The non-mean field version of the Kuramoto model is given by

$\dot \theta_i = \omega_i + \sum_j K_{ij} \sin(\theta_j-\theta_i)$

and its study is of considerable interest for understanding the synchronization of chaotic systems ($K_{ij}$ is called the coupling matrix).

I am interested in a special case* of the form $\omega_i \equiv c_i \omega_0$, $K_{ij} \equiv c_i K_0$, viz.

$\dot \theta_i = c_i \left[ \omega_0 + K_0 \sum_j \sin(\theta_j-\theta_i) \right]$

Has this case been treated in the literature for $c_i$ specified in advance or sampled from some distribution? References dealing with this case would be very helpful. I am ignorant of the literature but have not been able to find anything after looking.

Perhaps this is better suited to physics.stackexchange, but in principle it seems like more of a mathematics problem to me.

[*Specifically, I have reason to think that a system of this or similar form, in which the individual components are identical up to their individual rates, would (for suitably large $K_0$) drive its components to a common frequency that is (at least related to) the arithmetic mean of the $\omega_i$. Results speaking to the synchronization frequency are therefore particularly interesting to me.]