About the Kuramoto Model
The Kuramoto model (1975) describes the synchronization of coupled oscillators, a phenomenon observed across nature: fireflies flashing in unison, neurons firing together, metronomes synchronizing on a shared platform, and superconducting Josephson junctions. Yoshiki Kuramoto showed that even weak coupling can drive a system of diverse oscillators toward collective synchronization through a sharp phase transition.
Each oscillator i has phase theta_i evolving as: d(theta_i)/dt = omega_i + (K/N) * SUM_j sin(theta_j - theta_i), where omega_i is the natural frequency drawn from a Gaussian distribution with width Delta, K is the coupling strength, and N is the number of oscillators. The order parameter r * exp(i*psi) = (1/N) * SUM_j exp(i*theta_j) measures global coherence: r=0 means incoherent, r=1 means fully synchronized. The critical coupling for a Gaussian distribution is approximately K_c = 2*Delta*sqrt(2/pi).
Below the critical coupling K_c, oscillators run independently and r remains near zero. At K_c, a phase transition occurs: a cluster of oscillators locks together and r rises sharply. Above K_c, more oscillators join the synchronized cluster and r approaches 1. This collective emergence is analogous to phase transitions in statistical mechanics, making the Kuramoto model a paradigm for understanding synchronization in complex systems.
Use the coupling strength slider K to control synchronization. Start with K=0 (disorder) and gradually increase K to observe the phase transition. The phase circle shows each oscillator as a colored dot (blue=slow, red=fast). When synchronized, dots cluster dramatically. The order parameter time series r(t) tracks global coherence in real time. Try the presets to jump between regimes. Adjust the frequency spread and oscillator count to explore how diversity and population size affect synchronization thresholds.