关于两种混沌映射的有限乘积性质

首先在一般度量空间上给出有限积映射是Li-Yorke混沌的一个判据,并且用反倒展示:当有限积映射是Li-Yorke混沌时,未必一定存在因子映射是Li-Yorke混沌的.然后,利用上述判据,在[0,1]N上证明有限积映射有不可数scrsmbled集的一个充要条件.进而,推出关于有限积映射为Li-Yorke 混沌的一组等价...     

不含混沌真子系统的Li-Yorke混沌

研究了一类Li-Yorke混沌系统,该系统没有真子系统是Li-Yorke混沌的,我们称之为混沌极小系统.本文证明混沌极小系统是拓扑传递的,而且该系统每个非空开集都包含一个不可数混乱集.混沌极小系统不一定是极小的,本文构造了一个这样的反例.特别地,我们考察了线段连续自映射,指出该类系统都不是混沌极小的,线段上混沌极小子系统的存在性和该系统有正熵是等价的.     

离散系统的广义混沌控制

针对确定性离散动力学系统的混沌控制与反控制问题,从配置Lyapunov指数出发,提出一种实现混沌控制与反控制的一般性方法.首先给出了受控系统混沌判断的特征值条件,满足该条件的系统,将产生Devaney意义下的混沌和Li-Yorke意义下的混沌.然后通过引入非对角型反馈来调整系统雅可比矩阵元素,灵活配置系统Lyapunov指数的数值和符号,从而实现离散系统的混沌控制或反控制.给出了必要的证明和仿真实例,仿真结果表明了算法的有效性.     

Bouncing Ball模型的弱混沌性

用异于传统的方法，作出Bouncing Ball映射不变流形的对称流形，从而成功地将稳定流形与不稳定流形的位置进行比较。应用[1]关于弱横截与弱混沌的有关概念及定理，给出了Borncing Ball映射产生弱混沌的较为一般的参数区域，进一步提示了Bouncing Ball映射的动力学行为。     

广义符号动力系统的混沌性

本文给出了符号动力系统的一般数学模型,它是离散时空系统的一种特殊情形.在现有离散时空系统的混沌概念和研究方法的基础上.研究了这类广义符号动力系统的混沌性,得到了一类在Devaney意义下新的广义符号混沌动力系统,从而推广了现有符号动力系统混沌性的研究范围.     

半流及其逆极限的混沌

该文对连续动力系统研究了Devaney意义下的混沌的不变性质．证明了：（1）半流是混沌的（resP，ω混沌的）当且仅当它的逆极限是混沌的（resp,ω混沌的）；（2）自映射是混沌的（resp.ω混沌的）当且仅当它的扭扩半流是混沌的（resp．ω混沌的）；（3）自映射逆极限的扭扩流拓扑共轭于其扭扩半流的逆极限．从（2）和（3）可知，结论（1）是对自映射的推广．     

Smale横截同宿定理的推广与Lozi映射的混沌现象

该文首先将平面上的λ入引理及Ｓｍａｌｅ横截同宿定理推广到映射力局部不可微的情形，进而讨论了Ｌｏｚｉ映射的混沌现象，得到了一组保证该映射产生混沌的充分条件，详见图５．     

Banach 空间上的离散混沌

研究Banach 空间上连续 Frechét可微映射导出 的离散动力系统之混沌. 建立一个由正则非退化同宿轨道产生混沌的判定定理, 并对n维实空间上的离散动力系统的混沌进行了讨论, 建立了两个由非退化返回扩张不动点产生混沌的判定定理, 其中一个为Marotto 定理的修正定理. 特别地, 分别给出了一般 Banach 空间及n维实空间上的连续可微映射不动点为扩张的充分必要条件, 彻底解决了多年以来人们对n上连续可微映射不动点的扩张性与其 Jacobi矩阵特征值之间关系的困惑.     

完备度量空间中的混沌判定

本文研究完备度量空间上的离散动力系统的混沌标准,证明了如果完备度量空间X上的连续映射f具有正则非退化返回排斥子或连接不动点的正则非退化异宿环,则存在f的不变闭子集A,使得f限制在此不变闭子集上的子系统与两个符号的符号动力系统拓扑共轭,从而获得具有这类结构的连续映射f具有Devaney混沌、分布混沌、正拓扑熵及ω-混沌,此结果改进了已有的相关结果.     

