Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains

We prove the existence of a compact random attractor for the stochastic Benjamin-Bona-Mahony equation defined on an unbounded domain. This random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity.     

Pullback attractors for asymptotically compact non-autonomous dynamical systems

First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, Schmalfuss, amongst others) which means the existence of a (random or time-dependent) family of compact attracting sets. Next, we prove a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. This attractor is minimal and, in most practical applications, it is unique. Finally, we illustrate the theory with a 2D Navier–Stokes model in an unbounded domain.     

无界区域上具线性阻尼的二维 Navier-Stokes 方程的拉回吸引子

该文首先介绍拉回渐近紧非自治动力系统的概念, 给出非自治动力系统拉回吸引子存在定理. 最后证明了无界区域上具线性阻尼的二维Navier-Stokes 方程的拉回吸引子的存在性, 并给出了其Fractal维数估计.     

Existence of backwards-compact pullback attractors for non-autonomous lattice dynamical systems

A pullback attractor is called backward compact if the union of attractors over the past time is pre-compact. We show that this kind of attractor exists for the first-order non-autonomous lattice dynamical system when the external force is backwards tempered and backwards asymptotically tail-null.     

Dynamical study and robustness for a nonlinear wastewater treatment model

In this work we deal with a wastewater treatment by using the activated sludge process. The problem is formulated as a nonlinear dynamical system. Firstly, we develop the dynamical study of the model when all parameters are well known. Hence, basic properties of invariance and dissipation are established and, under a suitable condition on parameters, a globally asymptotically stable equilibrium point occurs. Secondly, when the bacterium growth function and the substrate concentration in the feed stream are not well known, the robustness analysis provides the existence of an attractor domain to which all trajectories of the system converge. Finally, we prove that we can reduce the size (volume) of this attractor domain by increasing the recycle rate to a maximum fixed level.     

Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equations

A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynamical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.     

A note on multistep methods and attracting sets of dynamical systems

Summary We consider a dynamical system described by an autonomous ODE with an asymptotically stable attractor, a compact set of orbitrary shape, for which the stability can be characterized by a Lyapunov function. Using recent results of Eirola and Nevanlinna [1], we establish a uniform estimate for the change in value of this Lyapunov function on discrete trajectories of a consistent, strictly stable multistep method approximating the dynamical system. This estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in Kloeden and Lorenz [3, 4] for 1-step methods.This work was supported by the U.S. Department of Energy Contract DE-A503-76 ER72012     

关于吸引子的某些性质

Conley在[1]中讨论拓扑空间X上动力系统的吸引子——排斥子对时,给出了吸引子存在的两个充分条件．本文证明这两个条件实际上也是必要的,进而证明Conley所定义的吸引子是渐近稳定的．     

关于吸引子的某些性质

Conley在[1]中讨论拓扑空间X上动力系统的吸引子——排斥子对时，给出了吸引子存在的两个充分条件．本文证明这两个条件实际上也是必要的，进而证明Conley所定义的吸引子是渐近稳定的．     

Approximation on attraction domain of Cohen-Grossberg neural networks

In this paper, approximations of attraction domains of the asymptotically stable equilibrium points of some typical Cohen-Grossberg neural networks are achieved. Most Cohen-Grossberg neural networks are highly nonlinear systems which makes it difficult to approximate their attraction domain. Under some weak assumptions, we are allowed to employ the optimal Lyapunov method to obtain a Lyapunov function for asymptotically stable equilibrium points of a given Cohen-Grossberg neural network. With the help of this Lyapunov function, we approximate the corresponding attraction domain by the iterative expansion approach. Numerical simulations also illustrate that the approximation obtained is really part of the attraction domain.     

Dynamics of systems on infinite lattices

The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction-diffusion equation has a global attractor in a weighted l² space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction-diffusion equation is approached by finite-dimensional systems.     

Global Attractor of Hindmarsh-Rose Equations in Neurodynamics

Global dynamics for a new mathematical model in neurodynamics of the diffusive Hindmarsh-Rose equations on a bounded domain is investigated in this paper. The existence of a global attractor and its regularity are proved through uniform estimates showing the dissipative properties and the asymptotically compact and smoothing characteristics.     

