Asymptotic methods of nonlinear mechanics developed by N. M. Krylov and N. N. Bogoliubov originated new trend in perturbation theory. They pene- trated deep into various applied branches (theoretical physics, mechanics, ap- plied astronomy, dynamics of space flights, and others) and laid the founda- tion for lrumerous generalizations and for the creation of various modifications of thesem. E!f, hods. A great number of approaches and techniques exist and many differen. t classes of mathematical objects have been considered (ordinary differential equations, partial differential equations, delay diffe, 'ential equations and others). The stat. e of studying related problems was described in mono- graphs and original papers of Krylov N. M., Bogoliubov N. N. [1], [2], Bogoli- ubov N. N [1J, Bogoliubov N. N., Mitropolsky Yu. A. [1], Bogoliubov N. N., Mitropol- sky Yu. A., Samoilenko A. M. [1], Akulenko L. D. [1], van den Broek B. [1], van den Broek B., Verhulst F. [1], Chernousko F. L., Akulenko L. D. and Sokolov B. N. [1], Eckhause W. [l], Filatov A. N. [2], Filatov A. N., Shershkov V. V. [1], Gi- acaglia G. E. O. [1], Grassman J. [1], Grebennikov E. A. [1], Grebennikov E. A., Mitropolsky Yu. A. [1], Grebennikov E. A., Ryabov Yu. A. [1], Hale J . K. [I]' Ha- paev N. N. [1], Landa P. S. [1), Lomov S. A. [1], Lopatin A. K. [22]-[24], Lykova O. B.