Comments on the general dynamics of the nonlinear oscillator x¨+(1+x˙2)x=0

Velocity-dependent-frequency (VDF) oscillators are represented (for one-degree-of-freedom systems) by the differential equation x¨+f(x˙)x=0, f(0)>0f(0)>0. However, the standard methods for determining approximations to the periodic solutions may breakdown in their corresponding first-order calculations. In particular, the frequencies may become singular for finite values of the amplitude. A technique combining averaging and first-order iteration is shown to resolve this difficulty.