Shifted Van der Pol amplitude evolution ($r_T=\frac{c}{2}(k^2-1)r+\frac{c}{8}{r}^3$)

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.