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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.

Write $A=x+iy$. Then $x_{\tau }=(\mu +\beta -\lambda (x^2+y^2))x$ and $y_{\tau }=(\mu -\beta -\lambda (x^2+y^2))y$. These are a gradient flow for $V=-\frac{1}{2}(\mu +\beta )x^2-\frac{1}{2}(\mu -\beta )y^2+\frac{\lambda }{4}(x^2+y^2{)}^2$, so local minima give stable equilibrium points. At the origin the two eigenvalues are $\mu +\beta$ and $\mu -\beta$. The origin has exponential asymptotic stability if $\mu <-|\beta |$, retains asymptotic stability with algebraic decay at $\mu =-|\beta |$, and is unstable if $\mu >-|\beta |$. At equality, ${\rho }^2=x^2+y^2$ obeys $({\rho }^2{)}_{\tau }\le -2\lambda {\rho }^4$, since both linear coefficients are nonpositive. Integrating this inequality proves attraction even in the zero-eigenvalue direction.

For $\beta >0$ the stable nonzero equilibrium points are real; for $\beta <0$ they are imaginary:The real branch has Jacobian matrix eigenvalues $-2(\mu +\beta ),-2\beta$; the imaginary branch has $-2(\mu -\beta ),2\beta$. The oppositely aligned branch, when it exists, is a saddle equilibrium. No mixed real-imaginary nonzero equilibrium is possible when $\beta \ne 0$.

For $\beta =0$, the origin is stable for $\mu \le 0$, with algebraic decay at zero. If $\mu >0$, the circle $|A{|}^2=\mu /\lambda$ is radially attracting. Each point has Lyapunov stability but has a neutral phase direction, so it does not have individual asymptotic stability; the circle has orbital stability. This is the literal model's unpinned family. The general nonzero-$\beta$ branches instead exhibit phase locking to one of two phases separated by $\pi$.

Introduce $T=ϵt$ and write $x_0=A(T)\cos [t+\theta (T)]$. Averagingover one fast period gives $A{A}_T=-3A^4/8$ and ${\theta }_T=0$. The initial data therefore give the method of multiple scales resultthrough $t=O({ϵ}^{-1})$.

For the replacement damping, $\sin (\dot{x})\dot{x}={\dot{x}}^2-{\dot{x}}^4/6+\cdots$ is even in $\dot{x}$. Every term has zero resonant projection onto the fundamental over a complete orbit, so the $O(ϵ)$ slow amplitude and phase equations vanish. Thusthrough $t=O({ϵ}^{-1})$: there is no first-order secular damping, although bounded mean and higher-harmonic corrections occur.

A phase modulation changes the local position of a periodic pattern. Its local wavenumber is shifted by derivatives of the phase. In a method of multiple scales, a correction profile depending on the translated coordinate $\xi =x+\varphi (Y)$ must be differentiated with ${\mathcal{D}}_Y={\partial }_Y+{\varphi }_Y{\partial }_{\xi }$. Omitting this chain rule can change the nonlinear coefficient of a reduced amplitude equation even when its linear coefficients are unchanged.

For $y^{\prime \prime }+({\omega }^2+ϵ\cos t)y=0$, use the method of multiple scales with $T=ϵt$ and $y_0=a(T)\cos (t/2)+b(T)\sin (t/2)$. Removing secular terms gives $a_T=(k-1/2)b$ and $b_T=-(k+1/2)a$. The slow eigenvalues satisfy ${\lambda }^2=1/4-k^2$, so an exponentially growing mode exists inside the displayed leading instability interval. At its leading endpoints the slow matrix is nilpotent; higher-order corrections are needed to determine the exact edges. Floquet theory distinguishes this exponential instability from algebraic growth at a repeated multiplier.

For weak quadratic coupling between frequencies one and two, the first oscillator's complex amplitude can obey $2i{A}_T+\alpha A+b\overline{A}=0$ with a constant real second-oscillator amplitude $b$. In real coordinates this is a linear slow system with squared growth rate $(b^2-{\alpha }^2)/4$. Exponential amplification occurs when the modulation amplitude exceeds detuning, $|b|>|\alpha |$. The method of multiple scales is valid over bounded slow time while the physical amplitudes remain in the weak-coupling regime. At equality the slow matrix can have a Jordan block and produce linear growth for some initial phases.

For initial amplitude one, the method of multiple scales gives $x\simeq A(ϵt)\sin t$ with $A_T=-4A^2/(3\pi )$, hence $A=1/(1+4ϵt/(3\pi ))$. Positive damping causes algebraic decay. Negative damping yields a formal finite-time amplitude pole, but the weak-damping assumption fails before it because $|ϵ|A$ becomes order one. A leading approximation has order-$|ϵ|$ displacement error on fixed slow-time intervals away from that pole.

In a Parker dynamo wave model, an alpha effect of the form $\alpha ={ϵ}^{-1}{\alpha }_0\sin (\omega t/ϵ)$ has zero time average but can generate a finite averaged coupling. The method of multiple scales gives $\tilde{A}=-({\alpha }_0/\omega )\cos (\omega \tau )\overline{B}$ and $\tilde{B}=-(\Omega {\alpha }_0/{\omega }^2)\sin (\omega \tau ){\partial }_x\overline{B}$, where $\tau =t/ϵ$. Consequently $C=\Omega {\alpha }_0^2/(2{\omega }^2)$. With diffusion operator ${\partial }_x^2-{\ell }^2$, the averaged Fourier mode growth exponents are $-k^2-{\ell }^2\pm |k|\sqrt{\Omega C}$. For freely adjustable real $k$, growth requires $\Omega C>4{\ell }^2$. The result is an asymptotic averaged model for fixed nonzero $\omega$, not an exact replacement at finite $ϵ$.

A secular term grows in a nominal perturbation correction and eventually invalidates an otherwise ordered expansion. In the method of multiple scales, a solvability condition removes resonant forcing that would generate such growth.

For $x^{\prime \prime }+ϵc(1-x^2)x^{\prime }+x=k$, use $T=ϵt$ and $x=k-r(T)\cos (t+\varphi (T))$. Removal of secular terms gives ${\varphi }_T=0$ and the displayed amplitude equation. With $R=r^2$, $R_T=c(k^2-1)R+(c/4)R^2$, which is explicitly integrable. If $c$ is replaced by a slowly varying coefficient $C(T)$, the same solution uses the accumulated time ${\int }_0^{T}C(s)ds$. The method of multiple scales is uniform on bounded slow-time intervals only while its amplitude and correction estimates remain bounded.

A slow time resolves evolution occurring over many periods of a faster process. It is treated as an independent variable in the method of multiple scales.