An introduction to the theory of resonance systems

An asymptotic theory of ordinary differential equations with a small parameter is presented which uses the averaging principle. A method for constructing the first and higher approximations, representing solutions to the generalized Krylov-Bogoliubov equation, is described in detail, and consecutive algorithms are presented for implementing the method on a computer. The discussion focuses on multifrequency systems with slow and fast phase variables permitting resonance relations between the frequencies. A detailed classification of such systems is given; the averaging method is demonstrated for resonance systems, and various applications of the proposed methods in nonlinear mechanics are examined.