It is often useful to consider the 'collective rhythm' of the system, or the 'centre point' of the phases of the oscillators as depicted on the unit circle in the complex plane. This value, the complex order parameter, is defined as [8];
On the right, the top plot shows a number of unsynchronised oscillators spread over the circle, which gives an order parameter with r small (~0.3) and Φ at approximately π. The bottom plot shows a number of oscillators that are closely grouped with an order parameter with r very close to 1 and &Phi located at the centre of the cluster.
Where N and θi are defined as before, Φ is the argument of the order parameter and r is the modulus of the order parameter.When plotting the oscillators around a unit circle in the complex plane, the order parameter is often depicted as an arrow from the origin to the point (r, Φ) or as a circle (radius r) with a marker at the point (r, Φ) [8]. Small r implies the system is incoherent without any oscillators in synchrony and r=1 implies total synchronisation.
On the right, the top plot shows a number of unsynchronised oscillators spread over the circle, which gives an order parameter with r small (~0.3) and Φ at approximately π. The bottom plot shows a number of oscillators that are closely grouped with an order parameter with r very close to 1 and &Phi located at the centre of the cluster.
