  It was recently shown that a coherent oscillation of an axion can cause an
efficient parametric resonance, leading to a prominent emission of the
gravitational waves (GWs). In this paper, conducting the Floquet analysis, we
investigate the parametric resonance instability, which potentially triggers
the emission of the GWs from axions. Such a resonance instability takes place,
when the time evolution of the background field significantly deviates from the
harmonic oscillation. Therefore, the resonance instability cannot be described
by the Mathieu equation, whose stability/instability chart is well known. In
this paper, introducing an explicitly calculable parameter $\tilde{q}$, which
can be used to classify different types of the parametric resonance described
by the general Hill's equation, we investigate the stability/instability chart
for the general Hill's equation. This can also apply to the case where the
background oscillation is anharmonic. We show that the flapping resonance
instability, which takes place for $\tilde{q}=O(1)$, typically leads to the
most significant growth of the inhomogeneous modes among the self-resonance
instability. We also investigate whether the flapping resonance takes place for
the cosine potential or not.
