sensitivity, the notion recently introduced by C.~Good, R.~Leek, and J.~Mitchell. As a consequence, we prove a version of Auslander-Yorke dichotomy for the case of not necessarily compact phase spaces. Finally, we raise some questions about the relation between Devaney chaoticity and the notion of topological sensitivity, which was recently introduced by C.~Good and C.~Macías.
We study topological entropy of exactly Devaney chaotic maps on totally regular continua, i.e. on (topologically) rectifiable curves. After introducing the so-called P-Lipschitz maps (where P is a finite invariant set) we give an upper bound for their topological entropy. We prove that if a non-degenerate totally regular continuum X contains a free arc which does not disconnect X or if X contains arbitrarily large generalized stars then X admits an exactly Devaney chaotic map with arbitrarily small entropy. A possible application for further study of the best lower bounds of topological entrop…
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。
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.
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。
In this note, we give several equivalent definitions of Devaney's chaos
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 …
Let $T$ be a continuous self-map of a compact metric space $X$. The transformation $T$ induces natural a continuous self-map $T_K$ on the hyperspace $K(X)$ of all non-empty closed subsets of $X$. In this paper, we show that the system $(K(X),T_K)$ on the hyperspace is Devaney chaotic if and only if $(K(X),T_K)$ is an HY-system if and only if $(X,T)$ is an HY-system, where a system $(Y,S)$ is called an HY-system if it is totally transitive and has dense small periodic sets.
Li-Yorke and Devaney chaotic uniform dynamical systems amongst weighted shifts
该研究刻画有限离散域上加权广义移位所诱导一致动力系统的 Li–Yorke 混沌与 Devaney 混沌性质，并分别给出充要条件。
We give sufficient conditions for sensitivity of continuous group actions on uniform spaces.
On a class between Devaney chaotic and Li-Yorke chaotic generalized shift dynamical systems
本文证明广义移位动力系统为稠密混沌当且仅当底映射无拟周期点，从而确立该类严格介于Devaney混沌与Li-Yorke混沌之间，并给出多种敏感性与混沌性质的等价刻画。
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。
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.
In this article, we show that a chaotic behavior can be found on a cube with arbitrary finite dimension. That is, the cube is a quasi-minimal set with Poincare chaos. Moreover, the dynamics is shown to be Devaney and Li-Yorke chaotic. It can be characterized as a domain-structured chaos for an associated map. Previously, this was known only for unit section and for Devaney and Li-Yorke chaos.
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。
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。
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。
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。
Existence and Configuration of Invariant Sets in $C^\infty([a,b])$ on which the Differential Operator Exhibits Devaney's Chaos
本文研究了微分算子在光滑函数空间上的混沌行为，通过构造同胚和证明不变集的稠密配置，展示了其在$C^\infty([a,b])$上的混沌特性。
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。
In this article, we provide a sufficient condition which gives Devaney chaos and distributional chaos for Cowen-Douglas operators. In fact, we obtain a distributionally chaotic criterion for bounded linear operators on Banach spaces.
Motivated by C*-algebra theory, ultragraph edge shift spaces generalize shifts of finite type to the infinite alphabet case. In this paper we study several notions of chaos for ultragraph shift spaces. More specifically, we show that Li-Yorke, Devaney and distributional chaos are equivalent conditions for ultragraph shift spaces, and characterize this condition in terms of a combinatorial property of the underlying ultragraph. Furthermore, we prove that such properties imply the existence of a compact, perfect set which is distributionally scrambled of type 1 in the ultragraph shift space (a r…
Morphogenesis, as it is understood in a wide sense by René Thom, is considered for various types of chaos. That is, those, obtained by period-doubling cascade, Devaney's and Li-Yorke chaos. Moreover, in discussion form we consider inheritance of intermittency, the double-scroll Chua's attractor and quasiperiodical motions as a possible skeleton of a chaotic attractor. To make our introduction of the paper more clear, we have to say that one may consider other various accompanying concepts of chaos such that a structure of the chaotic attractor, its fractal dimension, form of the bifurcation di…
Extending Chaos Theory: The Role of Nonlinearity in Multiple Mappings
本研究将 Devaney 混沌定义扩展到多映射系统，提出新定理表明单映射的非线性可引发集体混沌，并用 MATLAB 算法验证，为复杂系统非线性动力学研究奠定基础。
Many research works deal with chaotic neural networks for various fields of application. Unfortunately, up to now these networks are usually claimed to be chaotic without any mathematical proof. The purpose of this paper is to establish, based on a rigorous theoretical framework, an equivalence between chaotic iterations according to Devaney and a particular class of neural networks. On the one hand we show how to build such a network, on the other hand we provide a method to check if a neural network is a chaotic one. Finally, the ability of classical feedforward multilayer perceptrons to lea…
In this paper we introduce some weak dynamical properties by using subbases for the phase space. Among them, the notion of light chaos is the most significant. Severalexamples, which clarify the relationships between this kind of chaos and some classical notions, are given. Particular attention is also devoted to the connections between the dynamical properties of a system and the dynamical properties of the associated functional envelope. We show, among other things, that a continuous map f : X ! X, where X is a metric space, is chaotic (in the sense of Devaney) if and only if the associated …
Chaotic behaviour on invariant sets of linear operators
该研究探讨带不变集的拓扑向量空间上算子的多种混沌与混合性质，建立了不变集上的相关性质与其闭张成空间上类似性质的关联，并给出示例说明结果。
This paper introduces a class of polynomial maps in Euclidean spaces, investigates the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets, studies the chaotic dynamical behavior and strange attractors, and shows that some maps are chaotic in the sense of Li-Yorke or Devaney. This type of maps includes both the Logistic map and the Hénon map. For some maps in three-dimensional spaces under certain conditions, if the expansion dimension is equal to one or two, it is shown that there exist a Smale horseshoe and a uniformly hyperbolic invariant set on which…
Weak chaos in difference equations
本文研究巴拿赫和希尔伯特空间上离散动力学的有界解存在条件，在齐次差分方程具有离散二分性时给出充要条件，并处理共振情形，将可容许性转化为广义或伪可逆性，应用于e-三分性系统。
A simple chaos indicator based on the Lagrangian descriptor difference of neighboring orbits
本文提出了一种基于邻近轨道拉格朗日描述符差值的简单混沌指示器，通过对比实验验证其在检测混沌行为方面的有效性。
We study the fractional maps of complex order, $α_0e^{i r π/2}$ for $0<α_0<1$ and $0\le r<1$ in 1 and 2 dimensions. In two dimensions, we study H{é}non and Lozi map and in $1d$, we study logistic, tent, Gauss, circle, and Bernoulli maps. The generalization in $2d$ can be done in two different ways which are not equivalent for fractional-order and lead to different bifurcation diagrams. We observed that the smooth maps such as logistic, Gauss, and H{é}non maps do not show chaos while discontinuous maps such as Lozi, Bernoulli, and circle maps show chaos. The tent map is continuous but not diffe…
Pantograph Difference Equations: Stability, Multistability, and Chaos
该研究将缩放仪差分方程作为离散动力学映射，分析其稳定性、多稳态与混沌特性，区分齐次与非齐次情形，通过理论分析与数值实验揭示其动力学行为。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。