Limit behavior and on the attractor for the equations of motion of Oldroyd fluids

One constructs the attractor m, of a two-dimensional initial-boundary-value problem for the equations of motion of Oldroyd fluids, one proves the finite-dimensionality of the dynamical problem on m and the finiteness of the Hausdorff dimension of the attractor m.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 67–71, 1986.The authors express their gratitude to O. A. Ladyzhenskaya for her interest in the paper and for useful discussions.     

弱阻尼广义KdV方程的整体吸引子

证明了具有弱阻尼项的广义KdV方程约周期初边值问题解的存在唯—性及整体吸引子存在性,最后获得了吸引子的Hausdorff维数和分形维数的上界估计．     

Finite dimensionality of global attractors for a non-classical reaction-diffusion equation with memory

In this paper, we consider a periodic boundary value problem for a non-classical reaction-diffusion equation with memory. In other paper, we use the ω-limit compactness of the solution semigroup {S(t)}_t≥0 to get the existence of a global attractor. The main goal here is to give an estimate of the fractal dimension of the global attractor. By the fractal dimension theorem given by A.O. Celebi et al., we obtain that the fractal dimension of the global attractor for the problem is finite; this makes the results for the non-classical reaction-diffusion equations more substantial and perfect.     

非自治半线性热黏弹性问题的一致吸引子的维数(英文)

In this paper, we prove the existence of a uniform attractor for the process associated with a non-antonomous semilinear thermoelastic problem. And under the certain parameter, we obtain an upper bound for the Hausdorff dimension of the uniform attractor.     

Attractor dimension estimate for plane shear flow of micropolar fluid with free boundary

This research is motivated by a problem from lubrication theory. We consider a free boundary problem of a two‐dimensional boundary‐driven micropolar fluid flow. The existence of a unique global‐in‐time solution of the problem and the global attractor for the associated semigroup are known. In this paper we estimate the dimension of the global attractor in terms of the given data and the geometry of the domain of the flow by establishing a new version of the Lieb–Thirring inequality with constants depending explicitly on the geometry of the domain. We also obtain some new estimates for the Navier–Stokes shear flows. Copyright © 2005 John Wiley & Sons, Ltd.     

An upper bound on the attractor dimension of a 2D turbulent shear flow in lubrication theory

We consider a two-dimensional Navier–Stokes shear flow. There exists a unique global-in-time solution of the considered problem as well as the global attractor for the associated semigroup.Our aim is to estimate from above the dimension of the attractor in terms of given data and geometry of the domain of the flow. First we obtain a Kolmogorov-type bound on the time-averaged energy dissipation rate, independent of viscosity at large Reynolds numbers. Then we establish a version of the Lieb–Thirring inequality for a class of functions defined on the considered non-rectangular flow domain.This research is motivated by a problem from lubrication theory.     

Finite dimensional global and exponential attractors for a class of coupled time-dependent Ginzburg-Landau equations

We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover.First,we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses a global attractor.Then we establish the existence of an exponential attractor.As a consequence,we show that the global attractor is of finite fractal dimension.     

Pullback attractor for heat convection problem in a micropolar fluid

We consider a two-dimensional micropolar fluid flow heated from below. We assume that the temperature of the lower part of the boundary is a function of time. That leads to the non-autonomous system of equations. We show the existence of the pullback attractor for the problem. Next, the dimension of the attractor is estimated from above.     

Global attractor for heat convection problem in a micropolar fluid

The article is devoted to describe asymptotics in the heat convection problem for a micropolar fluid in two dimensions. We show the existence and the uniqueness of global in time solutions and then prove the existence of a global attractor for considered model. Next, the Hausdorff dimension of the global attractor is estimated. Copyright © 2006 John Wiley & Sons, Ltd.     

Attractors of Navier-Stokes systems and of parabolic equations,and estimates for their dimensions

One investigates the problem of the existence of an attractor α of the semi-group S_t, generated by the solutions of the nonlinear nonstationary equations $$\frac{{\partial u}}{{\partial t}} = A(u), u|_{t = 0} = u_0 (x); S_t u_0 \equiv u(t)$$ . One proves a very general theorem on the existence of an attractor α of the semigroup S_t for t→∞. One gives examples of differential equations having attractors: a second-order quasilinear parabolic equation, a two-dimensional Navier—Stokes system, a monotone parabolic equation of any order. One proves a theorem on the finiteness of the Hausdorff dimension of the attractor α. One gives an estimate for the Hausdorff dimension of the attractor α for a two-dimensional Navier—Stokes system.     

