The stability of attractors for non-autonomous perturbations of gradient-like systems |
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Authors: | José A Langa Antonio Suárez |
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Institution: | a Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain b Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK c Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain |
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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 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→∞. 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. |
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