轨道结构的层次与拓扑半共轭

对离散系统提出轨道结构存在3个层次的观念,引进拟弱几乎周期点的概念和关于拓扑半共轭的极小覆盖的概念,讨论了轨道结构3个层次与点的回复性之间的关系以及某些性质在拓扑半共轭下的保持问题.     

熵极小动力系统的复杂性

设X是一个紧致度量空间,f:X→X是一个连续映射,(X,f)是熵极小的.该文首先证明了f是强遍历的;另外,如果还假设X中存在f的一个真的(拟)弱几乎周期点,则得到f具有正拓扑熵且对任意的n1,f~n是遍历敏感依赖的.因此,f在Li-Yorke和Takens-Ruelle意义下是混沌的.该文所得结论改进和推广了最近的一些结论.     

关于拟弱几乎周期点的一些结论和公开问题(英文)

本文主要在离散动力系统中,概括了有关于拟弱几乎周期点的一些结论并提出了一些公开问题.     

几乎周期性与SS混沌集

引进正则移位不变集的概念,证明了有正则移位不变集的紧致系统在几乎周期点集中存在SS混沌集,特别地,具有正拓扑熵的区间映射在几乎周期点集中存在SS混沌集.     

几乎周期性与SS混沌集

引进正则移位不变集的概念,证明了有正则移位不变集的紧致系统在几乎周期点集中存在SS混沌集,特别地,具有正拓扑熵的区间映射在几乎周期点集中存在SS混沌集.     

一个单峰函数族的模型

总的说来,人们对单峰函数族的动态性质已有较深刻的理解.如 MSS 序列的提出,周期窗口的发现,以及从不同角度对其动态性质的探讨都表明了这点.但是对参数所对应的函数拓扑熵大于零的混沌区域的结构还没有彻底研究清楚,本文讨论一个典型的单峰函数族,发现其混沌区域具有分层的结构,即可按层的次序把这个区域中的周期点、拟周期点和其他点的分布规律刻划出来.     

弱几乎周期点和测度中心

在动力系统讨论中往往出现干扰或假象.为了排除它们,本文引进测度中心概念,而为了决定测度中心的结构,又引进弱几乎周期点的概念.我们完全决定了测度中心的结构,并给出例子说明,测度中心可以真包含于运动中心以及存在在非游荡集上紊动但有零熵的系统.     

多间隙二级齿轮非线性振动分岔特性研究

采用集中质量法,建立了多间隙二级齿轮系统的五自由度非线性振动模型.模型考虑了各齿轮副间变刚度、齿侧间隙、支承间隙以及传动误差等非线性因素,推导出系统量纲振动微分方程,并利用分岔图、Poincaré截面图,全面地分析了系统转速、阻尼比对系统分岔特性的影响.结果发现系统在各种非线性因素的综合影响下,表现出丰富复杂的分岔特性.系统随着参数的变化先后出现短周期运动、长周期运动、拟周期运动及混沌运动.在不同阻尼比下,系统随着转速的逐渐减小,由稳定的周期1运动,倍化分岔变为稳定的周期2运动,再经过Hopf分岔变为拟周期运动,通过激变又变为稳定的周期1运动,最终通过Hopf分岔-锁相进入混沌.随着转速的逐渐增大,系统随阻尼比变化的混沌运动范围减小,出现稳定的周期1运动、长周期和拟周期运动,并且长周期和拟周期运动范围逐渐变小而稳定的周期1运动的范围逐渐变大.     

具有弱specification性质的amenable群作用的拟弱几乎周期点的回复层次（英文）

设(X,G)是一个G-系统,{F_n}_n=1^∞是G的一个F?lner序列.本文证明了如果(X,G)具有强specification性质,则在(X,G)上存在一个相对于G的F?lner序列{F_n}_n=1^∞的真拟弱几乎周期点x和一个由x沿着{F_n}_n=1^∞的轨道生成的不变测度μ,使得μ的支撑与x的相对于F?lner序列{F_n}_n=1^∞的极小吸引中心相同.此外,如果(X,G)有弱specification性质且(X,G)的周期测度在(X,G)的不变测度集合M(X,G)中稠密,则在(X,G)上存在一个相对于G上的F?lner序列{F_n}_n=1^∞的真拟弱几乎周期点y,使得每一个由y的轨道沿F?lner序列{F_n}_n=1^∞...     

