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. © 2006 Elsevier Inc. All rights reserved.
Langa, J. A., Robinson, J. C., Suárez, A., & Vidal-López, A. (2007). The stability of attractors for non-autonomous perturbations of gradient-like systems. Journal of Differential Equations, 234(2), 607–625. https://doi.org/10.1016/j.jde.2006.11.016