Generalization of the Second Bogolyubov's Theorem for Non-Almost   Periodic Systems
  The article is devoted to the generalization of the second Bogolyubov's theorem to non-almost periodic dynamical systems. We prove the analog of the second Bogolyubov's theorem for recurrent or pseudo recurrent dynamical systems in Banach spaces. Namely, we obtain the relation between a recurrent dynamical system and its averaged dynamical system. We also study existence of recurrent and pseudo recurrent motions (including special cases of periodic, quasi-periodic and almost periodic motions) in related nonautonomous systems. 
David N. Cheban, Jinqiao Duan, Anatoly Gherco