Phase Oscillators (under construction)

We can observe many rhythmic systems in nature. Such systems are described by dynamical systems mathematically, although most of them have often highly many degrees of freedom. The phase reduction technique [1] reduces a limit-cycle system even with external forces to a phase oscillator system with only one phase variable, which helps our mathematical and numerical analysis.
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.

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