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For $x\prime \prime +ϵf(x)x\prime +x=0$, write $x_0=A(T)\cos (t+\theta )$ and $T=ϵt$. Removing secular terms gives $A_T=-A⟨f(A\cos \psi ){\sin }^2\psi ⟩$ and ${\theta }_T=0$, where the brackets mean a period average. This is the first-order solvability condition in the method of multiple scales, with a slowly evolving amplitude and no first-order frequency correction.

Introduce the slow time $T=ϵt$ and expand $y=y_0(t,T)+ϵy_1(t,T)+\cdots$. At leading order writeThe first-order equation isIts resonant sine and cosine components must vanish by the solvability condition in the method of multiple scales. Let angle brackets denote a full-period average. The cosine projection of the nonlinear term is zero, since $⟨{\sin }^n\psi \cos \psi ⟩=0$. The sine projection givesThis is the averaged oscillator damping by a power of velocity.

If $n$ is odd, defineThen $R_T=-C_n{R}^n$, givingFor every fixed $n$, these leading multiple-scale expansions are uniform on bounded slow-time intervals $0\le ϵt\le T_0$. The unforced phase remains $t$ to this order. A bounded first-order correction supplies the small adjustment to the initial derivative and initial displacement; it does not change this leading result.

If $n$ is even, the average of the odd power ${\sin }^{n+1}\psi$ vanishes. Thus $R_T=0$ andThe first-order forcing has only a constant and even harmonics, so it produces no fundamental resonance. Its bounded particular solution can be supplemented by homogeneous sine and cosine terms to satisfy the initial conditions. There can be an $O({ϵ}^2)$ frequency correction, but its phase accumulation is only $O(ϵ)$ on the present time scale. In contrast to the odd case, the even-power force is invariant under velocity reversal and the equation is reversible; calling it positive damping for both signs of velocity would be wrong. The mechanical energy identity makes the distinction clear:This is nonpositive for odd $n$ and changes sign for even $n$; its cycle average reproduces the amplitude equation above.

For large odd $n$, the central binomial coefficient givesThis form reveals several time regimes. For $L_nϵt\ll 1$, $R=1-C_nϵt+\cdots$; at $t=O(1/(ϵ\sqrt{n}))$, the amplitude change is only $O(1/n)$. For $L_nϵt\gg 1$ but $\log (L_nϵt)\ll n$, the leading change is $R\simeq 1-\log (L_nϵt)/n$. In particular, at $t=O(1/ϵ)$ it is only $O(\log n/n)$, despite the long elapsed time. High powers damp mainly near velocity maxima, and even a small amplitude reduction strongly suppresses subsequent damping.

An order-one amplitude reduction in the averaged law requires $\log (L_nϵt)=O(n)$. Formally, $t=e^{(n-1)\sigma }/(ϵL_n)$ gives $R\sim e^{-\sigma }$, and for fixed odd $n>1$ its late envelope is an algebraic power $t^{-1/(n-1)}$. These exponential-in-$n$ time scales are predictions of extending the averaged law; the fixed-$n$, bounded-$ϵt$ multiple-scale error estimate alone does not establish uniform accuracy that far. A joint large-$n$, small-$ϵ$ approximation also needs control of the increasingly sharp nonlinear peaks, rather than assuming fixed-$n$ error constants are uniform. Taking $n$ fixed first and then examining the large-$n$ envelope avoids that unsupported interchange of limits.

Put $T=ϵt$ and use the method of multiple scales with $x=x_0(t,T)+ϵx_1+\cdots$. At leading order, $x_{0,tt}+x_0=k$, so write $x_0=k-r(T)\cos (t+\varphi (T))$ with $r(0)=k$, $\varphi (0)=0$. The order-$ϵ$ equation isUsing ${\cos }^{2}t\sin t=(\sin t+\sin 3t)/4$, its fundamental sine coefficient is $-2r_T-cr(1-k^2-r^2/4)$, and its fundamental cosine coefficient forces ${\varphi }_T=0$. Removing these resonant forcings is the solvability condition in the method of multiple scales. Thus the shifted Van der Pol amplitude evolution isWith $a=k^2-1$ and $R=r^2$, solve $R_T=caR+c{R}^2/4$ by putting $W=1/R$. This givesThe uniformly valid leading approximation is $x(t)=k-r(ϵt)\cos t+O(ϵ)$ on bounded slow-time intervals for which $r$ remains bounded. The exact initial derivative is recovered by the first-order correction; the displayed leading expression already satisfies the leading initial data.

