Attractors For Non-autonomous Dynamical Systems Research Articles

Necessary and sufficient criteria for existence, regularity, and asymptotic stability of enhanced pullback attractors with applications to 3D primitive equations

We introduce several new concepts called enhanced pullback attractors for nonautonomous dynamical systems by improving the compactness and attraction of the usual pullback attractors in strong topology spaces uniformly over some infinite time intervals. Then we establish several necessary and sufficient criteria for the existence, regularity and asymptotic stability of these enhanced pullback attractors under general theoretical frameworks which are expected to be applied to many kinds of nonautonomous PDEs. These abstract results are applied to the famous 3D primitive equations modeling the large-scale ocean and atmosphere dynamics. We establish the existence, regularity and asymptotic stability of the enhanced pullback attractors for the nonautonomous primitive equations in [Formula: see text] and [Formula: see text] when the forces satisfy some mild conditions. The uniform pullback asymptotic compactness of the solution operators in [Formula: see text] and [Formula: see text] are established by deriving the uniform “flattening effects” of the solutions in [Formula: see text] and the uniform [Formula: see text]-Hölder continuity of the solutions from [Formula: see text] to [Formula: see text] with respect to the initial data, respectively. Our approaches are applicable to prove the existence of global attractors for the autonomous primitive equations in [Formula: see text] provided [Formula: see text].

Stability of regular pullback attractors for non-autonomous dynamical systems: theoretical results and applications

Stability of regular pullback attractors for non-autonomous dynamical systems: theoretical results and applications

Measure and Statistical Attractors for Nonautonomous Dynamical Systems

Various inequivalent notions of attraction for autonomous dynamical systems have been proposed, each of them useful to understand specific aspects of attraction. Milnor’s notion of a measure attractor considers invariant sets with positive measure basin of attraction, while Ilyashenko’s weaker notion of a statistical attractor considers positive measure points that approach the invariant set in terms of averages. In this paper we propose generalisations of these notions to nonautonomous evolution processes in continuous time. We demonstrate that pullback/forward measure/statistical attractors can be defined in an analogous manner and relate these to the respective autonomous notions when an autonomous system is considered as nonautonomous. There are some subtleties even in this special case–we illustrate an example of a two-dimensional flow with a one-dimensional measure attractor containing a single point statistical attractor. We show that the single point can be a pullback measure attractor for this system. Finally, for the particular case of an asymptotically autonomous system (where there are autonomous future and past limit systems) we relate pullback (respectively, forward) attractors to the past (respectively, future) limit systems.

Local Approximate Forward Attractors of Nonautonomous Dynamical Systems

Pullback dynamics of nonautonomous dynamical systems has been considerably developed. However, it is still a tough job to study forward dynamics of nonautonomous dynamical systems, since forward attractors were only obtained in some particular cases. In the paper, under some reasonable conditions, it is shown that closing to a local pullback attractor, there is an approximate forward attractor. Specifically, let ϕ be a cocycle semiflow on a Banach space X with driving system θ on a base space P. Suppose that the base space P is compact and ϕ is uniformly asymptotically compact. Let A(∙) be a local pullback attractor with being compact. We prove that every e-extended neighborhood Ae(∙) of A(∙) will forward attract every bounded set B(∙) that is pullback attracted by A(∙). We then call Ae(∙) an approximate forward attractor of ϕ.

Approximate Forward Attractors of Non-Autonomous Dynamical Systems

In this paper the forward asymptotical behavior of non-autonomous dynamical systems and their attractors are investigated. Under general conditions, the authors show that every neighborhood of pullback attractor has forward attracting property.

Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing

In this paper, we study the squeezing property and finite dimensionality of cocycle attractors for non-autonomous dynamical systems (NRDS). We show that the generalized random cocycle squeezing property (RCSP) is a sufficient condition to prove a determining modes result and the finite dimensionality of invariant non-autonomous random sets, where the upper bound of the dimension is uniform for all components of the invariant set. We also prove that the RCSP can imply the pullback flattening property in uniformly convex Banach space so that could also contribute to establish the asymptotic compactness of the system. The cocycle attractor for 2D Navier-Stokes equation with additive white noise and translation bounded non-autonomous forcing is studied as an application.

Continuity of pullback and uniform attractors

We study the continuity of pullback and uniform attractors for non-autonomous dynamical systems with respect to perturbations of a parameter. Consider a family of dynamical systems parameterized by λ∈Λ, where Λ is a complete metric space, such that for each λ∈Λ there exists a unique pullback attractor Aλ(t). Using the theory of Baire category we show under natural conditions that there exists a residual set Λ⁎⊆Λ such that for every t∈R the function λ↦Aλ(t) is continuous at each λ∈Λ⁎ with respect to the Hausdorff metric. Similarly, given a family of uniform attractors Aλ, there is a residual set at which the map λ↦Aλ is continuous. We also introduce notions of equi-attraction suitable for pullback and uniform attractors and then show when Λ is compact that the continuity of pullback attractors and uniform attractors with respect to λ is equivalent to pullback equi-attraction and, respectively, uniform equi-attraction. These abstract results are then illustrated in the context of the Lorenz equations and the two-dimensional Navier–Stokes equations.

