= Solution
Away from the first <parametric resonance>, one forcing interaction changes a harmonic <frequency> by one and produces no resonant coupling of the two conjugate oscillations. Two interactions can couple <frequency> $+1$ to <frequency> $-1$, so the next <parametric resonance> is near $\omega=1$ and its effect first appears at order $\epsilon^2$. This gives \b[$p=2$] and the appropriate slow time $T=\epsilon^2t$.
Write $\omega=1+\kappa\epsilon^2$ and $y_0=a(T)\cos t+b(T)\sin t$. The first correction satisfies $(\partial_t^2+1)y_1=-\cos t\,y_0$, whose bounded particular solution is
$$
y_1=-\frac a2+\frac a6\cos2t+\frac b6\sin2t.
$$
At second order,
$$
(\partial_t^2+1)y_2=-2\partial_t\partial_Ty_0-2\kappa y_0-\cos t\,y_1.
$$
The resonant part of $-\cos t\,y_1$ is $(5a/12)\cos t-(b/12)\sin t$. Applying the <solvability condition in the method of multiple scales> gives
$$
\boxed{a_T=(\kappa+1/24)b,\qquad b_T=(5/24-\kappa)a.}
$$
The squared slow <eigenvalue> is
$$
\lambda^2=(\kappa+1/24)(5/24-\kappa)
=\frac1{64}-(\kappa-1/12)^2.
$$
The <second instability tongue of a weak Mathieu oscillator> therefore has the leading interval
$$
\boxed{-\frac1{24}<\kappa<\frac5{24},\qquad
1-\frac{\epsilon^2}{24}+o(\epsilon^2)<\omega<1+\frac{5\epsilon^2}{24}+o(\epsilon^2).}
$$
Here the two little-$o$ terms describe the respective exact edge locations; the strict inequality in $\kappa$ is the leading interior criterion. In particular $\omega=1$ is unstable, with physical growth rate $\epsilon^2\sqrt5/24+o(\epsilon^2)$. The general leading solution for bounded $T$ again has $y_0=a(T)\cos t+b(T)\sin t$, where the <matrix exponential> of the displayed two-by-two <amplitude> system evolves arbitrary $a(0),b(0)$. Adding $\epsilon y_1$ identifies the first nonresonant harmonic correction. The leading expression alone has error $O(\epsilon)$ on $t=O(\epsilon^{-2})$.
For comparison, at fixed $\omega\ne1/2,1$, write $y_0=A(T)e^{i\omega t}+\overline A(T)e^{-i\omega t}$. The first forced harmonics have coefficients $A/[2(2\omega+1)]$ and $A/[2(1-2\omega)]$. Their return to the original harmonic at the next order gives
$$
A_T=\frac{i}{4\omega(1-4\omega^2)}A.
$$
It is a phase correction, not <amplitude> growth. Consequently, away from the first two resonance neighborhoods, a general leading solution valid for bounded $\epsilon^2t$ is
$$
y=C\cos(\Omega t)+D\sin(\Omega t)+O(\epsilon),\qquad
\Omega=\omega+\frac{\epsilon^2}{4\omega(1-4\omega^2)}.
$$
Narrower higher-order instability intervals are invisible on this time scale. The detuned resonant calculation above is needed near $\omega=1$ because both the phase shift and the conjugate-mode coupling then matter.
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