There is an important limitation to the source's unqualified time-range wording. Starting from $r(0)=k$, the amplitude decays for $k^2<4/5$, is constant at leading order for $k^2=4/5$, and grows for $k^2>4/5$. In the latter case the denominator vanishes atA bounded-amplitude multiple-scale expansion is uniform for $0\le T\le C<T_b$, not through its predicted large-amplitude breakdown. For the decaying case every fixed bounded slow-time interval is available; the leading unstable equilibrium at $k^2=4/5$ also has bounded amplitude over such intervals. The envelope singularity is a breakdown warning, not a proof that the exact original solution has precisely this blow-up time. The coefficient in the PDF is $ϵc$, correcting the converted TeX's duplicated $ϵ$.

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 $ϵ\to 0$.

The linear damped harmonic oscillator has characteristic exponents $-ϵ\pm i\sqrt{1-{ϵ}^2}$. Imposing both initial conditions gives, for $|ϵ|<1$,For completeness, at $|ϵ|=1$ its continuous limiting form is $t{e}^{-ϵt}$; for $|ϵ|>1$ replace $\sin (\omega t)/\omega$ by $\sinh (\sqrt{{ϵ}^2-1}t)/\sqrt{{ϵ}^2-1}$. The initial derivative is one, as the original PDF specifies; the converted TeX loses this derivative and corrupts the time derivatives in both oscillator equations.

The separate nonlinear oscillator has quadratic drag. Introduce the slow time $T=ϵt$ and use the method of multiple scales,The order-$ϵ$ equation isFor positive amplitude, write $\vartheta =t+\varphi$. Its right side is $-2A_T\cos \vartheta +2A{\varphi }_T\sin \vartheta -A^2|\cos \vartheta |\cos \vartheta$. The cosine coefficient of the last angular function iswhile its sine coefficient is zero. The solvability condition in the method of multiple scales removes the resonant harmonics, givingThe leading initial conditions select $A(0)=1$, $\varphi (0)=0$. Therefore the quadratically damped oscillator hasThe same sign follows from the exact energy balance $d[({\dot{x}}^2+x^2)/2]/dt=-ϵ|\dot{x}{|}^3$. Averaging $|\cos \vartheta {|}^3$ gives $4/(3\pi )$, consistent with the amplitude equation.

For $ϵ>0$, the amplitude decays algebraically and stays at most one. On each fixed interval $0\le ϵt\le T_{\ast }$, the omitted displacement correction is $O(ϵ)$, so this approximation is valid through times of order ${ϵ}^{-1}$. The effective drag strength $ϵA$ decreases rather than increases, and the envelope predicts $A\sim 3\pi /(4ϵt)$ at later times. There is no finite-time pole in the positive-drag approximation. A uniform error assertion for arbitrarily long intervals is stronger than the finite-slow-time expansion just derived; the stated $O(ϵ)$ estimate applies to the specified $ϵt=O(1)$ range.

For $ϵ<0$, write $\delta =1-4|ϵ|t/(3\pi )$. The amplitude grows as $1/\delta$, with a formal pole atOn fixed slow-time intervals strictly before that pole, the displacement error remains $O(|ϵ|)$. This estimate is not uniform as $\delta \to 0$: nonresonant corrections have size comparable to $|ϵ|A^2$, and the required weak-drag condition is $|ϵ|A=|ϵ|/\delta \ll 1$. Thus the approximation necessarily fails when $\delta =O(|ϵ|)$, corresponding to a distance of order one in fast time from the formal pole, and may require additional error control when approaching it. The plot therefore stops short of the pole. Its position is a prediction of the leading amplitude equation, not an exact blow-up time for the original nonlinear equation; continuing the formula past it is invalid.

