Coupling in the circle map I'm currently investigating Arnold tongues (areas in parameter space with rational rotation numbers $\rho$, ie. $\rho(\Omega, K) \in \mathbb{Q}$) arising when iterating the circle map 
$$ \theta_{n+1} = \theta_n + \Omega + \frac{K}{2\pi}\sin{2\pi\theta_n} \pmod 1$$ 
It is stated in the article on wikipedia (and elsewhere, eg. Gilmore & Lefranc, 2012) that the parameter $K$ represent a coupling constant - so presumably, the last term in the circle map represents a coupling.
How can one assert that this is true?
EDIT (Clarification):
Gilmore and Lefranc motivate the circle map by considering two oscillators $\theta$ and $\theta'$ with frequencies $v$ and $v'$ so that $\theta(t) = vt \pmod 1$ and $\theta'(t) = v't \pmod 1$. The first oscillator is Poincaré sampled at $t = n/v', n \in \mathbb{N}$ so that 
$$\theta_{n+1} = \theta_n + \Omega \pmod 1,$$
where $\Omega = v/v'$. 
So what I'm after is an explanation of why these two oscillators are coupled by the last term in the circle map.
 A: This is really some sort of terminology but it has roots in simple examples.
Suppose you have two simple rotations of circle: 
$\varphi^{(1)}_{n+1} = \Omega^{(1)} + \varphi^{(1)}_{n}$ and $\varphi^{(2)}_{n+1} = \Omega^{(2)} + \varphi^{(2)}_{n}$ (all maps are w.r.t. $\mod 2\pi$). The dynamics of each map doesn't depend on dynamics of another map and vice versa. 
Coupling is a way to add interaction between two objects (think about spring and masses system, for example). Some classical examples of coupling force for rotations/oscillations has dependency on $\varphi^{(2)} - \varphi^{(1)}$ 
via some odd function $f(x) = -f(-x)$.
In this case coupled dynamics looks like:
$$\begin{align} \varphi^{(1)}_{n+1} = \Omega^{(1)} + \varphi^{(1)}_{n} + K \cdot f(\varphi^{(1)}_n - \varphi^{(2)}_n), \\
\varphi^{(2)}_{n+1} = \Omega^{(2)} + \varphi^{(2)}_{n} + K \cdot f(\varphi^{(2)}_n - \varphi^{(1)}_n)
\end{align}$$
$K$ will have a meaning of coupling strength: when $K=0$ there is no coupling 
at all, otherwise two dynamics interact.
There are two simplest examples of such coupling force: it's a linear coupling
$f(x) = Cx$ and coupling via sine function $f(x) = C\sin x$.
Sometimes people  (when studying synchronization questions, for example) may care only about the difference of these two motions. And here comes dynamics of difference:
$$ \theta_{n+1} = \Omega + \theta_n + 2K \cdot f(\theta_n),$$
where $\theta_n = \varphi^{(2)}_n - \varphi^{(1)}_n$ and 
$\Omega = \Omega^{(2)} - \Omega^{(1)}$.
You may see that this is exactly the form that you provided at the beginning.
