[1] Yoshiki Kuramoto, "Chemical Oscillations, Waves, and Turbulence" (Dover, New York, 2003).

A coupled limit-cycle system is reduced to a coupled phase-oscillator system by the phase reduction, and as a representative model the Kuramoto model has been extensively studied.

weak coupling ← → strong coupling

In the Kuramoto model, the phase oscillators evolve according to the ordinary differential equations:

The oscillators have the natural frequencies, which are in some cases assumed to follow the Lorentz distribution:

The above figures show that the system undergoes the synchronization transition when the coupling strength K is increased. That is, with small K the oscillators move incoherently while with strong coupling they cooperate and tend to have similar phases.