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.

Apply the method of multiple scales with independent fast time $t$ and slow time $T=\epsilon t$, and write $x=x_0(t,T)+\epsilon x_1(t,T)+\cdots$. The chain rule gives $d/dt={\partial }_t+\epsilon {\partial }_T$. At leading order,At first order,A forcing component at the natural frequency produces a secular term in $x_1$. Thus the solvability condition in the method of multiple scales is orthogonality to both $\sin \theta$ and $\cos \theta$. Define the fast-phase average, holding $R,\varphi ,T$ fixed, bySince $⟨{\sin }^2\theta ⟩=⟨{\cos }^2\theta ⟩=1/2$ and $⟨\sin \theta \cos \theta ⟩=0$, the amplitude-phase equations areThey are leading averaged equations, not exact equations for a purely sinusoidal solution of a general nonlinear oscillator.

For the cubic conservative force, $f=-R^3{\cos }^3\theta$, odd symmetry gives $⟨{\cos }^3\theta \sin \theta ⟩=0$, while $⟨{\cos }^4\theta ⟩=3/8$. HenceIf $x(0)=x_{\ast }$ and $x^{\prime }(0)=v_{\ast }$, leading initial matching givesand thereforeon bounded slow time intervals for fixed bounded initial data. The positive frequency shift agrees with the increased restoring force of $x^{\prime \prime }+x+\epsilon x^3=0$. The zero initial state is the exact zero solution, with irrelevant phase. If a first waveform correction is desired, the remaining first-order forcing is $-R_0^3\cos 3\theta /4$, so $x_1=R_0^3\cos 3\theta /32$ plus a homogeneous correction chosen to match the initial data at the next order. This exhibits explicitly why the leading harmonic ansatz is approximate.

Assume $0<ϵ\ll 1$ and $X,Y,\alpha =O(1)$. For $n=2$, use the method of multiple scales with $T=ϵt$ and complex amplitudesAt first order the forcing equations areThe product has frequencies $\pm 1,\pm 3$. Its $e^{it}$ coefficient is $\overline{A}B$, while it has no $e^{2it}$ coefficient. Removing the secular terms givesThe leading initial conditions give $A(0)=X/2$ and $B=Y/2$. Put $b=Y/2$, $A=p+iq$. The primary parametric resonance of a two-to-one oscillator pair is governed byFor $b^2>{\alpha }^2$, choose $\lambda >0$. Solving with $p(0)=X/2,q(0)=0$ yieldsIf $b^2<{\alpha }^2$, let $\nu =\sqrt{{\alpha }^2-b^2}/2$; replace $\cosh (\lambda T)$ by $\cos (\nu T)$ and $\sinh (\lambda T)/\lambda$ by $\sin (\nu T)/\nu$. On the boundary $b^2={\alpha }^2$, take the continuous limit:Thus exponential growth occurs when $|Y|>2|\alpha |$ and $X\ne 0$, on the scale $t=O({ϵ}^{-1})$. Equality is a marginal case that can produce linear slow growth: for $b=\alpha \ne 0$ the displayed initial phase excites it, whereas for $b=-\alpha$ it does not. If $X=0$, $x$ stays identically zero and the other oscillator is exactly uncoupled. The leading $y$ amplitude is constant; its backreaction is a higher-order effect on this time scale.

These are leading approximations for bounded slow time $T$, hence $t=O({ϵ}^{-1})$, while the amplitudes stay $O(1)$. First-order corrections can enforce the initial velocities exactly; the leading formula alone has an $O(ϵ)$ initial-velocity mismatch. In the unstable case the expansion must not be extrapolated through arbitrarily large amplification, where weak coupling and omitted terms cease to be uniform.

