Generalized decomposition methods for nonlinear oscillators

Two generalized decomposition methods for nonlinear oscillators are presented and the second one is applied to a nonlinear ordinary differential equation whose velocity is bounded between −1 and +1. The first generalized decomposition method proposed here is based on the introduction of a linear stiffness term, a change of independent variable, the introduction of an artificial parameter and the expansion of the solution in power series of this parameter, and yields, at first-order, a periodic solution in the new independent variable. By requiring that the forcing be an analytic function of its arguments, higher-order approximations may be obtained by either solving an integral or a linear ordinary differential equation. For oscillators with periodic solutions, the frequency of oscillation is also expanded in power series of the artificial parameter in the second generalized decomposition method which is set to unity at the end of the calculations, and further approximations to the frequency can be obtained by requiring that, at each order in the expansion, the solution be free from secular terms. It is shown that the second generalized decomposition method is identical to an artificial parameter-Linstedt–Poincaré method recently introduced by the author and it predicts a frequency of oscillation, at first-order, which is more accurate than that provided by a first-order harmonic balance procedure.