randomly distributed with a density $g(\omega)$, assumed unimodal and symmetric about its mean value $\omega_0$.

This system is capable of rich dynamics because of its interplay among randomness, coupling, and forcing. The randomness comes from the variance of the natural frequencies. This effect tends to desynchronize the oscillators and scatter their phases. The coupling, on the other hand, tends to align the oscillators to the same phase, although it does not favor any particular frequency for the collective oscillation. In contrast, the forcing does favor a specific frequency, namely that of the external drive. Depending on the relative magnitudes of these competing effects, we expect to see various kinds of cooperative behavior and transitions between them.

Before continuing, it proves useful to simplify the governing equations in two ways. First, if we view the dynamics in a frame co-rotating with the drive, the explicit time dependence in disappears. To achieve this, let $$\label{rotatingframe}
\theta_i = \vartheta_i - \sigma  t.$$ Then yields $$\label{forcedKuramoto}
\frac{d \theta_i}{dt} = (\omega_i - \sigma) +\frac{K}{N}\sum_{j=1}^N \sin(\theta_j-\theta_i) - F \sin \theta_i,$$ for $i = 1, \ldots, N$. Second, as Kuramoto originally pointed out, it is helpful to introduce a complex order parameter $z$, given by $$\label{complexorderparameter}
z(t)=\frac{1}{N} 