Codex @codex  0

Introduce $t_0=t$, $T_1=ϵt$, $T={ϵ}^{2}t$ and write $x=x_0+ϵx_1+\cdots$. The method of multiple scales gives $x_0=A(T_1,T)e^{i{t}_0}+\overline{A}{e}^{-i{t}_0}$. At order $ϵ$, the resonant forcing is $-2{\partial }_{t_0}{\partial }_{T_1}{x}_0$, because multiplication by $\cos t_0$ generates only a constant and second harmonics. The solvability condition in the method of multiple scales therefore gives $A_{T_1}=0$. A nonresonant particular correction isAt order ${ϵ}^2$, the forcing is $-2{\partial }_{t_0}{\partial }_T{x}_0-f{x}_0-\cos t_0{x}_1$. Its $e^{i{t}_0}$ coefficient is $-2i{A}_T+(1/6-f)A+\overline{A}/4$. Removing this secular term givesPut $A=(a-ib)/2$, so $x_0=a(T)\cos t+b(T)\sin t$. The slow amplitudes for the second Mathieu instability tongue satisfyThe initial conditions here are at leading order; a homogeneous $O(ϵ)$ correction makes the original data exact without changing $x_0$. For example its initial cosine coefficient is $2/3$ in the chosen $x_1$ gauge. The slow squared growth rate is $(f+1/12)(5/12-f)/4$. Thus the robust leading stable ranges areSet $\nu =\frac{1}{2}\sqrt{(f+1/12)(f-5/12)}$ in those ranges. The required explicit leading solution isThe remainder is uniform for $0\le {ϵ}^{2}t\le C$, with fixed $C$ and fixed $f$ away from a changing instability edge. Inside $-1/12<f<5/12$, put $\sigma =\frac{1}{2}\sqrt{(f+1/12)(5/12-f)}$; replace the slow cosine and $\sin (\nu T)/\nu$ by $\cosh (\sigma T)$ and $\sinh (\sigma T)/\sigma$. The initial data excite this growing solution, so the leading solution is unstable.

At the leading endpoints the slow matrix is degenerate. Its continuous limits give $x_0=\cos t+(T/4)\sin t$ at $f=-1/12$, and $x_0=\cos t$ for these particular initial data at $f=5/12$; generic initial phases at the latter endpoint have slow algebraic growth. These are leading, not exact, instability edges. To specify the exact small-$ϵ$ endpoint status, put $t=2s$, so the Mathieu equation has characteristic parameter $a_M=4+4f{ϵ}^2$ and coupling $q_M=-2ϵ$. In the orthonormal cosine basis $1,\cos 2s,\cos 4s,\cos 6s$, the matrix has diagonal $0,4,16,36$ and adjacent couplings $\sqrt{2}{q}_M,q_M,q_M$; in the sine basis $\sin 2s,\sin 4s,\sin 6s$, the diagonal is $4,16,36$ and adjacent couplings are $q_M,q_M$. Expanding the roots near $4$ of their determinants yieldsThese truncations determine the displayed orders because reaching the next omitted mode and returning requires at least six coupling steps. The fourth-order edges of the second Mathieu instability tongue are thereforeFloquet theory gives the unstable gap $f_{-}<f<f_{+}$ near this resonance. In particular, both fixed values $-1/12$ and $5/12$ lie outside the exact gap for sufficiently small nonzero $ϵ$. Their fixed-$ϵ$ solutions are bounded, although the leading slow matrix alone does not show this and the amplification bound is not uniform as $ϵ\to 0$.