An artificial parameter–Linstedt–Poincare´ method for oscillators with smooth odd nonlinearities

An artificial parameter method for obtaining the periodic solutions of oscillators with smooth odd nonlinearities is presented. The method is based on the introduction of a linear stiffness term and a new dependent variable both of which are proportional to the unknown frequency of oscillation, the introduction of an artificial parameter and the expansion of both the solution and the unknown frequency of oscillation in series of the artificial parameter. The method results in linear ordinary differential equations at each order in the parameter. By imposing the nonsecularity condition at each order in the expansion, the method provides different approximations to both the solution and the frequency of oscillation. The method does not require any minimization procedure; neither does it require the expansion of constants in terms of the artificial parameter. It is shown that the method presented here is also a decomposition technique and a homotopy perturbation method provided that in these techniques the unknown frequency of oscillation is expanded in terms of an artificial parameter and the nonsecularity condition is imposed at each order in the expansion procedure. It is also shown by means of six examples that the first approximation to the frequency of oscillation coincides with that obtained by means of harmonic balance methods, two- and three-level iterative techniques, and modified Linstedt–Poincare´ procedures based on the expansion of the solution and constants that appear in the differential equation in terms of an artificial parameter.