Bogolyubov averaging and normalization procedures in nonlinear mechanics. I

Анотація

A new method of asymptotic analysis of nonlinear dynamical systems is developed with extensive use of group-theoretical methods. The technique of normalization, which is named an "asymptotic decomposition" by the authors, is developed in the context of Bogolyubov averaging. In this paper, we also discuss how this technique helps to understand and develop the averaging method for systems in standard form and systems with several fast variables. The new method treats a centralized system as a direct analog of an averaged system according to Bogolyubov. The operation of averaging is interpreted as a Bogolyubov projector in the construction of the projection of an operator onto the algebra of the centralizer.