# Synchronization frequency of the Kuramoto model for coupling matrix with constant rows

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.]

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Found (something at least very close to) the answer in theorem V.1 of arxiv.org/abs/0910.5673 –  Steve Huntsman Jul 8 '11 at 4:43

As far as literature goes:

"From Kuramoto to Crawford..." - S Strogatz: http://omnis.if.ufrj.br/~monica/ACMSM2008/StrogatzPD2000.pdf

"Synchronization in Complex Networks" - Arenas et al http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.2976v3.pdf

However,

The special case you mention can be reduced to the mean field kuramoto model by introducing a complex order parameter. $re^{i \Psi} = \frac{1}{n}\sum\limits_{j=1}^n e^{i\theta_j}$

So $\dot\theta_i = \omega_i + K_ir\sin(\Psi - \theta_i)$

Also, if you let $\omega_i \rightarrow \Omega + \Delta\omega_i$ and $\theta_i \rightarrow \theta_i + \Omega t$ where $\Omega$ is your mean frequency

you get

$\dot\theta_i = \Delta\omega_i + K_ir\sin(\Psi - \theta_i)$

with the mean of all $\Delta\omega_i$ are zero and most importantly, $\dot\theta_i = 0$ now corresponds to that oscillator running at your mean frequency.

Now if $K_i >\frac{2}{\pi g(0)} \forall i$ with $g$ the p.d.f for your $\Delta\omega_i$'s, then you'll definately get a large proportion of oscillators synching to your mean frequency. (Strogatz' paper covers this derivation for constant $K$)

How many is a tough question, depending on your p.d.f. For general $K_i$ i think Arenas' paper should provide a starting point.

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Thanks, see also my comment above for a reference that I found after another half-day of digging. –  Steve Huntsman Jul 12 '11 at 22:08