= Oscillator coupled to slow damping feedback
{title2=$R_T=-YR,\quad Y_T=\langle g(R\cos\theta)\rangle$}
For $x''+x=-2\varepsilon yx'$ and $y'=\varepsilon g(x)$, the <method of multiple scales> gives the displayed <amplitude-phase equations>, together with constant leading phase. For $g(x)=x$, $Y$ is constant and $R$ changes exponentially. For $g(x)=\log|x|$ and positive amplitude, the integrable phase average is $\log(R/2)$. Thus $L=\log(R/2)$ obeys $L_{TT}+L=0$ with $L_T=-Y$. The identically zero oscillator is inadmissible for logarithmic feedback, whose force is undefined there.
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