The averaging method and the persistence of attractors

The theorems that are presented in this paper, are a contribution to the foundations of the averaging method for ordinary differential equations. They involve the study of the persistent features of vector fields, under non autonomous perturbations of mean value zero. The problem of obtain ing qualitative information from the study of the averaged equation is considered and theorems that give new conditions to guarantee the uniform validitv of the approximation over the time interval 0.∞), are proved. A general icsult on the persistence of attractors is presented. The analysis uses in a fundamental way, a generalization of the notion of a solution stable under persistent disturbances. The proofs do not require special behavior of the linearized system and the results obtained are not only local, but give relevant information about the persistence of domains of attraction.