The structure of global attractors for non-autonomous perturbations of gradient-like dynamical systems (Q785365)

This paper provides a complete description of the structure of compact global (forward) attractors for time-variant perturbations of autonomous gradient-like dynamical systems. The unperturbed autonomous differential equation is assumed to have a finite number of equilibria. It is shown that the perturbed nonautonomous system has exactly the same number of invariant sections (reflecting the particular time-dependence and solutions). Moreover, it is established that the compact global (forward) attractor of the nonautonomous perturbed system coincides with the union of the unstable manifolds of this finite number of invariant sections. The approach is based on a skew-product formulation allowing in particular periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent (in the sense of Birkhoff) time-dependence.