I am currently trying to solve the kuramato model numerically using the runge-kutta 4th order method. I consider the kuramato model:
$\ddot{\theta_i} = P_i - \alpha\omega_i + K \underset{i \neq j}{\sum}sin(\theta_i - \theta_j)$
I then split the differential equation into two first orders like:
$$\dot{\omega_i} = P_i - \alpha\omega_i + K \underset{i \neq j}{\sum}sin(\theta_i - \theta_j)$$ $$\dot{\theta_i} = \omega_i$$
And from here I am unsure how to proceed. I reckon that I need to define $$f(\omega_i, \theta_i) = P_i - \alpha\omega_i + K \underset{i \neq j}{\sum}sin(\theta_i - \theta_j)$$ And: $$g(\omega_i, \theta_i) = \omega_i$$ And use the RK4 approximation scheme on the two functions, but I can't quite figure out how to do this on a coupled system. Any help would be deeply appreciated.
Cheers,
Fabric
I think the problem is conceptual. The important thing to realize is that you have produced a single system consisting of, say, $2n$ coupled ordinary differential equations, rather than two coupled systems of $n$ equations each.
You have a vector equation of the form \begin{equation} \ddot{\theta} = f(\theta, \dot{\theta}) \end{equation} and by making the identification $\omega = \dot{\theta}$, you arrive at the system \begin{equation} \frac{d}{dt} \begin{pmatrix} \theta \\ \omega \end{pmatrix} = \begin{pmatrix} \omega \\ f(\theta, \omega) \end{pmatrix} =: F( \theta, \omega) \end{equation} If you define a new state vector, say, $x = (\theta,\omega)^T \in \mathbb{R}^{2n}$, then your new system can be written in the familiar form \begin{equation} \dot{x} = F(x), \quad F : \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n} \end{equation} and you can now apply a Runge-Kutta method.