Given parameters $(a,k,A) \in \mathbb{R}^3$, we consider on $\mathbb{S}^1$ the $2\pi$-periodic ODE $$ \dot{\theta} \ = \ - a\sin(\theta) + k + A\cos(t) \hspace{4mm} \mathrm{mod} \ 2\pi. $$ Identifying $\mathbb{S}^1$ with the one-point compactification $\hat{\mathbb{R}}$ of the reals via the stereographic projection $\tan(\frac{\cdot}{2})$, this equation becomes the Riccati equation $$ \dot{y} \ = \ -ay + \tfrac{1}{2}(k+A\cos(t))(1+y^2). $$ As shown in this question, the time-$t$ map is a Möbius transformation for every $t$, and so we have the following: Either

(a) all solutions are neutrally stable,

(b) there are exactly two periodic solutions, one stable and one unstable, or

(c) there is one periodic solution, and this is globally attractive but not stable;

moreover, in cases (b) and (c), the periodic solutions must be $2\pi$-periodic; in case (a), if there is a $2\pi$-periodic solution then all solutions are $2\pi$-periodic.

Is it the case that for every $(a,k,A)$ for which there exists at least one $2\pi$-periodic solution, there are parameters $(\tilde{a},\tilde{k},\tilde{A})$ arbitrarily close to $(a,k,A)$ for which (b) holds?

**Remark.** If I understand correctly, the set $M$ of orientation-preserving real Möbius transformations of $\hat{\mathbb{R}}$ is a 3-dimensional connected noncompact Lie group (which is naturally embedded into the 3-dimensional complex Lie group of all Möbius transformations of $\hat{\mathbb{R}}$), with each $f \in M$ admitting the local chart
$$ \frac{f(\cdot) + \alpha}{\beta f(\cdot) + \gamma} \ \mapsto \ (\alpha,\beta,\gamma). $$
[This chart is defined on the set of all $\tilde{f} \in M$ for which $\tilde{f}{}^{-1}(0) \neq f^{-1}(\infty)$.]

The set $M$ can be partitioned as \begin{align*} M \ &= \ U_1 \cup U_2 \cup \bar{S} \\ &\textrm{with } \ \bar{S} \ = \ \partial U_1 \ = \ \partial U_2 \ = \ S \cup \{e\} \end{align*} where [using notation in which an expression involving $x$ is intended to mean that expression as a function of $x$]: $$ U_1 \ = \ \left\{ \lambda - \frac{\mu}{x+\nu} : 0 < \mu < \left( \frac{\lambda+\nu}{2} \right)^{\!2} \right\} \ \cup \ \big\{\mu x + \nu : \mu \in (0,\infty) \setminus \{1\} \big\} $$ is the open set consisting of all maps with a stable and an unstable fixed point, $$ U_2 \ = \ \left\{ \lambda - \frac{\mu}{x+\nu} : \mu > \left( \frac{\lambda+\nu}{2} \right)^{\!2} \right\}$$ is the open set consisting of all maps that are topologically conjugate to a nontrivial circle rotation, $$ S \ = \ \left\{ \lambda - \frac{\frac{1}{4}(\lambda+\nu)^2}{x+\nu} : (\lambda,\nu) \neq (0,0) \right\} \ \cup \ \big\{ x + \nu : \nu \neq 0 \big\} $$ is the set of all maps admitting a unique fixed point (in which case all trajectories are homoclinic), and $e$ is the identity function.

Now let $\Phi \colon \mathbb{R}^3 \to M$ be the map sending a tripet $(a,k,A)$ onto the time-$2\pi$ mapping of the Riccati equation above. As in Anthony's idea, this map will (I believe) be an analytic map. The question then becomes:

Is it the case that $\Phi^{-1}(\bar{U}_1) \subset \overline{\Phi^{-1}(U_1)}\,$?

[Equivalently, since $\partial U_1 = \bar{S}$: Is it the case that $\Phi^{-1}(\bar{S}) \subset \overline{\Phi^{-1}(U_1)}\,$?]

I had hoped that perhaps $\Phi$ is a local diffeomorphism, in which case the result would be clear. But **$\Phi$ is clearly not a local diffeomorphism**: indeed, it only has rank $1$ at points $(0,k,0)$, and has rank at most $2$ at points $(0,k,A)$.