Li-Yorke敏感的乘积性和复合性

讨论Li-Yorke敏感的乘积性质以及它的迭代不变性.主要证明了Li-Yorke敏感在乘积运算下是保持的,以及在一致连续意义下,它的复合运算也是保持的.同时,举例说明该结论对于一般的连续自映射不成立.     

Recent development of chaos theory in topological dynamics

We give a summary on the recent development of chaos theory in topological dynamics,focusing on Li–Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.     

Noncontinuous maps and Devaney’s chaos

Vu Dong Tô has proven in [1] that for any mapping f: X → X, where X is a metric space that is not precompact, the third condition in the Devaney’s definition of chaos follows from the first two even if f is not assumed to be continuous. This paper completes this result by analysing the precompact case. We show that if X is either finite or perfect one can always find a map f: X → X that satisfies the first two conditions of Devaney’s chaos but not the third. Additionally, if X is neither finite nor perfect there is no f: X → X that would satisfy the first two conditions of Devaney’s chaos at the same time.     

Discrete chaos induced by heteroclinic cycles connecting repellers in Banach spaces

This paper is concerned with chaos induced by heteroclinic cycles connecting repellers for maps in Banach spaces. Several criteria of chaos are established in general Banach spaces and finite-dimensional spaces, respectively, by employing the coupled-expansion theory. All the maps presented in this paper are proved to be chaotic in the sense of both Li-Yorke and Devaney or in the sense of both Li-Yorke and Wiggins or in the sense of Li-Yorke. An illustrative example is provided with computer simulations.     

Omega-limit sets and invariant chaos in dimension one

Omega-limit sets play an important role in one-dimensional dynamics. During last fifty year at least three definitions of basic set has appeared. Authors often use results with different definition. Here we fill in the gap of missing proof of equivalency of these definitions. Using results on basic sets we generalize results in paper [P. Oprocha, Invariant scrambled sets and distributional chaos, Dyn. Syst. 24 (2009), no. 1, 31–43.] to the case continuous maps of finite graphs. The Li-Yorke chaos is weaker than positive topological entropy. The equivalency arises when we add condition of invariance to Li-Yorke scrambled set. In this note we show that for a continuous graph map properties positive topological entropy; horseshoe; invariant Li-Yorke scrambled set; uniform invariant distributional chaotic scrambled set and distributionaly chaotic pair are mutually equivalent.     

Devaney’s chaos implies distributional chaos in a sequence

Let X be a complete metric space without isolated points, and let f:X→X be a continuous map. In this paper we prove that if f is transitive and has a periodic point of period p, then f is distributionally chaotic in a sequence. Particularly, chaos in the sense of Devaney is stronger than distributional chaos in a sequence.     

Distributional chaos for backward shifts

We provide sufficient conditions which give uniform distributional chaos for backward shift operators. We also compare distributional chaos with other well-studied notions of chaos for linear operators, like Devaney chaos and hypercyclicity, and show that Devaney chaos implies uniform distributional chaos for weighted backward shifts, but there are examples of backward shifts which are uniformly distributionally chaotic and not hypercyclic.     

How to minimize the control frequency to sustain transient chaos using partial control

In any control problem it is desirable to apply the control as infrequently as possible. In this paper we address the problem of how to minimize the frequency of control in presence of external perturbations, that we call disturbances, when the goal is to sustain transient chaos. We show here that the partial control method, that allows to find the minimum control required to sustain transient chaos in presence of disturbances, is the key to find such minimum control frequency. We prove first for the paradigmatic tent map of slope greater than 2 that for a constant value of the disturbances, the control required to sustain transient chaos decreases when the control is applied every k iterates of the map. We show that the combination of this property with the fact that the disturbances grow with k implies that there is a minimum control frequency and we provide a procedure to compute it. Finally we give evidence of the generality of this result showing that the same features are reproduced when considering the Hénon map.