= Solution
At leading order,
$$
u=R\cos\theta,
\qquad
u_t=-R\sin\theta,
$$
so
$$
f=-u(u_t)^2
=-R^3\cos\theta\sin^2\theta.
$$
The required period averages are
$$
\langle\cos\theta\sin^3\theta\rangle=0,
\qquad
\langle\cos^2\theta\sin^2\theta\rangle=\frac18.
$$
Hence
$$
R'=0,
\qquad
R\phi'=\frac{R^3}{8}.
$$
The initial data give $R(0)=1$ and $\phi(0)=0$, so
$$
\boxed{R(T)=1,
\qquad
\phi(T)=\frac T8}.
$$
Thus the perturbation produces a slow frequency shift but no leading amplitude decay:
$$
\boxed{u(t)\sim\cos\left(t+\frac{\epsilon t}{8}\right).}
$$
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