Cocycle attractors in nonautonomously perturbed differential equations. (English) Zbl 0905.34047

The authors deal with the asymptotic behaviour of nonautonomous ordinary differential equations \( \frac{d u}{d t} = F(u) + \varepsilon g(t,u) \) obtained by nonautonomous perturbations \(g(t,u)\) of autonomous differential equations \( \frac{d u}{d t} = F(u) \) 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 \(F(u)\) 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.


34D45 Attractors of solutions to ordinary differential equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure