Approximate solutions to Kuramoto synchronization model

[Danny Boy]

According to the wiki entry 'Kuramoto Model', if we consider the $N=2$ case then the governing equations are $$\frac{d \theta_1}{dt} = \omega_i + \frac{K}{2}\sin(\theta_2 - \theta_1)~~~\text{and}~~~\frac{d \theta_2}{dt} = \omega_i + \frac{K}{2}\sin(\theta_1 - \theta_2),$$
where $\theta_i$ are the phases of the oscillator, $\omega_i$ are the natural frequencies and $K$ is the coupling constant.

To find solutions I then propose (as in the wiki entry) using the transformation $re^{i \psi} = \frac{1}{2}(e^{i \theta_1} + e^{i \theta_2})$. Thus allowing us to rewrite the coupled equations as $$\frac{d \theta_i}{dt} = \omega_i + Kr\sin(\psi - \theta_i)~~~~~~~\text{for }i = 1,2~~~~~~~~(1)$$
If we further assume the statistical averages of phases is zero (i.e. $\psi = 0$), the governing equations then become $$\frac{d \theta_i}{dt} = \omega_i -Kr \sin(\theta_i).~~~~~~~~~~~~~~~~~~(2)$$

Questions:

- Am I correct in stating that $\psi=0$ assumption is quite limiting in that as $\psi$ is also a function of time $t$, hence even if $\psi(0) = 0$ it does not necessarily imply that $\psi(t) = 0$?

- What would be a feasible recommended approximation method to attempt to solve the type of ODE in (1)?

Thanks for any assistance.

 

[IsometricPion]

Yes, setting $\psi(t)=0$ without any other changes seems pretty limiting to me. However, I think it might be more reasonable to transform all of the angular variables (i.e., the $\theta_i$'s and $\psi$) in such a way that they all start at zero, and such that the sum/average of the $\omega_i$'s is zero. Since $\psi$ is sort of an average of the angular variables, $\psi(t)=0$ might not be such a bad approximation after such a transformation.

I think that looking for stationary solutions and then expanding about them in configuration or phase space is typically a good way to gain some information on the behavior of a non-linear system of ODEs.

 

[Danny Boy]

@IsometricPion Thanks for your response and recommendations. I think the $\psi(t) = 0$ follows from transforming into a rotating reference frame where it rotates such that the average phase is always zero. Hence I think part of my misunderstanding was that I was still thinking in terms of the non-rotating inertial reference frame. It is interesting how little the equations change after the transformation into a rotating reference frame (it goes from $\frac{d \theta_i}{dt} = \omega_i + Kr\sin(\psi - \theta_i)$ to $\frac{d \theta_i}{dt} = \omega_i -Kr \sin(\theta_i)$).