Quadratic oscillator phase-modulation condition ($A_T=-\frac{5i}{3}|A{|}^{2}A$)

For $u^{\prime \prime }+u=ϵu^2$, a leading oscillation $U_0=A(T)e^{it}+\overline{A}(T)e^{-it}$ has slow time $T={ϵ}^{2}t$. The first-order forcing contains only zero and double frequencies. At second order the fundamental forcing is $(10/3)|A{|}^{2}A-2i{A}_T$; eliminating resonance gives the displayed equation. Its modulus is constant and its phase drifts. A leading real amplitude $R=2|A|$ therefore has frequency $1-5R^2{ϵ}^2/12+O({ϵ}^3)$. This replaces a secular phase term by a uniformly modulated oscillation.