Pullback and uniform exponential attractor for non-autonomous Boissonade system

In this paper, we first improve the existing sufficient conditions for the existence of pullback and uniform exponential attractors for non-autonomous dynamical systems. Then we prove the existence, continuity and boundedness of fractal dimension of a pullback exponential attractor for Boissonade system effected by time-dependent forces. Finally, we prove the existence and boundedness of fractal dimension of a uniform exponential attractor for Boissonade system driven by quasi-periodic external forces.

Uniform global attractors for non-autonomous dissipative dynamical systems

In this paper we consider sufficient conditions for the existence of uniform compact global attractor for non-autonomous dynamical systems in special classes of infinite-dimensional phase spaces. The obtained generalizations allow us to avoid the restrictive compactness assumptions on the space of shifts of non-autonomous terms in particular evolution problems. The results are applied to several evolution inclusions.

Pullback attractors of reaction-diffusion inclusions with space-dependent delay

Inspired by biological phenomena with effects of switching off (maybe just for a while), we investigate non-autonomous reaction-diffusion inclusions whose multi-valued reaction term may depend on the essential supremum over a time interval in the recent past (but) pointwise in space. The focus is on sufficient conditions for the existence of pullback attractors. If the multi-valued reaction term satisfies a form of inclusion principle standard tools for non-autonomous dynamical systems in metric spaces can be applied and provide new results (even) for infinite time intervals of delay. More challenging is the case without assuming such a monotonicity assumption. Then we consider the parabolic differential inclusion with the time interval of delay depending on space and extend the approaches of norm-to-weak semigroups to a purely metric setting. This provides completely new tools for proving pullback attractors of non-autonomous dynamical systems in metric spaces.

Morse Decomposition of Attractors for Non-autonomous Dynamical Systems

Abstract A Morse decomposition of a global attractor describes its internal dynamics, i.e., the dynamics on invariant compact sets in the attractor and the connections between them. When we deal with non-autonomous dynamical systems, the concept of pullback attractor for the associated skew-product flow appears as a powerful tool to analyze the asymptotic behaviour of these systems. In this paper we develop a Morse decomposition theory for pullback attractors of non-autonomous dynamical systems in Banach spaces with compact base space which, in particular, defines a (non-autonomous) Lyapunov functional on the attractor describing a decaying energy level on the evolution of trajectories.

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

Some questions concerning attractors for non-autonomous dynamical systems

We compare various concepts of attractor in the context of non-autonomous dynamical systems. Then, we prove an appropriate version of the Pliss reduction principle for non-autonomous differential systems with rapidly oscillating coefficients.

On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems

For an abstract dynamical system, we establish, under minimal assumptions, the existence of D -attractor, i.e. a pullback attractor for a given class D of families of time varying subsets of the phase space. We relate this concept with the usual attractor of fixed bounded sets, pointing out its usefulness in order to ensure the existence of this last attractor in particular situations. Moreover, we prove that under a simple assumption these two notions of attractors generate, in fact, the same object. This is then applied to a Navier–Stokes model, improving some previous results on attractor theory.

Lower semicontinuity of attractors for non-autonomous dynamical systems

AbstractThis paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

Nonautonomous attractors of switching systems

Switching systems formed by switching between a finite number of autonomous ordinary differential equations are formulated as abstract nonautonomous dynamical systems driven by the shift operator on the space of switching controls. A metric is proposed for such switching controls and the space of switching controls with a given positive dwell time is shown to be compact. This allows known results on pullback attractors for nonautonomous dynamical systems to be applied to switching systems including conditions which ensure that a pullback attractor exists and is also a forward attractor. In particular, this provides a conceptual framework for investigating the asymptotic behaviour of switching systems beyond the usual case in which the asymptotic stability of the zero solution under switching is considered.

Pullback attractors of nonautonomous dynamical systems

We present the necessary and sufficient conditions and a new method to study the existence of pullback attractors of nonautonomous infinite dimensional dynamical systems. For illustrating our method, we apply it to nonautonomous 2D Navier-Stokes systems. We also show that the parametrically inflated pullback attractors and uniform attractors are robust with respect to the perturbations of both cocycle mappings and driving systems. As an example, we consider the nonautonomous 2D Navier-Stokes system with rapidly oscillating external force.

Approximation of attractors of nonautonomous dynamical systems

This paper is devoted to the numerical approximation of attractors. For general nonautonomous dynamical systems we first introduce a new type of attractor which includes some classes of noncompact attractors such as unbounded unstable manifolds. We then adapt two cell mapping algorithms to the nonautonomous setting and use the computer program GAIO for the analysis of an explicit example, a two-dimensional system of nonautonomous difference equations. Finally we present numerical data which indicate a bifurcation of nonautonomous attractors in the Duffing-van der Pol oscillator.

Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: An example

The subject of this paper is the asymptotic behavior of a class of nonautonomous, infinite-dimensional dynamical systems with an underlying unbounded domain. We present an approach that is able to overcome both the law of compactness of the trajectories and the continuity of the spectrum of the linear part of the equations under consideration, providing nevertheless existence of uniform attractors. Moreover, our approach allows us to estimate the Hausdorff dimension of attractors of nonautonomous equations in terms of physical parameters. c 2000 John Wiley & Sons, Inc.

Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: An example

Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: An example