= Quadratic oscillator phase-modulation condition
{title2=$A_T=-\frac{5i}{3}|A|^2A$}
For $u''+u=\epsilon u^2$, a leading oscillation $U_0=A(T)e^{it}+\overline A(T)e^{-it}$ has slow time $T=\epsilon^2t$. The first-order forcing contains only zero and double frequencies. At second order the fundamental forcing is $(10/3)|A|^2A-2iA_T$; eliminating <resonance> gives the displayed equation. Its modulus is constant and its phase drifts. A leading real amplitude $R=2|A|$ therefore has frequency $1-5R^2\epsilon^2/12+O(\epsilon^3)$. This replaces a secular phase term by a uniformly modulated oscillation.
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