关于时滞微分方程生成的半流的不动点

本文主要讨论了一类时滞微分方程生成的半流的不动点，并得到其相关性质。     

Weak-mixing implies sensitive dependence

Let X be a metric space, B the σ-algebra of Borel subsets of X, and μ a probability measure on (X,B). In this note, for a measure-preserving map T (respectively a measure-preserving semi-flow φ) on (X,B,μ), we prove that if suppμ=X, and T (respectively φ) is weak-mixing, then T (respectively φ) has sensitive dependence.     

Asymptotic stability of discrete cocycles

We introduce the notion of asymptotic stability of sequences of multifunctions associated with discrete cocycles. Some sufficient conditions for existence of attracting sets are given. The use of the topological (Kuratowski's) limits, as less complicated as commonly used Hausdorff metric, let us to weaken many standard assumptions. We show that in considered case existence of attractor is a property of a cocycle mapping itself and does not depend on properties of a parameter nor a state space. The obtained results generalize earlier on iterated function systems and can be applied for non-autonomous as well as random dynamical systems.     

On approximations of the Euler-Peano scheme for multivalued stochastic differential equations

It is known that a unique strong solution exists for multivalued stochastic differential equations under the Lipschitz continuity and linear growth conditions. In this paper we apply the Euler-Peano scheme to show that existence of weak solution and pathwise uniqueness still hold when the coefficients are random and satisfy one-sided locally Lipschitz continuous and an integral condition (i.e. Krylov's conditions put forward in On Kolmogorov's equations for finite-dimensional diffusions, Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998), Lecture Notes in Math., 1715, Springer, Berlin, 1999, pp. 1–63). When the coefficients are nonrandom and possibly discontinuous but only satisfy some integral conditions, the sequence of solutions of the Euler-Peano scheme converges weakly, and the limit is a weak solution of the corresponding MSDE. As a particular case, we obtain a global semi-flow for stochastic differential equations reflected in closed, convex domains.     

连续自映射及其扭扩半流的不变测度

本文研究了紧致度量空间上连续自映射及连续半流的不变测度,并且证明了如下结论:(1)在拓扑等价的无不动点的连续半流的不变测度之间以及在连续自映射及其扭扩半流的不变测度之间存在一一对应;(2)作为(1)的应用,给出如下结论(见[2,定理2.1]):“环面上无不动点的连续流是唯一遍历的当且仅当它至多有一条周期轨”一个易接受的证明.     

Dynamics of a delayed Lotka–Volterra competition model with directed dispersal

We consider a diffusive Lotka–Volterra competition system with stage structure, where the intrinsic growth rates and the carrying capacities of the species are assumed spatially heterogeneous. Here, we also assume each of the competing populations chooses its dispersal strategy as the tendency to have a distribution proportional to a certain positive prescribed function. We give the effects of dispersal strategy, delay, the intrinsic growth rates and the competition parameters on the global dynamics of the delayed reaction diffusion model. Our result shows that competitive exclusion occurs when one of the diffusion strategies is proportional to the carrying capacity, while the other is not; while both populations can coexist if the competition favors the latter species. Finally, we point out that the method is also applied to the global dynamics of other kinds of delayed competition models.     

半流的原像熵

本文对紧致度量空间上的连续半流引入了几类原像熵的定义,并对它们的性质进行了研究,证明了对于无不动点的连续半流而言,这些熵具有一定程度的拓扑共轭不变性,对这些熵的关系进行了研究并得到了联系这些熵的不等式,还证明了连续半流与其时刻1映射具有相同的拓扑熵和原像熵。     

具有跟踪性质的连续半流

在这篇论文中，我们给出了连续半流的跟踪性质与其逆极限的跟踪性质之间的一些等价条件，并且做为应用，我们证明了具有强跟踪性质的连续半流在其游荡集上的限制也具有强跟踪性质以及连续半流的谱分解定理。