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Two fundamental harmonics of frequency one multiply into only a constant and second harmonics. The first-order solvability condition in the method of multiple scales therefore has no amplitude-changing resonant coupling. Their first-order particular corrections can mix back into the fundamental at second order, selecting the possible scale $t=O({ϵ}^{-2})$. Detuning may suppress the resulting resonance, so identifying this scale alone does not prove instability for every initial state.

For $x^{\prime \prime }+[1+ϵ\cos t+f{ϵ}^2]x=0$, let $T={ϵ}^{2}t$ and $x_0=a(T)\cos t+b(T)\sin t$. First-order forcing produces a constant and second harmonics, not a resonant harmonic. Their feedback at second order gives the displayed solvability condition in the method of multiple scales. The slow squared growth rate is $(f+1/12)(5/12-f)/4$. Consequently the leading instability interval is $-1/12<f<5/12$; outside its closure the slow amplitudes oscillate. The endpoints are degenerate and require the fourth-order edges of the second Mathieu instability tongue for exact small-parameter stability.