The Method of Integral Manifolds in the Theory of Nonlinear Oscillations

Abstract

This chapter discusses the method of integral manifolds in the theory of nonlinear oscillations. An interesting trend is the extension of the integral manifolds methods to the study of infinite systems of differential equations. This is a pressing problem in connection with the study of oscillations in systems with distributed parameters described by partial differential equations and particularly in connection with the application of asymptotic methods to the study of single-frequency oscillations in electrodynamic problems. The development and application of the theory of integral manifolds to the study of systems of differential equations with a small parameter attached to the highest derivatives are sophisticated. The theory of integral manifold describes the relaxational oscillatory processes. For a degenerate oscillatory system, it is comparatively simple to find an analytical expression for the limiting cycle. At the same time, the solution of problem even for a series of particular cases is of great interest in connection with numerous problems in physics and engineering.