Ginzburg－Landau－Newell模型的动力学行为

木文对Ｇｉｎｚｂｕｒｇ－Ｌａｎｄａｕ－Ｎｅｗｅｄ模型的动力学行为进行了讨论，得到了该模型的整体吸引子的存在性，同时得到了此吸引子维数的下界估计和该吸引子的Ｈａｕｓｄｏｒｆｆ维数和Ｗａｃｔａｌ（分形）维数的上界估计．     

半线性抛物方程在L2p空间中全局吸引子的存在性

本文研究了半线性抛物方程所生成的半群{S（t）}t≥0的吸引子的存在性.利用文献[1]中证明吸引子正则性的思想,分别得到半群{S（t）}t≥0在L^2p（Ω）空间中具有一个有界吸收集和一个全局吸引子.     

三维无界区域上弱阻尼非线性Schrodinger方程的指数吸引子

文［１］中证明了弱阻尼非线性Ｓｃｈｒｄｉｎｇｅｒ方程在无界区域ＲＮ（Ｎ≤３）上存在一个最大的紧吸引子．本文在此基础上得到了Ｒ３上指数吸引子的存在性     

耗散Zakharov方程组柯西问题的指数吸引子

文［１］中证明了耗散Ｚａｋｈａｒｏｖ方程组的最大吸引子的存在性．该文采用算子分解技术和构造犎２×犎１×犔２（犚）中渐进紧不变集的方法,得到了一维无界区域上耗散Ｚａｋｈａｒｏｖ方程组柯西问题的指数吸引子．     

一类吸引子的语法复杂性

本文研究了Ｆｅｉｇｅｎｂａｕｍ吸引子和周期窗口中Ｆｅｉｓｅｎｂａｕｍ吸引子决定的形式语言，讨论了它们的语法复杂性．证明了这类吸引子都是ＥＴＯＬ语言，从而是上下文有关语言（ＣＳＬ）；而不是上下文无关语言（ＣＦＬ）。     

耗散KDV型方程Cauchy问题的整体吸引子

该文对耗散ＫＤＶ型方程的动力学行为进行了讨论,得到了该方程在Ｈ＾２（Ｒ＾１）上存在整体吸引子。     

Kuramoto-Sivashinsky方程的渐近吸引子

本文给出了一周期边界条件下Ｋ－Ｓ方程的有限维渐进吸引子（Ａｓｙｍｐｔｏｔｉｃ Ａｔｔｒａｃｔｏｒ）而非通常的近似吸引子（Ａｐｐｒｏｘｉｍａｔｅ Ａｔｔｒａｃｔｏｒ）。     

耗散MKdV方程的指数吸引子

讨论了无界区域R~1上的MKdV方程,运用带权空间构造一类紧算子和算子分解的方法,得到该方程在H~2（R~1）上指数吸引子的存在性.     

迭代函数系统的吸收子的维数估计

某些迭代函数系统（ＩＦＳ）的吸引子的Ｈａｕｓｄｏｒｆｆ维数的经典估计通常由求解两个非线性方程的解得到。要相同条件下，本文给出卫些显示不等式估计，使经典估计是其中的特殊情形，而且某些ＩＦＳ是吸引子的维数可以通过求解满足一定条件的线性方程而得出。     

迭代函数系统的吸引子的维数估计

某些迭代函数系统（ＩＦＳ）的吸引子的Ｈａｕｓｄｏｒｆ维数的经典估计通常由求解两个非线性方程的解得到．在相同条件下，本文给出了一些显式不等式估计，使经典估计是其中的特殊情形．而且某些ＩＦＳ的吸引子的维数可以通过求解满足一定条件的非线性方程而得出     

Measure attractors and random attractors for stochastic partial differential equations

In the theory of stochastic differential equations we can distinguish between two kinds of attractors. The first one is the attractor (measure attractor) with respect to the Markov semigroup generated by a stochastic differential equation. The second meaning of attractors (random attractors) is to be understood with respect to each trajectory of the random equation. The aim of this paper is to bring together the two meanings of attractors. In particular, we show the existence of measure attractors if random attractors exist. We can also show the uniqueness of the stationary distributions of the stochastic Navier-Stokes equation if the viscosity is large     

Dissipative relativistic standard map: Periodic attractors and basins of attraction

The dissipative relativistic standard map, introduced by Ciubotariu et al. [Ciubotariu C, Badelita L, Stancu V. Chaos in dissipative relativistic standard maps. Chaos, Solitons & Fractals 2002;13:1253–67.], is further studied numerically for small damping in the resonant case. We find that the attractors are all periodic; their basins of attraction have fractal boundaries and are closely interwoven. The number of attractors increases with decreasing damping. For a very small damping, there are thousands of periodic attractors, comprising mostly of the lowest-period attractors of period one or two; the basin of attraction of these lowest-period attractors is significantly larger compared to the basins of the higher-period attractors.     