Dynamics of Thermally-Insulated Nonequilibrium Stefan Problem

We study a two-phase Stefan problem with kinetics. Here we prove existence of a finite-dimensional attractor for the problem without heat losses. Fot the most part we use a more elegant technique of energetic type estimates in appropriately defined weighted Sobolev spaces as opposite to the parabolic potentials of [9]. We demonstrate existence of compact attractors in the Sobolev spaces and prove that the attractor consists of sufficiently regular functions. This allows us to show that the Hausdorff dimension of the attractor is finite.     

Correlation dimension for iterated function systems

The correlation dimension of an attractor is a fundamental dynamical invariant that can be computed from a time series. We show that the correlation dimension of the attractor of a class of iterated function systems in is typically uniquely determined by the contraction rates of the maps which make up the system. When the contraction rates are uniform in each direction, our results imply that for a corresponding class of deterministic systems the information dimension of the attractor is typically equal to its Lyapunov dimension, as conjected by Kaplan and Yorke.

     

Hausdorff dimension of self-similar sets with overlaps

We provide a simple formula to compute the Hausdorff dimension of the attractor of an overlapping iterated function system of contractive similarities satisfying a certain collection of assumptions. This formula is obtained by associating a non-overlapping infinite iterated function system to an iterated function system satisfying our assumptions and using the results of Moran to compute the Hausdorff dimension of the attractor of this infinite iterated function system, thus showing that the Hausforff dimension of the attractor of this infinite iterated function system agrees with that of the attractor of the original iterated function system. Our methods are applicable to some iterated function systems that do not satisfy the finite type condition recently introduced by Ngai and Wang.      

Model based method for estimating an attractor dimension from uni/multivariate chaotic time series with application to Bremen climatic dynamics

In this paper, a method for estimating an attractor embedding dimension based on polynomial models and its application in investigating the dimension of Bremen climatic dynamics are presented. The attractor embedding dimension provides the primary knowledge for analyzing the invariant characteristics of the attractor and determines the number of necessary variables to model the dynamics. Therefore, the optimality of this dimension has an important role in computational efforts, analysis of the Lyapunov exponents, and efficiency of modeling and prediction. The smoothness property of the reconstructed map implies that, there is no self-intersection in the reconstructed attractor. The method of this paper relies on testing this property by locally fitting a general polynomial autoregressive model to the given data and evaluating the normalized one step ahead prediction error. The corresponding algorithms are developed in uni/multivariate form and some probable advantages of using information from other time series are discussed. The effectiveness of the proposed method is shown by simulation results of its application to some well-known chaotic benchmark systems. Finally, the proposed methodology is applied to two major dynamic components of the climate data of the Bremen city to estimate the related minimum attractor embedding dimension.     

Iterated function systems a rising from recursive estimation problems

Summary We show that under suitable conditions the one-step predictor of a finite-state Markov chain from noisy observations has a unique stationary law which is supported by a self-similar set, called the attractor. Under additional symmetry conditions such attractor is either connected, or totally disconnected and perfect. In this latter case the predictor keeps an infinite memory of the past observations. The main problem of interest is to identify those values of the parameters of the chain and the observation process for which this happens. In the binary case, the problem is completely solved. In higher dimension the problem is harder: a complete solution is presented for ternary chains in the completely symmetric persistent case.     

Lyapunov functions in the attractors dimension theory

The effectiveness of constructing Lyapunov functions in the attractors dimension theory is theory of the dimension demonstrated. Formulae for the Lyapunov dimension of the Lorenz, Hénon and Chirikov attractors are derived and proved. A hypothesis regarding the formula for the dimension of the Rössler attractor is formulated.     

A note on the dimension of the global attractor for an abstract semilinear hyperbolic problem

We study a semilinear hyperbolic problem, written as a second-order evolution equation in an infinite-dimensional Hilbert space. Assuming existence of the global attractor, we estimate its fractal dimension explicitly in terms of the data. Despite its elementary character, our technique gives reasonable results. Notably, we require no additional regularity, although nonlinear damping is allowed.     

有阻尼Sine－Gordon方程的全局吸引子的维数

本文通过引入新范数,得到有阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的Ｄｉｒｉｃｈｌｅｔ问题的全局吸引子的维数的一个估计．结果表明：当“阻尼”与“扩散”同时增大或正弦项系数减小时,吸引子的维数减小．特别地,得到了零维吸引子存在的参数条件．     

有阻尼Sine－Gordon方程的全局吸引子的维数

本文通过引入新范数，得到有阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的Ｄｉｒｉｃｈｌｅｔ问题的全局吸引子的维数的一个估计．结果表明：当“阻尼”与“扩散”同时增大或正弦项系数减小时，吸引子的维数减小．特别地，得到了零维吸引子存在的参数条件．     

Estimates on the dimension of an exponential attractor for a delay differential equation

We study the long time behavior of delay differential equation, considered in a bounded domain in ? ^d . Using the short trajectory method to prove the existence of the exponential attractor. Also we have estimates on the fractal dimension of an exponential attractor.