For $n=1$, both leading oscillations have frequencies $\pm 1$, so their quadratic product contains only frequencies $0,\pm 2$. There is no first-order resonant coupling: the solvability condition in the method of multiple scales gives $A_T=i\alpha A/2$ and $B_T=0$, a frequency shift without first-order amplitude growth. The constant and second-harmonic corrections can feed back into the fundamental at second order. Consequently the first possible growth scale from this second-order resonance from quadratic oscillator coupling isThis identifies the possible slow scale, not a guaranteed instability for every parameter or initial state. In particular, the first-order $\alpha$ frequency shift may detune the second-order resonance unless appropriately tuned; a second-order amplitude analysis would decide actual growth.

The method of multiple scales introduces independent variables for the fast oscillation and its slow modulation, for example $t_0=t$, $T=ϵt$, and, when needed, $T_2={ϵ}^{2}t$. The derivative becomes $d/dt={\partial }_{t_0}+ϵ{\partial }_T+{ϵ}^2{\partial }_{T_2}+\cdots$. One expands the solution in $ϵ$ while allowing its leading amplitudes and phases to depend on the slow times.

A solvability condition in the method of multiple scales removes resonant forcing of each fast homogeneous mode. Otherwise a correction contains a secular term, such as $ϵt\sin t$, and an ordinary asymptotic expansion fails on long times. The resulting slow amplitude equations incorporate that accumulated effect in the leading solution. One must still check that the amplitudes remain within the assumed weak-coupling regime and that omitted slower effects have not accumulated.

Introduce the slow time $T=\epsilon t$ and treat $t,T$ independently in the method of multiple scales. WriteAt the next order,For fixed $R$ define the period average byHere $f$ inside the averages is evaluated at the leading position and velocity $R\cos \varphi ,-R\sin \varphi$. Orthogonality to both fundamental harmonics is the solvability condition in the method of multiple scales: it removes the resonant forcing that would otherwise generate a secular term. Since the squared sine and cosine averages are $1/2$, the amplitude-phase equations areEquivalently $\dot{R}=-\epsilon ⟨f\sin (t+\theta )⟩$ and $\dot{\theta }=-\epsilon ⟨f\cos (t+\theta )⟩/R$ at first order. These equations describe a bounded, weakly perturbed oscillation over $t=O({\epsilon }^{-1})$ while the amplitude remains in a range where the expansion is ordered. At zero amplitude the oscillation phase coordinate is singular; Cartesian harmonic coefficients or an equilibrium analysis should replace it.

Use the same method of multiple scales and its solvability condition in the method of multiple scales. The position-dependent damping function is odd, so the lemma that odd position-dependent damping has zero first-order amplitude drift applies. In its wave amplitude average, $\psi \mapsto \psi +\pi$ changes $\sin (A\cos \psi )$ to its negative while preserving ${\sin }^2\psi$, so the average is zero. Hence $A^{\prime }=0$; the wave phase average again gives ${\theta }^{\prime }=0$.

Equivalently, expand the position-dependent damping in its uniformly convergent power series for bounded $A$. Each term ${\cos }^{2m+1}\psi \sin \psi$ has only even sine Fourier harmonics, so none resonates with the unit-frequency oscillator. With the initial wave amplitude and wave phase unchanged, the leading uniform approximation isThere is no first-order slow drift. Effects at the next order accumulate only an $O(ϵ)$ correction on this time scale.

Introduce the slow time $T=ϵt$ and use the method of multiple scales with $x_0=A(T)\cos \psi$, $\psi =t+\theta (T)$. The initial data give $A(0)=\sqrt{2}$ and $\theta (0)=-\pi /4$. At first order the equation for the correction isThe averaged amplitude for position-dependent damping follows from the solvability condition in the method of multiple scales, which removes the resonant sine and cosine terms. With brackets denoting a full-period average, this givesThe wave phase average vanishes because $f(A\cos \psi )\sin \psi \cos \psi$ changes sign under $\psi \mapsto 2\pi -\psi$.