Random Point Attractors Versus Random Set Attractors

The notion of an attractor for a random dynamical system withrespect to a general collection of deterministic sets is introduced.This comprises, in particular, global point attractors and globalset attractors. After deriving a necessary and sufficient conditionfor existence of the corresponding attractors it is proved thata global set attractor always contains all unstable sets ofall of its subsets. Then it is shown that in general randompoint attractors, in contrast to deterministic point attractors,do not support all invariant measures of the system. However,for white noise systems it holds that the minimal point attractorsupports all invariant Markov measures of the system.     

Attractors for Multi-valued Non-autonomous Dynamical Systems: Relationship,Characterization and Robustness

In this paper we study cocycle attractors, pullback attractors and uniform attractors for multi-valued non-autonomous dynamical systems. We first consider the relationship between the three attractors and find that, under suitable conditions, they imply each other. Then, for generalized dynamical systems, we find that these attractors can be characterized by complete trajectories, which implies that the uniform attractor is lifted invariant, though it has no standard invariance by definition. Finally, we study both upper and lower semi-continuity of these attractors. A weak equi-attraction method is introduced to study the lower semi-continuity, and we show with an example the advantages of this method. A reaction-diffusion system and a scalar ordinary differential inclusion are studied as applications.     

Uniform attractors for non-autonomous motion equations of viscoelastic medium

Sufficient conditions for existence of minimal uniform trajectory attractors and uniform global attractors of non-autonomous evolution equations in Banach spaces are obtained. It is not assumed that the symbol space of an equation is a compact metric space and that the family of trajectory spaces corresponding to this symbol space is translation-coordinated or closed in any sense. Using these results, existence of minimal uniform trajectory attractors and uniform global attractors for weak solutions of the boundary value problem for motion equations of an incompressible viscoelastic medium with the Jeffreys constitutive law is shown.     

Generation and circuit implementation of fractional-order multi-scroll attractors

This paper considers the generating of multi-scroll chaotic attractors for a new fractional-order linear system by using the piecewise-linear function. Multi-scroll chaotic attractors are generated by extending the number of saddle equilibrium points with index 2. Poincaré map and maximum Lyapunov exponents are applied to verifying the chaotic behaviors of the generated multi-scroll chaotic attractors. A circuit for the multi-scroll attractor is designed and simulated. Moreover, physical experiment of 3-scroll attractors and 5-scroll attractors are implemented. The numerical simulation, the circuit simulation and hardware experimental results are in accordance with each other, which verifies the effectiveness and physical realization of the approach.     

Fractional standard map: Riemann–Liouville vs. Caputo

Properties of the phase space of the standard maps with memory obtained from the differential equations with the Riemann–Liouville and Caputo derivatives are considered. Properties of the attractors which these fractional dynamical systems demonstrate are different from properties of the regular and chaotic attractors of systems without memory: they exist in the asymptotic sense, different types of trajectories may lead to the same attracting points, trajectories may intersect, and chaotic attractors may overlap. Two maps have significant differences in the types of attractors they demonstrate and convergence of trajectories to the attracting points and trajectories. Still existence of the most remarkable new type of attractors, “cascade of bifurcation type trajectories”, is a common feature of both maps.     

Bounds for attractors and the existence of homoclinic orbits in the lorenz system

Frequency estimates are derived for the Lyapunov dimension of attractors of non-linear dynamical systems. A theorem on the localization of global attractors is proved for the Lorenz system. This theorem is applied to obtain upper bounds for the Lyapunov dimension of attractors and to prove the existence of homoclinic orbits in the Lorenz system.     

Attractors and Dimensions for Discretizations of a NLS Equation with a Non-local Nonlinear Term

In this paper we consider a semi-dicretized nonlinear Schrödinger (NLS) equation with local integral nonlinearity. It is proved that for each mesh size, there exist attractors for the discretized system. The bounds for the Hausdorff and fractal dimensions of the discrete attractors are obtained, and the various bounds are independent of the mesh sizes. Furthermore, numerical experiments are given and many interesting phenomena are observed such as limit cycles, chaotic attractors and a so-called crisis of the chaotic attractors.     

一类反应扩散方程组最大吸引子的正则性和维数估计

反应扩散方程组的最大吸引子通常是在不变区域内研究,如果不具有不变区域,或者去掉不变区域的限制而在全空间考虑这类问题,其结果如何？本文将证明一类反应扩散方程组在全空间最大吸引子的存在性,并对该吸引子的正则性进行了详细讨论,还给出了该吸引子的维数估计．