The two variable expansion procedure for the approximate solution of certain nonlinear differential equations

Abstract

A method is presented for deriving the asymptotic representation valid for large times for the motion of a particle under the influence of a predominantly linear restoring force and small nonlinear perturbations. It is shown that such an asymptotic representation must be a function of two time variables, in order to depict the behavior of the solution. The basic ideas are explained by the liberal use of simple examples, and the method is also applied to two idealized problems in celestial mechanics. (Author)