Article
The stability of attractors for non-autonomous perturbations of gradient-like systems
Author/s | Langa Rosado, José Antonio
![]() ![]() ![]() ![]() ![]() ![]() ![]() Robinson, James C. Suárez Fernández, Antonio ![]() ![]() ![]() ![]() ![]() ![]() ![]() Vidal López, Alejandro |
Department | Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico |
Publication Date | 2007-03-15 |
Deposit Date | 2016-10-05 |
Published in |
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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 ... 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 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 → ∞. 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 → −∞, 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. |
Project ID. | MTM2005-01412
![]() BFM2003-06446 ![]() MTM2006-07932 ![]() BFM2003-03810 ![]() |
Citation | Langa Rosado, J.A., Robinson, J.C., Suárez Fernández, A. y Vidal López, A. (2007). The stability of attractors for non-autonomous perturbations of gradient-like systems. Journal of Differential Equations, 234 (2), 607-625. |
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