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.     

Uniformly exponentially stable approximations for a class of damped systems

We consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable. This result is then combined with previous ones on space semi-discretizations to derive similar results on fully-discrete approximation schemes. Our method is mainly based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. Our methods also allow to deal directly with stabilization properties of fully discrete approximation schemes without numerical viscosity, under a suitable CFL type condition on the time and space discretization parameters.     

Existence of Attractors for Approximations to the Bingham Model and Their Convergence to the Attractors of the Initial Model

Considering the Bingham fluid motion model, we study the approximation problem, prove its unique solvability, and the existence of attractors. We show that the attractors of the approximation problem converge to the attractors of the Bingham model in the sense of the Hausdorff semidistance in the corresponding metric space as the approximation parameter vanishes.

     

Attractors and approximations for lattice dynamical systems

We present a sufficient condition for the existence of a global attractor for general lattice dynamical systems, then consider the existence of attractors and their approximation for second-order and first-order lattice systems which, in particular case, can be regarded as the spatial discretizations of corresponding wave equations and reaction-diffusion equations in R^k.     

Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations

The reaction-diffusion equations are approximated by a fully discrete system: a Legendre-Galerkin approximation for the space variables and a semi-implicit scheme for the time integration. The stability and the convergence of the fully discrete system are established. It is also shown that, under a restriction on the space dimension and the growth rate of the nonlinear term, the approximate attractors of the discrete finite dimensional dynamical systems converge to the attractor of the original infinite dimensional dynamical systems. An error estimate of optimal order is derived as well without any further regularity assumption.     

Approximation of bone remodeling models

We present convergence results and error estimates concerning the numerical approximation of a class of bone remodeling models, that are elastic adaptive rod models. These are characterized by an elliptic variational equation, representing the equilibrium of the rod under the action of applied loads, coupled with an ordinary differential equation with respect to time, describing the physiological process of bone remodeling. We first consider the semi-discrete approximation, where only the space variables are discretized using the standard Galerkin method, and then, applying the forward Euler method for the time discretization, we focus on the fully discrete approximation.     

Regularity of pullback attractors for non-autonomous stochastic coupled reaction-diffusion systems

We consider the dynamical behavior of the typical non-autonomous autocatalytic stochastic coupled reaction-diffusion systems on the entire space $\mathbb{R}^n$. Some new uniform asymptotic estimates are implemented to investigate the existence of pullback attractors in the Sobolev space $H^1(\mathbb{R}^n)^3$ for the three-component reversible Gray-Scott system.     

Pointed shape and global attractors for metrizable spaces

In this paper we consider two notions of attractors for semidynamical systems defined in metric spaces. We show that Borsuk's weak and strong shape theories are a convenient framework to study the global properties which the attractor inherits from the phase space.Moreover we obtain pointed equivalences (even in the absence of equilibria) which allow to consider also pointed invariants, like shape groups.     

Global attractors for singular perturbations of the Cahn-Hilliard equations

We consider the singular perturbations of two boundary value problems, concerning respectively the viscous and the nonviscous Cahn-Hilliard equations in one dimension of space. We show that the dynamical systems generated by these two problems admit global attractors in the phase space , and that these global attractors are at least upper-semicontinuous with respect to the vanishing of the perturbation parameter.     

Mean-square accuracy estimates of a projection-difference method for weakly solvable quasilinear parabolic equations

In a separable Hilbert space, we consider a weakly solvable quasilinear parabolic problem, which is solved by an approximate projection-difference method. With respect to time, we use a linear Euler method implicit in the leading part. We obtain coercive mean-square accuracy estimates for the approximate solutions. These estimates imply the convergence of the approximate method and provide the convergence rate accurate in the approximation order both in time and in space.     

On uniform attractors of explicit approximations

We prove a theorem stating that the uniform attractors of a family of semiprocesses that do not necessarily have a common time semigroup depend on the parameter uppersemicontinuously. We consider an explicit finite-difference scheme for a nonautonomous system of ordinary differential equations and an explicit spectral-difference scheme for the vorticity equation with time-dependent bounded right-hand side on a sphere. We obtain theorems on the existence of uniform attractors of numerical schemes and their closeness to true attractors of the original differential problems.     

A General Approximation Scheme for Attractors of Abstract Parabolic Problems

We consider semilinear problems of the form u′ = Au + f(u), where A generates an exponentially decaying compact analytic C ₀-semigroup in a Banach space E, f:E ^α → E is differentiable globally Lipschitz and bounded (E ^α = D((?A)^α) with the graph norm). Under a very general approximation scheme, we prove that attractors for such problems behave upper semicontinuously. If all equilibrium points are hyperbolic, then there is an odd number of them. If, in addition, all global solutions converge as t → ±∞, then the attractors behave lower semicontinuously. This general approximation scheme includes finite element method, projection and finite difference methods. The main assumption on the approximation is the compact convergence of resolvents, which may be applied to many other problems not related to discretization.     

具有白噪声的随机格点系统的随机吸引子的Kolmogorov熵

基于耗散的随机格点系统解的渐近行为理论,主要运用元素分解法与有限维空间中多面体球覆盖的拓扑性质,研究了具有白噪声的随机Klein-Gordon-Schrdinger格点动力系统的随机吸引子的Kolmogorov熵,并得到它的一个上界.     

多峰映射族中非双曲奇异吸引子的丰富性

本文考虑多峰映射族中非双曲奇异吸引子的丰富性,证明多维参数空间中存在正测度的参数集合,对应系统具有绝对连续的不变测度.     

Random attractors of stochastic partial differential equations: A smooth approximation approach

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.     

Approximation of attractors for evolution equations with the help of attractors for finite systems

The problem of approximation of attractors for semidynamical systems (SDS) in a metric space is considered. Let some (exact) SDS possessing an attractor M be inaccurately defined, i.e., another SDS, which is close in some sense to the exact one, be given. The problem is to find a set M that is close to M in the Hausdorff metric. A finite procedure for construction of M is suggested. The results obtained are suitable for numerical construction of attractors for a rather large class of systems, including the ones generated by the Lorenz equations. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 71–86, 1992. Translated by I. N. Kostin.     

Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition

In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.     

Approximations of quasi-stationary distributions for markov chains

We consider a simple and widely used method for evaluating quasi-stationary distributions of continuous time Markov chains. The infinite state space is replaced by a large, but finite approximation, which is used to evaluate a candidate distribution. We give some conditions under which the method works, and describe some important pitfalls.     

Regular and chaotic motions of a Chaplygin sleigh under periodic pulsed torque impacts

For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced three-dimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusive-like random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.     

On the Kneser property for some parabolic problems

In this paper we consider both a phase-field systems of equations and an abstract differential inclusion for which the uniqueness of the Cauchy problem fails. We prove that the Kneser property holds, that is, that the set of values attained by the solutions at every moment of time is compact and connected. These results are also applied for proving that the global attractors in both cases are connected. An application is given to a reaction–diffusion equation with discontinuous nonlinearity.