= Slow amplitudes for the second Mathieu instability tongue
{title2=$a_T=(f+1/12)b/2,\quad b_T=(5/12-f)a/2$}
For $x''+[1+\epsilon\cos t+f\epsilon^2]x=0$, let $T=\epsilon^2t$ 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.
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