= Solution
For <velocity-only forcing in averaged oscillator equations>, $f=f(\dot y)$, the <oscillation phase> average vanishes: $f(-R\sin\phi)\cos\phi$ is a periodic total <derivative>, using an antiderivative of $f$. Hence $\theta_T=0$ to first order, although the <amplitude> may change.
In the cubic case, $f=-R^3\sin^3\phi$, so
$$
R_T=R^3\langle\sin^4\phi\rangle=\frac38R^3.
$$
With $R(0)=R_0>0$, integration gives
$$
\boxed{R(T)=\frac{R_0}{\sqrt{1-3R_0^2T/4}},\qquad \theta(T)=\theta_0}.
$$
For positive $\varepsilon$ this is <cubic velocity anti-damping>, not damping: the exact <energy> <derivative> is $d(\dot y^2+y^2)/(2dt)=\varepsilon\dot y^4\geq0$. The slow solution grows and formally diverges at $T=4/(3R_0^2)$. The weakly nonlinear approximation loses validity before its <amplitude> becomes so large that $\varepsilon R^2$ is not small; its divergence is not a controlled prediction of the exact late-time solution. If $R_0=0$, the exact zero solution should be used separately.
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