An artificial parameter-decomposition method for nonlinear oscillators: Applications to oscillators with odd nonlinearities

Abstract

A method for obtaining series solutions of nonlinear second-order ordinary differential equations based on the introduction of an artificial parameter is presented and shown to be identical to the well-known Adomian's decomposition technique. The method is formulated in both integral and differential forms. For the determination of the limit cycle of oscillators with odd nonlinearities, two differential forms and one integral form of the artificial parameter method are presented. These versions are based on introducing a linear stiffness term with an unknown frequency, and the use of either the original independent variable or a new independent variable that depends linearly on the unknown frequency of the oscillator. The three formulations provide identical results, and their application to eight oscillators with odd nonlinearities shows that the artificial parameter technique presented in this paper predicts the same frequency of oscillation as the harmonic balance and iterative techniques as well as modified Linstedt–Poincaré methods. However, the method presented here is based on the introduction of an artificial parameter and does not require the presence of small perturbation parameters in the ordinary differential equation. It is also shown that two- and three-level iterative methods yield the same frequency of oscillation as the artificial parameter technique presented in this paper provided that the initial iterate of the former coincides with the leading-order solution of the latter and only one iteration of iterative techniques and only the second approximation of the artificial parameter method are determined.