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Cocycle attractors in nonautonomously perturbed differential equations. (English) Zbl 0905.34047
The authors deal with the asymptotic behaviour of nonautonomous ordinary differential equations du dt=F(u)+εg(t,u) obtained by nonautonomous perturbations g(t,u) of autonomous differential equations du dt=F(u) with a global attractor in 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.
MSC:
34D45Attractors
37C70Attractors and repellers, topological structure
37-99Dynamic systems and ergodic theory (MSC2000)