连续树映射非游荡集的拓扑结构

本文研究树（即不含有圈的一维紧致连通的分支流形）上连续自映射的非游荡集的拓扑结构．证明了孤立的周期点都是孤立的非游荡点；具有无限轨道的非游荡点集的聚点都是周期点集的二阶聚点，以及ω－极限集的导集等于周期点集的导集和非游荡集的二阶导集等于周期点集的二阶导集．     

拓扑熵为零的混乱映射的充分条件

线段上的连续自映射,当周期点集为闭集时,其轨道十分简单,当然,动力系统不会是混乱的,因此,研究周期点集的聚点的极限性态与混乱的关系,无疑可以进一步揭示混乱现象产生的原因。文[6]证明了当回归点集非闭,f是混乱的。本文则给出了周期点集非闭时f为混乱的充分条件。这说明了只要周期点集非闭动力系统就可能是混乱的。     

有非2方幂周期轨道的连续函数的特征

本文给出了区间I上的连续函数f的广义CR型异状点的定义。此定义是一些文献中已考虑过的几种类型的异状点概念的推广.通过对f或f~n上的RL~∞型单峰(或反单峰)轨道的存在性的探讨,本文证明了,f含有周期不等于2的(整数)方幂的周期轨道,其充要条件是f含有广义CR型异状点.对于满足这样的条件的连续函数f,通过对各种RL-序列的组合性质的分析,我们发现了在f~n(n等于2的某一方幂)的一个单峰(或反单峰)子系统f~n|Y上即存在着不可数个广义CR型异状点及不可数个具有某些其它性质的点.     

关于赋范空间上连续自映射的回归点

在赋范空间中讨论回归点的性质,主要得到了结果：（1）如果,是序列紧赋范空间X上的连续双射,x是f的任一回归点,则对于任意整数N〉0都存在f的回归点x0∈X使得f^n（x0）=x;（2）序列紧赋范空间上连续自映射的回归点集是f的强不变子集;（3）如果f是局部连通赋范空间X上的连续自映射,则f的每一个回归点或是类周期点或是类周期点的聚点．作为推论,在实直线段上得到了类似的结论．     

伪轨跟踪与伪移位不变集

本文证明了对于具有伪轨跟踪性质的紧致度量空间上的连续满射ｆ，存在某个正整数ｎ，有相对于ｆｎ的伪移位不变集的充要条件是存在非回归点的链回归点，或是存在非几乎周期点的回归点     

∑上的非弱几乎周期的回复点集和SS混沌集

该文对单边符号空间上的转移自映射进行了讨论,证明了在非弱几乎周期的回复点集上存在不可数的SS混沌集.     

单峰函数中具有指定的型的终于周期轨道的存在性

本文给出了捏制序列为ＢＣ的单峰函数具有ＢＣ＊Ｌ￣∞型周期轨道的几个充分条件。在此基础上，本文进一步讨论了一般的单峰函数ｆ中的终于周期轨道的存在性问题，证明了只要ｋ阶的终于ｎ－周期容许序列ＡＢ￣∞满足一定的要求，ｆ中就存在着ＡＢ￣∞型的ｋ阶终于ｎ－周期轨道。对广泛的一类单峰函数，本文减弱和简化了文献［１］的ＰａｒｔⅡ．３的主要定理的条件并加强了它的结论。对更一般的连续函数，本文亦充实了文献［８］的定理２的结论。     

标准序列逻辑系统S_3中公式的概率真度理论

利用势为3的非均匀概率空间的无穷乘积在三值标准序列逻辑系统中引入了公式的概率真度概念,证明了全体公式的概率真度值之集在[0,1]中没有孤立点;利用概率真度定义了概率相似度和伪距离,进而建立了概率逻辑度量空间,证明了该空间中没有孤立点,为三值命题的近似推理理论提供了一种可能的框架.     

Sarkovskii定理（续）

Ｓａｒｋｏｖｓｋｉ定理（续）万哲先５Ｓａｒｋｏｖｓｋｉ定理先证明一条引理引理８设ｆ是定义在闭区间Ｉ上的连续实函数，并且ｆ（Ｉ）Ｉ．再设ｆ有极小周期２ｎ＋１的周期点ｘ０，但是对于任意＜ｎ的自然数ｍ，ｆ都没有极小周期２ｍ＋１的周期点．设在含ｘ０的轨道中...     

关于一类自映射轨道的研究

1 概念及已有结果 设X为拓扑空间,f∈C0（X,X）,f0表示恒等映射,对任意自然数n,定义fn=fοfn-1. 称O（x,f）={fn（x）│n=0,1,2,… ;x∈X}为x的f轨道. 关于周期点、周期点集、周期、周期轨道,Sarkovskii序如通常定义,可参见[1].     

Homoclinic orbits for nonlinear difference equations containing both advance and retardation

In this paper we discuss how to use the critical point theory to study the existence of a nontrivial homoclinic orbit for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of homoclinic orbits is obtained.     

On the global dynamics of periodic triangular maps

This paper is an extension of an earlier paper that dealt with global dynamics in autonomous triangular maps. In the current paper, we extend the results on global dynamics of autonomous triangular maps to periodic non-autonomous triangular maps. We show that, under certain conditions, the orbit of every point in a periodic non-autonomous triangular map converges to a fixed point (respectively, periodic orbit of period p) if and only if there is no periodic orbit of prime period two (respectively, periodic orbits of prime period greater than p).     

Global dynamic behavior of a predator–prey model under ratio-dependent state impulsive control

This paper studies the global dynamic behavior of a prey–predator model with square root functional response under ratio-dependent state impulsive control strategy. It is shown that the boundary equilibrium point of the controlled system is globally asymptotically stable. An order-k periodic orbit is obtained by employing the Brouwer’s fixed point theorem. Furthermore, the critical values are determined for the existence of orbitally asymptotically stable order-1 and order-2 periodic orbits in finite time. These critical values play an important role in determining different kinds of order-k periodic orbits and can also be used for designing the control parameters to obtain the desirable dynamic behavior of the controlled prey–predator system. Moreover, it is found that the local equilibrium point is also globally asymptotically stable under the control strategy. Numerical examples are provided to validate the effectiveness and feasibility of the theoretical results.     

树映射的链等价集与拓扑熵

本文讨论了树映射f的链等价集的性质,得到了f具有零拓扑熵的几个等价条件,并证明了:如果 f的一个链等价集是个无限集,那么这个链等价集的任何孤立点都是f的非周期的终于周期点.     

The period on a family of non-linear oscillations and periodic motions of a perturbed system at a critical point of the family

Single-frequency oscillations of a reversible mechanical system are considered. It is shown that the oscillation period of a non-linear system usually only depends on a single parameter and it is established that, at a critical point of the family, at which the derivative of the period with respect to the parameter vanishes, due to the action of perturbations two families of symmetrical resonance periodic motions are produced. The oscillations of a satellite in an elliptic orbit, due to the action of gravitational and aerodynamic moments, are considered as an example. The operations in a circular orbit are investigated in detail initially, and then in an elliptical orbit of small eccentricity.     

Two-sided approximation to periodic solutions of ordinary differential equations

Summary A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered. First some results about variational equation systems for periodic solutions are obtained in Sect. 2. Then it is proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively. Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex.The Work is supported by the National Natural Science Foundation of China     

一种非线性函数的迭代动力分析

对一种非线性函数迭代的动力性质进行了分析,结果显示在函数的定义区间[－2,2]内,点χ的移动轨道具有如下性质：（i）点χ的移动轨道具有各种复杂的周期轨道,所有周期轨道形成的集合在区间[－2,2]内稠密,（ii）点χ的移动轨道或者在区间（－2,2）内稠密,或者经有限次映射后最终进入由所有的周期轨道形成的集合中。     

Recurrence and Almost Periodicity in Transformation Groups

In [5] Gottschalk comments that an interesting question is to find conditions under which pointwise recurrence in a transformation group (X,T) implies pointwise almost periodicity. In this paper we show that if in a transformation group (X,T), the phase space X is connected locally compact and Hausdorff and T is generative and each point xεX is of P-characteristic 0 for each replete semi-group P in T, then the existence of a single recurrent point implies that (X,T) is pointwise recurrent and each orbit closure is minimal. This implies of course from a general result of Gottschalk and Hedlund that the transformation group is pointwise almost periodic. The condition P-characteristic 0 is weaker than pointwise equicontinuity even when the phase space is compact--see example (4.6) at the end.     

On Quasi-weakly Almost Periodic Points of Continuous Flows

Let X be a compact metric space, F : X ×R→ X be a continuous flow and x ∈ X a proper quasi-weakly almost periodic point, that is, x is quasi-weakly almost periodic but not weakly almost periodic. The aim of this paper is to investigate whether there exists an invariant measure generated by the orbit of x such that the support of this measure coincides with the minimal center of attraction of x? In order to solve the problem, two continuous flows are constructed. In one continuous flow,there exist a proper quasi-weakly almost periodic point and an invariant measure generated by its orbit such that the support of this measure coincides with its minimal center of attraction; and in the other,there is a proper quasi-weakly almost periodic point such that the support of any invariant measure generated by its orbit is properly contained in its minimal center of attraction. So the mentioned problem is sufficiently answered in the paper.     

Homoclinic Orbits of Nonlinear Functional Difference Equations

In this paper, by using the critical point theory, we obtain the existence of a nontrivial homoclinic orbit which decays exponentially at infinity for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of nontrivial homoclinic orbits which decay exponentially at infinity is obtained.