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.
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.