Kuramoto Model for Phase Locking of Coupled Oscillators
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This Demonstration shows the collective synchronization of coupled oscillators according to the Kuramoto model , which is a modified version of the Winfree model  for the population of biological oscillators.[more]
Many systems in biology, physics and engineering show the phenomenon of collective synchronization, such as heart pacemaker cells, flashing fireflies, crickets chirping in unison, laser arrays and microwave oscillators.
The Kuramoto model is defined through the following set of time-dependent coupled differential equations
where is the time-dependent phase of the oscillator and is its natural frequency chosen from a probability density distribution (in this Demonstration, a Gaussian). The starting values of the are chosen from a uniform density distribution. The parameter is the coupling strength between each pair of oscillators. While is below a certain critical value , the oscillators behave completely incoherently: you can see the points giving the values spreading at random around the unit circle. When exceeds the critical value, the oscillator population locks in a partially synchronized group running around the unit circle at the same frequency. By further increasing , more and more oscillators are added into the synchronized group locked in phase.
This phase locking can thus be interpreted as a temporal phase transition, where the -dependent order parameter gives the amount of synchronization in the oscillator population as and time tend to infinity.
The degree of synchronization is given by the order parameter , which tends to 1 as for increasing and time, as you can check by running this Demonstration.
To optimize graphical performance, the solutions for the set of are precomputed in the initialization code for a given set of coupling parameters . In order to further explore the dependence of synchronization onset on model parameters (number of oscillators, coupling parameter, initial conditions), change the corresponding variables in the initialization code.[less]
Contributed by: Jessica Alfonsi (September 2016)
Open content licensed under CC BY-NC-SA
Snapshot 1: weak coupling parameter , incoherent behavior
Snapshot 2: critical coupling parameter , onset of partial synchronization, but incoherent behavior is still present
Snapshot 3: strong coupling parameter , synchronization reached after a short transient, order parameter close to 1
 Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Berlin: Springer-Verlag, 1984.
 A. T. Winfree, The Geometry of Biological Time, New York: Springer-Verlag, 1980.
 S. H. Strogatz, "From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators," Physica D: Nonlinear Phenomena, 143(1–4), 2000 pp. 1–20. doi:10.1016/s0167-2789(00)00094-4.
 J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, "The Kuramoto Model: A Simple Paradigm for Synchronization Phenomena," Reviews of Modern Physics, 77(1), 2005 pp. 137–185. doi:10.1103/RevModPhys.77.137.