The authors deal with the asymptotic behaviour of nonautonomous ordinary differential equations
obtained by nonautonomous perturbations
of autonomous differential equations
with a global attractor in
. In particular, they show that the perturbed system possesses a cocycle attractor in a neighbourhood of the global autonomous attractor, provided that the perturbation
is uniformly bounded, and both the vector field
and the perturbations
are uniformly Lipschitz continuous. Besides, one receives qualitative properties of cocyle attractors (like continuity, periodicity, constant Hausdorff dimension, asymptote to the corresponding autonomous attractor). The proofs are carried out using standard Lyapunov-function techniques. An one-dimensional example illustrates the presented theory. The paper represents a continuation of fundamental works of A. V. Babin, M. I. Vishik, J. Hale, G. R. Sell and T. Yoshizawa on qualitative asymptotic behaviour of semigroups.