The authors deal with the asymptotic behaviour of nonautonomous ordinary differential equations

$\frac{du}{dt}=F\left(u\right)+\epsilon g(t,u)$ obtained by nonautonomous perturbations

$g(t,u)$ of autonomous differential equations

$\frac{du}{dt}=F\left(u\right)$ with a global attractor in

${\mathbb{R}}^{d}$. In particular, they show that the perturbed system possesses a cocycle attractor in a neighbourhood of the global autonomous attractor, provided that the perturbation

$g(t,u)$ is uniformly bounded, and both the vector field

$F\left(u\right)$ and the perturbations

$g(t,u)$ 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.