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The stability of attractors for non-autonomous perturbations of gradient-like systems
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Langa, José A., Robinson, James C., Suarez, Antonio and Vidal-Lopez, Alejandro (2007) The stability of attractors for non-autonomous perturbations of gradient-like systems. Journal of Differential Equations, Vol.234 (No.2). pp. 607-625. doi:10.1016/j.jde.2006.11.016 ISSN 0022-0396.
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Official URL: http://dx.doi.org/10.1016/j.jde.2006.11.016
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 non-autonomous problems converge towards the autonomous attractor only in the Hausdorff semi-distance (upper semicontinuity), the assumption that the autonomous attractor has a 'gradient-like' structure (the union of the unstable manifolds of a finite 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 -> infinity. In finite-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 -> -infinity, 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 finite number of hyperbolic trajectories. (c) 2006 Elsevier Inc. All rights reserved.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Journal of Differential Equations | ||||
Publisher: | Academic Press | ||||
ISSN: | 0022-0396 | ||||
Official Date: | 15 March 2007 | ||||
Dates: |
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Volume: | Vol.234 | ||||
Number: | No.2 | ||||
Number of Pages: | 19 | ||||
Page Range: | pp. 607-625 | ||||
DOI: | 10.1016/j.jde.2006.11.016 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
Data sourced from Thomson Reuters' Web of Knowledge
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