ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

Krylov–Bogolyubov theorem about invariant measure

The averaging method, a pendulum with a vibrating suspension: N.N. Bogolyubov, A. Stephenson, P.L. Kapitza and others

The main moments of the historical development of one of the basic methods of nonlinear systems investigating (the averaging method) are traced. This method is understood as a transition from the so-called exact equation dx/dt = εX(t, x) (ε is small parameter), to the averaging equation dξ/dt = εX0(ξ) + ε2P2(ξ) + ... + εmPm(ξ) by corresponding variable substitution.Bogolyubov–Krylov’s approach to the problem of justifying the averaging method, based on the invariant measure theorem, is analyzed.