in contrast to other books devoted to the averaging method and the method of integral manifolds, in the present book we study oscillation systems with many varying frequencies. in the process of evolution, systems of this type can pass from one resonance state into another. this fact considerably complicates the investigation of nonlinear oscillations. in the present monograph, a new approach based on exact uniform estimates of oscillation integrals is proposed. on the basis of this approach, numerous completely new results on the justification of the averaging method and its applications are obtained and the integral manifolds of resonance oscillation systems are studied. this book is intended for a wide circle of research workers, experts, and engineers interested in oscillation processes, as well as for students and post-graduate students specialized in ordinary differential equations.