Uniform Nonautonomous Attractors under Discretization

A nonautonomous or cocycle dynamical system that is driven by an autonomous dynamical system acting on a compact metric space is assumed to have a uniform pullback attractor. It is shown that discretization by a one-step numerical scheme gives rise to a discrete time cocycle dynamical system with a uniform pullback attractor, the component subsets of which converge upper semi continuously to their continuous time counterparts as the maximum time step decreases to zero. The proof involves a Lyapunov function characterizing the uniform pullback attractor of the original system.