= Fourth-order edges of the second Mathieu instability tongue
{title2=$k_-(\epsilon),\ k_+(\epsilon)$}
For $u''+[1+k\epsilon^2+\epsilon\cos t]u=0$, set $t=2s$. The <Mathieu equation> has $a=4+4k\epsilon^2$ and $q=-2\epsilon$. Near the value $4$, the orthonormal even <Fourier series> basis $1,\cos2s,\cos4s,\cos6s$ gives diagonal entries $0,4,16,36$ and adjacent couplings $\sqrt2q,q,q$. The odd basis $\sin2s,\sin4s,\sin6s$ gives diagonal entries $4,16,36$ and adjacent couplings $q,q$. Higher modes require at least six coupling steps to affect the root near $4$, so these truncations determine its coefficients through $q^4$.
Substituting $a=4+c_2q^2+c_4q^4$ into the two <characteristic polynomials> and equating coefficients yields
$$
a_2=4+\frac5{12}q^2-\frac{763}{13824}q^4+O(q^6),\qquad
b_2=4-\frac1{12}q^2+\frac5{13824}q^4+O(q^6).
$$
Converting back gives the displayed gap edges with remainders $O(\epsilon^4)$ in $k$. In a fixed bounded neighborhood of this <parametric resonance>, the true unstable gap is $k_-<k<k_+$. The leading slow-amplitude calculation cannot decide exact stability at $k=-1/12$ or $5/12$; the fourth-order shifts put those fixed values just outside the gap for sufficiently small positive $\epsilon$. This does not provide a bound uniform in $\epsilon$ for their transient amplification.
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