\sum_{j=1}^N e^{i \theta_j(t)}.$$ Then the sum in reduces to $(K z e^{-i \theta_i})$, an identity which will prove useful later.

The order parameter also has a nice physical interpretation. Its amplitude $|z|$ quantifies the phase coherence of the population: an incoherent state has $z = 0$; a perfectly coherent state has $|z| = 1$. Furthermore, the argument of $z$ can be interpreted as the average phase of all the oscillators. So in a sense, the single complex number $z(t)$ serves as a proxy for the state of the population as a whole.

Sakaguchi \[1988\] was the first to study the periodically forced Kuramoto model. He derived a self-consistent equation for steady-state values of $|z|$, under the assumption that $z(t)$ was entrained by the external force (meaning that $z(t)$ appeared motionless in the rotating frame). In numerical simulations of Eq. , however, Sakaguchi found that this state of “forced entrainment” was not always attained. For some values of the parameters, the system could settle instead into a state of “mutual entrainment.” In this case a macroscopic fraction of the system self-synchronized at a different frequency from that of the drive, indicating that this sub-population had broken away and established its own 