= Solution
For <conservative forcing in averaged oscillator equations>, $f=f(y)$, the average determining $R_T$ vanishes. If $F'=f$, then
$$
f(R\cos\phi)\sin\phi=-\frac1R\frac{d}{d\phi}F(R\cos\phi),
$$
whose <period average> is zero. Thus $R=R_0$ to this order, while a position-dependent force generally shifts the <oscillation phase> and <frequency>. The exact equation also conserves the <energy> $\tfrac12\dot y^2+\tfrac12y^2-\varepsilon F(y)$, consistent with the absence of <amplitude> drift for bounded orbits.
For the cubic <position> force, $\langle f\cos\phi\rangle=R^3\langle\cos^4\phi\rangle=3R^3/8$. Therefore
$$
\boxed{R(T)=R_0,\qquad \theta(T)=\theta_0-\frac38R_0^2T}.
$$
The leading oscillation is $R_0\cos[(1-3\varepsilon R_0^2/8)t+\theta_0]$. For positive $\varepsilon$ it is a softening cubic oscillator, obtained by weakly perturbing a <harmonic oscillator>. The weak-force condition is $\varepsilon R_0^2\ll1$; the constant $R$ is the leading harmonic <amplitude>, not a claim that the exact waveform remains sinusoidal.
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