Comments on “investigation of the properties of the period for the nonlinear oscillator x''+(1+x^2)x'=0”

Abstract

Two Lidstedt-Poincaré perturbation based methods are used to solve the nonlinear differential equation of a nonlinear oscillator having the square of the angular frequency depends quadratically on the velocity. Mickens published two interesting papers [1, 2] about this oscillator and by using the harmonic balance method he found the approximate frequency is not defined for amplitudes of magnitude equal to or larger to two. We show these standard perturbation methods work better than the harmonic balance method. In particular, the modified Lindstedt-Poincaré method works well for the whole range of the oscillation amplitudes, and excellent agreement of the approximate frequency with the exact one has been demonstrated and discussed.