The harmonic balance method is commonly used to analyze the steady state periodic solutions of strongly non-linear systems. Mostly, only a few leading harmonics are used in the truncated Fourier series employed with this method. For the forced oscillations, this assumed solution can fail to uncover some periodic solutions which actually exist in the system response, and/or it may predict some extraneous solutions which have no counterpart in the actual response. It is shown that a qualitative failure of the harmonic balance method is neither restricted to a first approximation nor to the systems having an asymmetric potential well. It is also shown that the stability analysis of the predicted solutions or the residual values for the neglected higher harmonics cannot detect a qualitative failure of the harmonic balance method.
In symmetry breaking and period doubling pitchfork bifurcations, some new harmonics are brought into the system response. Some of these harmonics, with relatively small amplitude, are instrumental in deciding the sub-or supercritical nature of these bifurcations. The harmonic balance method can lead to erroneous results if one or more of these small but instrumental harmonics is neglected from the assumed solution.