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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.

Again put $T=ϵt$. To leading order the oscillator has form $x_0=k-r(T)\cos (t+\varphi (T))$, while the second variable has a slow part $Y(T)$. Averaging its equation over the fast period givesThe oscillatory forcing of the second variable produces only an order-$ϵ$ correction: its leading fast derivative is $-r(T)\cos t$, whose bounded primitive is $-r(T)\sin t$. Thus it creates no secular term.

The oscillator solvability condition is the same as in part (i), with $c$ replaced by $Y(T)$:Introduce accumulated slow timeThe shifted Van der Pol amplitude evolution therefore gives, for $a=k^2-1$,ConsequentlyAs before, uniformity for $t=O({ϵ}^{-1})$ means a fixed bounded slow-time interval on which the envelope stays bounded. For $k^2>4/5$ it fails when $S(T)$ reaches $\log (5-4/k^2)/(k^2-1)$, with limiting value $4$ at $k=1$. There is no bounded leading approximation covering all such slow times for arbitrary positive $k$.

Use the method of multiple scales, $T=\epsilon t$, and write $x_0=A(T)\cos \psi$, $\psi =t+\theta (T)$. At first order,Removing the fundamental secular terms gives the averaged amplitude for position-dependent damping:where the brackets mean a full-period average. The initial conditions set $A(0)=1$ and $\theta (0)=0$ at leading order. For quadratic damping, $⟨{\sin }^2\psi ⟩=1/2$ and $⟨{\cos }^2\psi {\sin }^2\psi ⟩=1/8$, givingThus the Van der Pol amplitude evolution yieldsThe leading amplitude approaches the stable value two. The small initial-velocity discrepancy introduced by the slow amplitude derivative is removed by the first-order correction and does not affect this leading approximation.

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 $ϵ\to 0$.

At fixed $t$, expand the exact damped harmonic oscillator solution using $e^{-ϵt}=1-ϵt+O({ϵ}^2)$, $\omega =1+O({ϵ}^2)$ and $1/\omega =1+O({ϵ}^2)$. Its Poincaré expansion through first order isThe correction $-ϵt\sin t$ is a secular term. When $ϵt$ is order one it is not small, and the truncated amplitude eventually changes sign and grows in magnitude, instead of reproducing positive damping. This loss of ordering is why a fixed-time regular perturbation is unsuitable on the damping timescale.

Set $T=ϵt$ and use the method of multiple scales:Since ${\omega }^2=1/4+kϵ+O({ϵ}^2)$, the first correction satisfiesThe resonant cosine and sine coefficients must vanish to avoid secular terms. Using the product-to-sum formulas givesThe remaining forcing has frequency $3/2$ and permits the bounded particular correction $y_1=(a/4)\cos (3t/2)+(b/4)\sin (3t/2)$. The two slow amplitudes carry the two arbitrary constants of the original equation.

For an explicit general leading solution, putThe matrix exponential givesThe error is uniform on bounded intervals of $T$, for bounded initial data. When $\lambda$ is imaginary this expression uses ordinary sine and cosine; at $\lambda =0$ its continuous limit is $I+TM$.

The primary instability tongue of a weak Mathieu oscillator has a positive slow eigenvalue precisely whenThese are the leading edges, rather than an exact finite-$ϵ$ interval. At the leading endpoints the amplitude matrix is a nilpotent matrix, so the leading solution can grow linearly in $T$ and does not decide the exact exponential instability. For completeness, set $\omega =1/2+\beta ϵ+\gamma {ϵ}^2$. The periodic or antiperiodic edge calculation uses a pure cosine when $\beta =-1/2$ and a pure sine when $\beta =1/2$, with the third-harmonic correction just found. The next resonant coefficient is $-(\gamma +3/8)y_0$, soThus the exact growing interval is between these shifted edges. In particular, for positive small $ϵ$ the fixed leading value $k=-1/2$ lies just inside it, whereas $k=1/2$ lies just outside. Interior detunings bounded away from the edges are already resolved by the leading method of multiple scales.

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.

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 $ϵ\to 0$.

The linear $\cos x$ mode has frequency squared $k^2-1$, so $k_c(0)=1$. To resolve the small frequency near threshold, writeThe leading equationand the initial data give

At $O({\epsilon }^2)$, the equation for ${\theta }_2$ has a forcing whose $\cos x$ component isThe Fredholm solvability condition removes this secular term and yields the slow amplitude equationIt has the conserved energyThe potential has maxima atA periodic orbit launched from $A(0)=1$, $A^{\prime }(0)=0$ exists only when that turning point lies inside the two maxima, namelyAt equality the orbit is the separatrix; below it, the assumed real periodic oscillation is lost. Therefore

The homogeneous solution is $A\cos x+B\sin x$. If $k\ne 1$, the method of undetermined coefficients permits a particular integral $C\cos kx$. Substitution gives $(1-k^2)C=1$ and henceFor $k=1$, the forcing is itself a homogeneous solution, so this trial fails by resonance in a differential equation. Multiplying the sine trial by $x$ givesThus the resonant general solution isThe secular term containing $x$ describes the increasing amplitude under undamped resonant forcing.

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.

A quadratic product of plane waves generates sums and differences of their wavenumbers and angular frequencies. A component drives a secular term when its generated pair also satisfies the linear dispersion relation. A solvability condition in the method of multiple scales then determines the coupled amplitude equations on a slow time.

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.