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.

1 Vector Fields, Algebras and Groups Generated by a System of Ordinary Differential Equations and their Properties.- 2 Decomposition of Systems of Ordinary Differential Equations.- 3 Asymptotic decomposition of systems of ordinary differential equations with a small parameter.- 4 Asymptotic Decomposition of Almost Linear Systems of Differential Equations with Constant Coefficients and Perturbations in the Form of Polynomials.- 5 Asymptotic Decomposition of Differential Systems with Small Parameter in the Representation Space of Finite-dimensional Lie Group.- 6 Asymptotic Decomposition of Differential Systems where Zero Approximation has Special Properties.- 7 Asymptotic Decomposition of Pfaffian Systems with a Small Parameter.- A: Lie series and Lie transformation.- B: The direct product of matrices.- B1: Definition.- B2: Systems of matrix equations.- C: Conditions for the solvability of systems of linear equations.- D: Elements of Lie group analysis of differential equations on the basis of the theory of extended operators.- D1: One-parameter group and its infinitesimal operator.- D2. Theory of extension.- Bibliographical Comments.- References.