Kuramoto Model
When weakly connected oscillators start moving as one.
The Kuramoto model shows how independent phases synchronize through coupling. It is a compact way to study consensus, neural rhythm, power-grid phase locking and market co-movement.
R(t)order parameter, from disorder to lock-in
Kcoupling strength between oscillators
Nphase agents with different natural frequencies
Interactive graphphase field
Model intuition
Synchronization is a phase transition.
Each oscillator has its own natural frequency. When coupling is low, phases drift apart. Past a critical region, the group forms a coherent rotating cluster and the order parameter rises toward one.
R close to 0 means phases are spread around the circle.
R close to 1 means most oscillators are phase locked.
The critical coupling grows when natural frequencies are more dispersed.
Noise can break local lock-in even when K is high.
Kuramoto / Python analysissimulation core
import numpy as np
def kuramoto(theta, omega, K, dt, steps):
"""Mean-field Kuramoto simulation."""
hist = np.zeros((steps, len(theta)))
order = np.zeros(steps)
for t in range(steps):
z = np.exp(1j * theta).mean()
R, psi = np.abs(z), np.angle(z)
theta += dt * (omega + K * R * np.sin(psi - theta))
hist[t] = np.mod(theta, 2 * np.pi)
order[t] = R
return hist, order
# Analysis:
# scan K, compute mean R after burn-in, find the lock-in threshold.