小波变换在分叉与混沌研究中的应用

非线性振动系统中的运动形式有三种可能：周期运动、拟周期运动和混沌·用Ｐｏｉｎｃａｒé映射可确定出系统周期运动，用谐波小波变换可区分拟周期运动和混沌·由此可准确地确定出参数空间中各种不同形式运动所对应的存在域·     

On quasi-weakly almost periodic points

The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) measures are important, such orbits form a full measure set for all invariant measures of the system, its closure is called the measure center of the system. To investigate this set, Zhou introduced the notions of weakly almost periodic point and quasi-weakly almost periodic point in 1990s, and presented some open problems on complexity of discrete dynamical systems in 2004. One of the open problems is as follows: for a quasi-weakly almost periodic point but not weakly almost periodic, is there an invariant measure generated by its orbit such that the support of this measure is equal to its minimal center of attraction (a closed invariant set which attracts its orbit statistically for every point and has no proper subset with this property)? Up to now, the problem remains open. In this paper, we construct two points in the one-sided shift system of two symbols, each of them generates a sub-shift system. One gives a positive answer to the question above, the other answers in the negative. Thus we solve the open problem completely. More important, the two examples show that a proper quasi-weakly almost periodic orbit behaves very differently with weakly almost periodic orbit.     

A note on quasi-weakly almost periodic point

Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597–606(2013)] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in[Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17, 493–502(2004)]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing.     

Recurrence of Transitive Points in Dynamical Systems with the Specification Property

Let T:X → X be a continuous map of a compact metric space X. A point x ∈ X is called Banach recurrent point if for all neighborhood V of x, {n ∈ N:Tⁿ(x) ∈ V } has positive upper Banach density. Denote by Tr(T), W(T), QW(T) and BR(T) the sets of transitive points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points of (X, T). If (X, T) has the specification property, then we show that every transitive point is Banach recurrent and ∅≠W(T) ∩ Tr(T) _≠^⊂W^*(T) ∩ Tr(T) _≠^⊂ QW(T) ∩ Tr(T) _≠^⊂ BR(T) ∩ Tr(T), in which W^*(T) is a recurrent points set related to an open question posed by Zhou and Feng. Specifically the set Tr(T) ∩ W^*(T)\W(T) is residual in X. Moreover, we construct a point x ∈ BR\QW in symbol dynamical system, and demonstrate that the sets W(T), QW(T) and BR(T) of a dynamical system are all Borel sets.     

TOPOLOGICAL ENTROPY AND IRREGULAR RECURRENCE

This paper is devoted to problems stated by Z. Zhou and F. Li in 2009. They concern relations between almost periodic, weakly almost periodic, and quasi-weakly almost periodic points of a continuous map $f$ and its topological entropy. The negative answer follows by our recent paper. But for continuous maps of the interval and other more general one-dimensional spaces we give more results; in some cases the answer is positive.     

一个A（f）≠M（f）的紧致系统（X，f）

本文证明，存在紧致系统，其几乎周期点集闭包不等于其测度中心．藉此，我们否定地回答了两个未解决的问题．     

一致空间上非自治系统的Banks定理

研究一致空间上非自治系统的敏感性,证明了定义在无限Hausdorff一致空间上有限生成的非自治系统满足拓扑传递性、周期点稠密以及存在两个不相交的不变周期点是初值敏感依赖的.从而推广了经典的Banks定理以及Li和Yang在2018年的一个结果.     

Two New Recurrent Levels for <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>-Flows

The central problem in dynamical systems is the asymptotic behavior or topological structure of the orbits. Nevertheless only orbits of points with certain recurrence and form a set of full measure are truly of importance. Of course, such a set is desired to be as small (in the sense of set inclusion) as possible. In this paper we discuss such two sets: the set of weakly almost periodic points and the set of quasi-weakly almost periodic points. While the two sets are different from each other by definitions, we prove that their closures both coincide with the measure center (or the minimal center of attraction) of the dynamical systems. Generally, a point may have three levels of orbit-structure: the support of an invariant measure generated by the point, its minimal center of attraction and its ω-limit set. We study the three levels of orbit-structure for weakly almost periodic points and quasi-weakly almost periodic points. We prove that quasi-weakly almost periodic points possess especially rich topological orbit-structures. We also present a necessary and sufficient condition for a point to belong to its own minimal center of attraction.     

半群作用Li-Yorke对的存在性

设X为紧度量空间,T为半群,本文研究了动力系统（X,T）上Li-Yorke对的存在性问题,证明了当（X,T）拓扑可迁且包含周期点时,在（X,T）上存在无限scrambled集.另外,列举了一些不包含Li-Yorke对的动力系统.     

On the Simplest Topologically Transitive Skew Products in the Plane

In this paper, we obtain (sufficient) conditions under which the set of periodic points of the topologically transitive skew product of mappings of an interval is dense in the phase space of the dynamical system considered.     

On the Shadowing Property and Shadowable Point of Set-valued Dynamical Systems

In this article, the authors introduce the concept of shadowable points for set-valued dynamical systems, the pointwise version of the shadowing property, and prove that a set-valued dynamical system has the shadowing property iff every point in the phase space is shadowable; every chain transitive set-valued dynamical system has either the shadowing property or no shadowable points; and for a set-valued dynamical system there exists a shadowable point iff there exists a minimal shadowable point. In the end, it is proved that a set-valued dynamical system with the shadowing property is totally transitive iff it is mixing and iff it has the specification property.