- #1
Danny Boy
- 49
- 3
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.
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.