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The structure of attractors in non-autonomous perturbations of gradient-like systems
TLDR
In this article, the authors study the stability of attractors under non-autonomous perturbations that are uniformly small in time and show that all trajectories converge to one of the hyperbolic trajectories as t! 1.Abstract:
We study the stability of attractors under non-autonomous perturbations that are uniformly small in time. While in general the pullback attractors for the nonautonomous problems converge towards the autonomous attractor only in the Hausdorfi semi-distance (upper semicontinuity), the assumption that the autonomous attractor has a ‘gradient-like’ structure (the union of the unstable manifolds of a flnite number of hyperbolic equilibria) implies convergence (i.e. also lower semicontinuity) provided that the local unstable manifolds perturb continuously. We go further when the underlying autonomous system is itself gradient-like, and show that all trajectories converge to one of the hyperbolic trajectories as t ! 1. In flnite-dimensional systems, in which we can reverse time and apply similar arguments to deduce that all bounded orbits converge to a hyperbolic trajectory as t ! i1, this implies that the ‘gradient-like’ structure of the attractor is also preserved under small non-autonomous perturbations: the pullback attractor is given as the union of the unstable manifolds of a flnite number of hyperbolic trajectories.read more
Citations
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Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system
TL;DR: In this article, the exact structure of the pullback attractors in non-autonomous problems with perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria is determined.
Journal ArticleDOI
On the continuity of pullback attractors for evolution processes
TL;DR: Li et al. as discussed by the authors showed that a non-autonomous regular perturbation of a gradient-like system produces a family of pullback attractors that are uniformly exponentially attracting, and they gave an explicit bound on the distance between members of this family.
Journal ArticleDOI
Lower semicontinuity of attractors for non-autonomous dynamical systems
TL;DR: In this paper, the authors studied the lower semicontinuity of attractors for semilinear and non-autonomous differential equations in Banach spaces, and showed that the unperturbed constant attractor can be given as the union of unstable manifolds of time-dependent hyperbolic solutions.
References
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Infinite-Dimensional Dynamical Systems in Mechanics and Physics
TL;DR: In this article, the authors give bounds on the number of degrees of freedom and the dimension of attractors of some physical systems, including inertial manifolds and slow manifolds.
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Asymptotic Behavior of Dissipative Systems
TL;DR: In this article, the authors consider a continuous dynamical system with a global attractor and describe the properties of the flow on the attractor asymptotically smooth and Morse-Smale maps.
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Attractors for Equations of Mathematical Physics
TL;DR: In this article, the authors define trajectories attractors of autonomous and non-autonomous equations as follows: Trajectory attractors are the attractors that follow a trajectory in Hausdorff spaces.
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Dynamical systems and numerical analysis
TL;DR: In this paper, the authors unify the study of dynamical systems and numerical solution of differential equations by formulating them as dynamical system and examining the convergence and stability properties of the methods.
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Attractors for Semi-groups and Evolution Equations
TL;DR: The number of determining modes and the fractal dimension of bounded invariant sets for the Navier-Stokes equations were estimated in this paper, where the authors also considered the evolution equations of hyperbolic type.