= Shifted Van der Pol amplitude evolution
{title2=$r_T=\tfrac c2(k^2-1)r+\tfrac c8r^3$}
For $x''+\epsilon c(1-x^2)x'+x=k$, use $T=\epsilon t$ and $x=k-r(T)\cos(t+\phi(T))$. Removal of <secular terms> gives $\phi_T=0$ 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.
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