2016/01/27 by Hoang, Luan T., Olson, Eric J., Robinson, James C.
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1601.07436
We study the continuity of pullback and uniform attractors for non-autonomous dynamical systems with respect to perturbations of a parameter. Consider a family of dynamical systems parameterised by a complete metric space Λ such that for each λ∈Λ there exists a unique pullback attractor \mathcal Aλ(t). Using the theory of Baire category we show under natural conditions that there exists a residual set Λ_*⊆Λ such that for every t∈\mathbb R the function λ↦\mathcal Aλ(t) is continuous at each λ∈Λ_* with respect to the Hausdorff metric. Similarly, given a family of uniform attractors \mathbb Aλ, there is a residual set at which the map λ↦\mathbb Aλ is continuous. We also introduce notions of equi-attraction suitable for pullback and uniform attractors and then show when Λ is compact that the continuity of pullback attractors and uniform attractors with respect to λ is equivalent to pullback equi-attraction and, respectively, uniform equi-attraction. These abstract results are then illustrated in the context of the Lorenz equations and the two-dimensional Navier-Stokes equations.