For the quadratic position-dependent damping, $⟨{\cos }^2\psi {\sin }^2\psi ⟩=1/8$ and $⟨{\sin }^2\psi ⟩=1/2$, soSolving this logistic equation with $A^2(0)=2$ gives $A(T)=2/\sqrt{1+e^{-T}}$. Thus the uniformly valid leading approximation isHere $T_{\ast }$ is any fixed finite positive constant. Bounded nonresonant corrections and the averaging estimate make the error uniform on this slow-time interval. The leading expression meets the position datum exactly and the velocity datum to leading order; the first-order correction supplies its $O(ϵ)$ adjustment.

Apply the method of multiple scales with independent fast time $t$ and slow time $T={ϵ}^{2}t$. A preliminary slow time $ϵt$ produces no resonant forcing at first order, so its leading amplitudes are constant; the first nontrivial modulation is at $T$. Write $x=x_0+ϵx_1+\cdots$, withAt order $ϵ$, the inhomogeneous linear differential equation is $({\partial }_t^2+1)x_1=-\cos t{x}_0$. Absorbing its homogeneous part into the amplitudes givesAt order ${ϵ}^2$, the forcing is $-2{\partial }_t{\partial }_T{x}_0-k{x}_0-\cos t{x}_1$. Its resonant Fourier series coefficients areThe solvability condition in the method of multiple scales sets both coefficients to zero, giving the amplitude equations. DefineFor bounded initial conditions $x(0)=x_{\ast }$, $\dot{x}(0)=v_{\ast }$, the requested leading approximation isThis holds on bounded intervals of $T$, hence $t=O({ϵ}^{-2})$, with the constant in the error allowed to depend on the bounded $T$ interval. If desired, retain the displayed $ϵx_1$ and adjust $A(0),B(0)$ by $O(ϵ)$ to match initial conditions at that order. No particular initial conditions were supplied.

Since $M^2=abI$, the matrix exponential is $I\cosh (\sigma T)+(M/\sigma )\sinh (\sigma T)$ when ${\sigma }^2=ab>0$. When $ab<0$, put $\omega =\sqrt{-ab}$ and replace this with $I\cos (\omega T)+(M/\omega )\sin (\omega T)$. At $ab=0$ it is $I+TM$. Thus the leading second instability tongue of a weak Mathieu oscillator has exponentially growing solutions for $-1/12<k<5/12$, and the leading modulation is bounded for every initial condition precisely whenAt either displayed endpoint, $M$ is a nonzero nilpotent operator; some initial conditions give a secular term proportional to $T$. This leading calculation alone must not be mistaken for an exact finite-$ϵ$ stability decision at the endpoints.

For an exact small-$ϵ$ interpretation, put $t=2s$. The standard Mathieu equation parameters are $a_{\mathrm{M}}=4+4k{ϵ}^2$ and $q=-2ϵ$. Its two relevant Mathieu characteristic values are obtained by projecting $-d^2/d{s}^2+2q\cos 2s$ onto even and odd Fourier series modes. Near the unperturbed value $4$, the even modes $1,\cos 2s,\cos 4s,\cos 6s$ have diagonal entries $0,4,16,36$ and adjacent couplings $\sqrt{2}q,q,q$ in an orthonormal basis; the odd modes $\sin 2s,\sin 4s,\sin 6s$ have diagonal entries $4,16,36$ and couplings $q,q$. Substituting $4+c_2{q}^2+c_4{q}^4$ into their characteristic polynomials givesHigher modes of the Fourier series first contribute at higher order. Therefore the fourth-order edges of the second Mathieu instability tongue areFloquet theory gives boundedness of every solution outside the closed gap, $k<k_{-}(ϵ)$ or $k>k_{+}(ϵ)$, in this fixed-$k=O(1)$ neighborhood. At the actual gap edges the second independent solution is generically unbounded. In particular, if $k$ is fixed exactly at $-1/12$ or $5/12$, it lies just outside the true gap and is stable for each sufficiently small positive $ϵ$. The strict boxed criterion is the robust leading-order, uniformly separated criterion; the endpoints require this higher-order refinement, and their amplification bounds need not remain uniform as $ϵ\to 0$.

