I have heard that differential equations on $\mathbb{S}^1$ of the form
$$ \dot{x}(t) \ = \ A\sin(2\pi x(t)) + g(t) \hspace{4mm} \mathrm{mod} \ 1, $$
where $A$ is a constant and $g$ is a continuous $1$-periodic function, cannot admit locally asymptotically stable periodic solutions of least period $N \geq 2$.

[In other words, if the rotation number of the time-$1$ map $f \colon \mathbb{S}^1\to\mathbb{S}^1$ is a non-zero rational, then all the periodic solutions are not locally asymptotically stable.]

Is the above claim true? More generally, are there any existing studies specifically on the above differential equation (outside the case of constant $g$)?