 Solution

For $f(x)=\sin x$, the period average $⟨\sin (A\cos \psi ){\sin }^2\psi ⟩$ vanishes: a half-period shift reverses the first factor and leaves the second unchanged. This is the zero first-order amplitude drift for odd damping. The averaged amplitude for position-dependent damping therefore has $A_T=0$, while its first-order phase equation again has ${\theta }_T=0$. With the initial conditions,The first-order forcing has no fundamental resonant component, so it produces bounded corrections rather than a secular amplitude change on this time scale. This statement concerns the leading approximation on $t=O({\epsilon }^{-1})$.