离散FitzHugh-Nagumo方程的整体吸引子和维数

该文对ＦｉｔｚＨｕｇｈ Ｎａｇｕｍｏ方程初边值问题用有限差分格式离散空间变量,证明了离散模型整体吸引子的存在性,并给出了与犿无关的Ｈａｕｓｄｏｒｆｆ维数和Ｆｒａｃｔａｌ维数上界估计。     

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．     

耦合Sine-Gordon方程组的渐近行为

本文考虑带Ｄｉｒｉｃｈｌｅｔ边界条件的耦合Ｓｉｎｅ－Ｇｏｒｄｏｎ方程组的渐近行为．证明了整体吸引子的存在性,并给出了整体吸引子的Ｈａｕｓｄｏｏｆ维数的上界估计．本文的估计与文［９］的结果相比有本质上的改进．在参数满足一定条件下,证明了整体吸引子恰好是系统的唯一平衡点     

Taylor-Couette流稳定性的谱方法研究

该文根据ｓｔｏｋｅｓ算子特征函数,利用谱方法研究了由轴对称Ｔａｙｌｏｒ Ｃｏｕｅｔｔｅ流导出的多模态方程．给出了三模态方程平衡点存在的条件,证明了它的吸引子的存在性,并给出其Ｈａｕｓ ｄｏｒｆｆ维数的上界的估计．     

一类二维非线性Schrodinger方程的吸引子及其维数

本文研究了一类二维非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程解的有限维行为，我们得到了此方程存在吸引子，并得到了此吸引子维数的上界估计     

Sine—Gordon方程的全局吸引子的维数估计

本文得到了阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的狄氏问题的全局吸引子的Ｈａｕｓｄｏｒｆｆ维数以偶数上界的参数条件，特别地，当阻尼与Ｌａｐｌａｅ算子的第一个特征值适当大时，全局吸引子是零维的，零维吸引子恰是系统的唯一平衡解并且指数吸引相空间的有界集。     

一类二维非线性Schroedinger方程的吸引子及其维数

本研究了一类二维非线性Schrodinger方程解的有限维行为，我们得到了此方程存在吸引子，并得到了此吸引子维数的上界估计     

带五次项的NLS方程及其谱逼近的整体吸引子的维数估计

通过给出一般发展方程和其近似方程解的整体吸引子的Hausdorff维数上界间的关系，继[1，2]的讨论，本文进一步得到了带五次项的NLS方程和半离散Fourier谱近似解的整体吸引子的Hausdorff维数的上界估计。     

广义耦合FitzHugh-Nagumo方程解的渐近行为

本文研究了广义耦合FitzHugh—Nagumo方程及广义离散耦合FitzHugh-Nagumo方程，分别证明了连续模型及离散模型整体吸引子的存在性，并给出了其Huasdorff维数估计，其中离散模型的Hausdorff维数上界与n无关．     

Belousov－Zhabotinskii化学反应Field－Noyes模型的整体吸引子和惯性流形

本文讨论Ｂｅｌｏｕｓｏｖ—Ｚｈａｂｏｔｉｎｓｋｉｉ化学反应Ｆｉｅｌｄ—Ｎｏｙｅｓ模型（三维的方程组）整体吸引子的存在性、维数估计以及惯性流形的存在性．     

Attractors and Dimensions for Discretizations of a Dissipative Zakharov Equations

Abstract In this paper, a dissipative Zakharov equations are discretized by difference method.We make priorestimates for the algebric system of equations. It is proved that for each mesh size,there exist attractors forthe discretized system.The bounds of the Hausdorff dimensions of the discrete attractors are obtained,and thevarious bounds are dependent of the mesh sizes.     

Attractors and Dimensions for Discretizations of a Generalized Ginzburg-Landau Equation

In this paper, we discretize the generalized Ginzburg-Landau equations with the periodic boundary condition by the finite difference method in spatial direction. It is proved that for each mesh size, there exist at tractors for the discretized systems. The bounds for the Hausdorff dimensions of the discrete attractors are obtained, and the various bounds are independtmt of tho mesh sizes.     

ATTRACTORS FOR DISCRETIZATION OF GINZBURG-LANDAU-BBM EQUATIONS

1. IntroductionIn this paper) we consider the following periodic initial value problem for the system ofGinzburg-Landau equation coupled with BBM equationwhere e(x,t) is a complex function, n(x, t) is a real scalar function, at a, 5, 7, al, a2, FI, adZare real constants, and gi (x), g200 are given real functions.This problem describes the nonlinear interactions between Langmuir wave and ion acousticwave in plasma physics, e(x, t) denotes electric field, n(x, t) the perturbation of density (…     

A linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg–Landau equation

In this paper, we study a linearized Crank–Nicolson Galerkin finite element method for solving the nonlinear fractional Ginzburg–Landau equation. The boundedness, existence and uniqueness of the numerical solution are studied in details. Then we prove that the optimal error estimates hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed. Finally, numerical tests are investigated to support our theoretical analysis.     

On the hydrodynamic limit of Ginzburg‐Landau wave vortices

In this paper, we study the hydrodynamic limit of the finite Ginzburg‐Landau wave vortices, which was established in [16]. Unlike the classical vortex method for incompressible Euler equations, we prove here that the densities approximated by the vortex blob method associated with the Ginzburg‐Landau wave vortices tend to the solutions of the pressure‐less compressible Euler‐Poisson equations. The convergence of such approximation is proved before the formation of singularities in the limit system as the blob sizes and the grid sizes tend to zero in appropriate rates. © 2002 John Wiley & Sons, Inc.     

Adiabatic Limit for Hyperbolic Ginzburg–Landau Equations

We study the adiabatic limit in hyperbolic Ginzburg–Landau equations which are Euler–Lagrange equations for the Abelian Higgs model. Solutions of Ginzburg–Landau equations in this limit converge to geodesics on the moduli space of static solutions in the metric determined by the kinetic energy of the system. According to heuristic adiabatic principle, every solution of Ginzburg–Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some geodesic. A rigorous proof of this result was proposed recently by Palvelev.     

Well-posedness of the fractional Ginzburg–Landau equation

In this paper, we investigate the well-posedness of the real fractional Ginzburg–Landau equation in several different function spaces, which have been used to deal with the Burgers’ equation, the semilinear heat equation, the Navier–Stokes equations, etc. The long time asymptotic behavior of the nonnegative global solutions is also studied in details.     

Mesh-dependent stability for finite element approximations of parabolic equations with mass lumping

In this paper, based on some mesh-dependent estimates on the extreme eigenvalues of a general finite element system defined on a simplicial mesh, novel and sharp bounds on the permissible time step size are derived for the mass lumping finite element approximations of parabolic equations. The bounds are dependent not only on the mesh size but also on the mesh shape. These results provide guidance to the stability of numerical solutions of parabolic problems in relation to the unstructured geometric meshing. Numerical experiments on both uniform meshes and adaptive meshes are presented to validate the theoretical analysis.     

Mesh-dependent stability for finite element approximations of parabolic equations with mass lumping

In this paper, based on some mesh-dependent estimates on the extreme eigenvalues of a general finite element system defined on a simplicial mesh, novel and sharp bounds on the permissible time step size are derived for the mass lumping finite element approximations of parabolic equations. The bounds are dependent not only on the mesh size but also on the mesh shape. These results provide guidance to the stability of numerical solutions of parabolic problems in relation to the unstructured geometric meshing. Numerical experiments on both uniform meshes and adaptive meshes are presented to validate the theoretical analysis.     

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.