Nonlinear Mechanics, Groups and Symmetry

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.