I have heard that differential equations on $\mathbb{S}^1$ of the form \begin{equation} \hspace{40mm} \dot{\theta}(t) \ = \ A\sin(\theta(t)) + g(t) \hspace{4mm} \mathrm{mod} \ 2\pi, \hspace{40mm} (1) \end{equation} 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 $g$)?

Just to add, in case this makes the problem easier: I'm particularly interested in the case that $g(t)=a + b\sin(\phi + 2\pi t)$.