Can a periodically additively perturbed sinusoidal vector field on the circle have a stable periodic orbit of higher least period? 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 $g(t)=a + b\sin(\phi + 2\pi t)$.
 A: I feel that this is important that I answer you that one third of my thesis was devoted to this equation. :) I have two published articles on this equation (one in Russian though) and I am not alone. Please, contact me and I will send you my thesis with its introduction mentioning all research I know of studying this equation! You can look at my article with A. Klimenko here (https://arxiv.org/pdf/1305.6746.pdf), there is some bibliography.
More precisely, we study the following family of  differential equations on the circle :
$$
\dot{x}= \frac{\cos x + a + b \cos t}{\mu}.
$$
We call this family "Josephson equation"(not the best term since there is another Josephson equation in superconductivity by Brian D. Josephson...) since it models the behaviour of Josephson junctions. I suppose your interests comes from there ?...
Here $a,b, \mu$ are parameters and we study the so-called Arnold tongues (open domains in the space of parameters where the rotation number is constant). As you say, it is indeed true and a remarkable property of this equation that Arnold tongues exist only for integer values of the rotation number. This "miracle" is indeed related to the fact the first-return map is Moebius.
In our article we replace $\cos t$ by any periodic function $g(t)$, and some claims still hold true. 
Moscow School did a lot on that in late decade: see the works of Buchstaber, Karpov, Tertychnyj as well as the seminar of Ilyashenko, works by Glutsyuk, Klimenko, myself, Schurov, Kleptsyn, Ryzhov, Fililomov and others...
The thing that was interesting to us, was to study the Arnold tongues of this equation, their precise geometric structure, I find the pictures quite beautiful ! The tongues, as opposed to the classical Arnold's picture for the perturbations of rotations, form some sort of braids...
Please tell me if you need more information.

1 V. Kleptsyn, I. Schurov, O. Romaskevich Josephson effect and slow-fast systems (in Russian), Nanostructures. Mathematical physics and modelling, 8:1, pp. 31–46, (2013); // [arxiv: 1305.6755]
[2] A. Klimenko, O. Romaskevich , Asymptotic properties of Arnold tongues and Josephson effect, Moscow Mathematical Journal, volume 14, issue 2, pp. 367-384, (2014), // [arxiv: 1305.6746]
