On Linstedt–Poincaré techniques for the quintic Duffing equation

An artificial parameter method for the quintic Duffing equation 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, and 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 that a small parameter be present in the governing equation, and its results are compared with those of Linstedt–Poincaré, modified Linstedt–Poincaré, harmonic balance and Galerkin techniques. It is shown that these four techniques predict the same first-order approximation to the frequency of oscillation as the artificial parameter method presented in this paper, and the latter introduces higher order corrections at second order, whereas similar corrections are introduced by the Linstedt–Poincaré and modified Linstedt–Poincaré methods at orders equal to and higher than three.