动力系统中的Lipschitz稳定性

本文讨论动力系统的Ｌｉｐｓｃｈｉｔｚ稳定性与吸引性之间的关系．给出动力系统中弱吸引性、吸引性和强吸引性这三个概念相互等价的条件，在一定的条件下证明了（弱、强）吸引子与全局（弱、强）吸引子是一致的．本文还讨论了度量Ｌｙａｐｕｎｏｖ稳定性和拓扑Ｌｙａｐｕｎｏｖ稳定性及它们与Ｌｉｐｓｃｈｉｔｚ稳定性之间的关系．     

Random attractor of stochastic complex Ginzburg–Landau equation with multiplicative noise on unbounded domain

The existence of a compact random attractor for the stochastic complex Ginzburg–Landau equation with multiplicative noise has been investigated on unbounded domain. The solutions are considered in suitable spaces with weights. According to crucial properties of Ornstein–Uhlenbeck process, using the tail-estimates method, the key uniform a priori estimates for the tail of solutions have been obtained, which give the asymptotic compactness of random attractors. Then the existence of a compact random attractor for the corresponding dynamical system is proved in suitable spaces with weights.     

Existence of traveling waves in compressible Euler equations with viscosity and capillarity

Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added. For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of first-order, which admits an asymptotically stable equilibrium point and a saddle point. We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point in fact lies on the boundary of this set. Then, we establish a saddle-to-stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point. This gives us a traveling wave of the viscous–capillary compressible Euler equations.     

Finite fractal dimension of pullback attractors for non-autonomous 2D Navier–Stokes equations in some unbounded domains

We study the asymptotic behaviour of non-autonomous 2D Navier–Stokes equations in unbounded domains for which a Poincaré inequality holds. In particular, we give sufficient conditions for their pullback attractor to have finite fractal dimension. The existence of pullback attractors in this framework comes from the existence of bounded absorbing sets of pullback asymptotically compact processes [T. Caraballo, G. ?ukaszewicz, J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal. 64 (3) (2006) 484–498]. We show that, under suitable conditions, the method of Lyapunov exponents in [P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 (1984) [5]] for the dimension of attractors can be developed in this new context.     

A uniformly exponential random forward attractor which is not a pullback attractor

A compact random set is constructed which is invariant with respect to a random dynamical system, and which is an exponential forward attractor, but fails to be a pullback attractor.     

Random attractors for stochastic 2D-Navier–Stokes equations in some unbounded domains

We show that the stochastic flow generated by the 2-dimensional Stochastic Navier–Stokes equations with rough noise on a Poincaré-like domain has a unique random attractor. One of the technical problems associated with the rough noise is overcomed by the use of the corresponding Cameron–Martin (or reproducing kernel Hilbert) space. Our results complement the result by Brze?niak and Li (2006) [10] who showed that the corresponding flow is asymptotically compact and also generalize Caraballo et al. (2006) [12] who proved existence of a unique attractor for the time-dependent deterministic Navier–Stokes equations.     

非自治Klein-Gordon-Schr(o)dinger格点动力系统的一致吸引子

首先证明了耗散的非自治Klein-Gordon-Schr(o)dinger格点动力系统的解确定的一族过程的紧一致吸引子的存在性.其次得到了该紧一致吸引子的Kolmogorov熵的一个上界.最后建立了该紧一致吸引子的上半连续性.     

渐近紧随机动力系统及其极限集的性质

通过对随机动力系统极限行为的研究,推广了传统动力系统的相关定义和理论.由于在无界区域上,Sobolevr紧嵌入的缺乏,Crauel,Debussche及Flancloli等人在研究有界区域上的随机演化系统时所引入的渐近紧的概念不再适用.通过对Ladyzlmnskaya,Rosa等人在对确定性系统的研究中所提出的渐近紧概念的推广,引入了随机动力系统渐近紧的概念,并以相应的示例及严格的理论推导证明了此概念的合理性和必要性.最后,作为这一新的概念的应用,证明了渐近紧随机动力系统极限集的性质.