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For $u^{\prime \prime }+[1+k{ϵ}^2+ϵ\cos t]u=0$, set $t=2s$. The Mathieu equation has $a=4+4k{ϵ}^2$ and $q=-2ϵ$. Near the value $4$, the orthonormal even Fourier series basis $1,\cos 2s,\cos 4s,\cos 6s$ gives diagonal entries $0,4,16,36$ and adjacent couplings $\sqrt{2}q,q,q$. The odd basis $\sin 2s,\sin 4s,\sin 6s$ 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 $a=4+c_2{q}^2+c_4{q}^4$ into the two characteristic polynomials and equating coefficients yieldsConverting back gives the displayed gap edges with remainders $O({ϵ}^4)$ 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 $ϵ$. This does not provide a bound uniform in $ϵ$ for their transient amplification.

The Mathieu equation is the displayed linear differential equation with periodic coefficients. Floquet theory describes its bounded and exponentially growing solutions, while its periodic Mathieu characteristic values mark the edges of parametric resonance gaps. The forcing normalization matters: replacing $t$ by $t/2$ rescales both the constant and periodic coefficients.

Floquet theory suggests a sequence of instability tongues nearAt these limiting frequencies the unperturbed Floquet multipliers $e^{\pm 2\pi i\omega }$ coincide. Each multiplication by $\cos t$ changes a Fourier frequency by one, so $m$ forcing interactions are needed to couple $+m/2$ and $-m/2$. This explains both the sequence of parametric resonances and the successive powers of $ϵ$ in their widths.

There is no implication that the exact half-integers themselves lie in every finite-$ϵ$ growing interval. The tongue centers generally shift by $O({ϵ}^2)$, which exceeds the width when $m>2$. It is the narrow intervals near the shifted centers that contain exponentially growing solutions. Frequencies staying a fixed distance from every half-integer are stable for sufficiently small $ϵ$.

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 ${ϵ}^2$. This gives $p=2$ and the appropriate slow time $T={ϵ}^{2}t$.

Write $\omega =1+\kappa {ϵ}^2$ 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 isAt second order,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 givesThe squared slow eigenvalue isThe second instability tongue of a weak Mathieu oscillator therefore has the leading intervalHere 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 ${ϵ}^2\sqrt{5}/24+o({ϵ}^2)$. 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 $ϵy_1$ identifies the first nonresonant harmonic correction. The leading expression alone has error $O(ϵ)$ on $t=O({ϵ}^{-2})$.

For comparison, at fixed $\omega \ne 1/2,1$, 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 givesIt is a phase correction, not amplitude growth. Consequently, away from the first two resonance neighborhoods, a general leading solution valid for bounded ${ϵ}^{2}t$ isNarrower 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.

The method of multiple scales applies when a small perturbation acts cumulatively over a long time, as in weakly perturbed harmonic oscillators or slowly modulated waves. A regular asymptotic expansion can contain secular terms, such as $ϵt\sin (\omega t)$, which cease to be small on the time scale of interest. Introduce independent fast and slow times, for example $t$ and $T=ϵt$, and allow the leading amplitude and phase to depend on $T$. The solvability condition in the method of multiple scales removes resonant forcing at each order and determines that slow evolution. For the Mathieu equation, these amplitude equations reveal parametric resonance and its instability intervals.