Eliminate the instantaneous Stokes flow velocity in favour of temperature. On a horizontal Fourier mode $k$, the Stokes temperature-slaving operator maps $\theta$ to $w={\mathcal{W}}_R\theta$, where $(D^2-k^2{)}^{2}w=R{k}^2\theta$ and $w=w^{\prime \prime }=0$. The temperature evolution has linear operator ${\mathcal{L}}_R=D^2-k^2+{\mathcal{W}}_R$ and bilinear map $\mathcal{N}(\varphi ,\chi )=-{\mathbf{u}}_R[\varphi ]\cdot \nabla \chi$. Under the homogeneous thermal Dirichlet boundary conditions, ${\mathcal{L}}_{R_c}$ is self-adjoint. Normalize its critical eigenfunction as $g(z)=\sin \pi z$ and set $s=k_c^2+{\pi }^2$; the critical vertical velocity is $f=sg$.

At order $ϵ$, the critical eigenfunction equation gives ${\mathcal{L}}_{R_c}(g{e}^{i{k}_{c}x})=0$. At order ${ϵ}^2$, the weakly nonlinear expansion contains the imposed second harmonic and the quadratic products of the critical mode: a horizontally uniform temperature correction proportional to $|A{|}^2$ and, in a general vertical-mode calculation, a second harmonic proportional to $A^2$. These corrections are found by solving the noncritical boundary value problems, with homogeneous thermal data except for the imposed forcing.

At order ${ϵ}^3$, the method of multiple scales produces the slow derivative $A_{\tau }$, the detuning term $R_2{\partial }_R{\mathcal{L}}_R$, and the two cross-advection terms involving first- and second-order fields. Project the $e^{i{k}_{c}x}$ component onto the adjoint eigenfunction using the vertical inner product. This is the solvability condition in the method of multiple scales: divide each resonant projection by $⟨g,g{⟩}_z$. The detuning supplies $\mu A$ with $\mu \propto R_2$; interactions of horizontal wavenumbers $2k_c$ and $-k_c$ permit $\beta A^{\ast }$ with $\beta \propto \gamma$; self-interaction through the slaved mean and second harmonic supplies $-\lambda |A{|}^{2}A$. Other products have the wrong horizontal wavenumber. Reflection $x\mapsto -x$ permits real coefficients with this cosine forcing. Thus the symmetry-allowed spatially forced convection amplitude equation isThere is a useful specialization that should not be silently missed. For the literal one-vertical-mode Stokes flow problem, the vanishing two-to-one forcing coefficient for Stokes convection makes $\beta =0$ at this order. To see this, write a positive second-harmonic forcing component as $(w,\theta )=(W(z),\Theta (z))e^{2i{k}_{c}x}$, incorporating the cosine's factor $1/2$. Its coupling to the negative critical harmonic has projected integrand, apart from sign and its factor $A^{\ast }$,The integral vanishes because $g=0$ at both plates, even though $\Theta (0)$ is nonzero. This proves the cancellation without solving the forced profiles. The permitted coefficient is therefore zero times $\gamma$; symmetry alone does not establish nonzero phase pinning for the equations actually supplied.

The same normalization makes the remaining coefficients explicit. Since ${\mathcal{W}}_{R}g=(R{k}_c^2/s^2)g$, $\mu =R_2{k}_c^2/s^2$. The quadratic second harmonic cancels for $f=sg$, while the uniform correction is ${\theta }_{20}=-s|A{|}^2\sin (2\pi z)/(2\pi )$. Projecting $-w_1{\partial }_z{\theta }_{20}$ gives $-s^2|A{|}^{2}A/2$. Thus for the literal model and this temperature normalization,A generic nonzero $\beta$ would require a nonvanishing projection in an amended physical model or a different forcing structure. It is still meaningful to classify the real-coefficient amplitude equation requested independently.