Cocycle attractors in nonautonomously perturbed differential equations

A nonautonomous dynamical system obtained by a small nonautonomous perturbation of an autonomous system with a global attractor is shown to have a cocycle attractor in a neighbourhood of a global autonomous attractor. The component sets of the cocycle attractor are continuous, have constant Hausdorff dimension, are T-periodic if the nonautonomous perturbations are T-periodic, and asymptote to the autonomous attractor in the asymptotically autonomous case.