Topological considerations and the method of averaging: a connection between local and global results

In the paper we present an approach to averaging of ordinary differential equations on an infinite time interval. This approach allows one to prove results concerning the existence of a solution of the non-averaged system that is not arbitrarily close to the corresponding solution of the averaged system, but only finitely close to it on an infinite time interval. At the same time, our assumptions on the right-hand side of the system are in some sense less restrictive comparing to the classical results on averaging.