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Matching powers of $ϵ$ gives $u_0^{\prime \prime }+u_0=0$, $u_1^{\prime \prime }+u_1=u_0^2$, and $u_2^{\prime \prime }+u_2=2u_0{u}_1$, with $u_0(0)=2$ and homogeneous initial data for every higher correction. ThusThere is no resonant fundamental forcing at first order. At second order,The first term resonates with the homogeneous oscillator. Solving with the initial data givesConsequently the regular perturbation approximation is $u=u_0+ϵu_1+{ϵ}^2{u}_2+O({ϵ}^3)$ for fixed $t$. The first secular term is $(10/3){ϵ}^{2}t\sin t$, at order ${ϵ}^2$. It makes the regular expansion nonuniform on times of order ${ϵ}^{-2}$.

For a weakly perturbed harmonic oscillator, expand both its periodic solution and its angular frequency, using the stretched time $\tau =\omega t$. At each order, choose the frequency correction to remove fundamental-harmonic forcing that would produce a secular term. For a self-excited periodic orbit, cancellation of the independent sine and cosine terms can determine both the leading amplitude and the frequency correction. Actual existence follows when the associated Poincaré return map has a simple displacement zero, by the implicit function theorem.

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.

A secular term is a perturbative correction whose growth makes an initially ordered asymptotic expansion fail on a longer scale. A forcing at the natural frequency of a harmonic oscillator gives the simplest example:is a particular solution, as direct differentiation verifies. Although $ϵu_1$ is small at fixed $t$, it can be order one at $t=O({ϵ}^{-1})$.

Such growth need not mean the exact solution grows. The exact bounded oscillation $\cos (\sqrt{1-ϵ}t)$ expands as $\cos t+(ϵt/2)\sin t+\cdots$ at fixed time: its secular term represents accumulating phase error. The Poincaré-Lindstedt method absorbs this into a corrected angular frequency, while the method of multiple scales evolves a slow phase or amplitude.

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 solvability condition in the method of multiple scales is the compatibility requirement allowing a bounded periodic correction at a given perturbation order. Forwith $2\pi$-periodic $F$ in the fast variable, the Fredholm solvability condition for a self-adjoint operator requiresNecessity follows by integration by parts against the two periodic homogeneous solutions. For smooth periodic $F$, sufficiency follows by Fourier series: all coefficients outside modes $\pm 1$ can be divided by $1-n^2$. The forbidden modes would instead generate linearly growing secular terms.

The forcing contains slow derivatives of lower-order amplitudes, so these two conditions become amplitude-phase equations. For a general non-self-adjoint operator, project onto its adjoint null space instead of assuming the same homogeneous modes work.^[ref]