The Kuramoto model is the canonical mean-field example of collective synchronization: N coupled phase oscillators with natural frequencies ω_i interact through an all-to-all sine coupling of strength K. Writing the complex order parameterZ = r e^{iψ} = N⁻¹ Σ_j e^{iθ_j}, each phase obeys θ̇_i = ω_i + K r sin(ψ − θ_i)—the same law as the full sum, but computed in O(N) time. This page integrates the ODEs with RK4, draws each oscillator as a colored dot on the unit circle (hue encodes ω_i), shows the mean-field pointer Z in gold, and plots r(t) as a simple coherence diagnostic. You can sweep K and the frequency spread to watch partial and nearly complete locking on a finite system; quantitative critical coupling depends on the frequency distribution and the N → ∞ limit.
Who it's for: Students in nonlinear dynamics, statistical physics, or networks who want a clean bridge between coupled oscillators and order-parameter thinking.
Key terms
Kuramoto model
Synchronization
Order parameter
Mean-field coupling
RK4
Phase oscillator
How it works
The Kuramoto model couples N phase oscillators with natural frequencies ω_i and all-to-all interaction strength K. It is the standard playground for synchronization: beyond a coupling threshold, a macroscopic fraction of oscillators locks to a common rhythm, tracked by the complex order parameter Z = N⁻¹ Σ e^{iθ_i}.
Key equations
θ̇_i = ω_i + (K/N) Σ_j sin(θ_j − θ_i). With Z = r e^{iψ} = N⁻¹ Σ_j e^{iθ_j}, the sum becomes K r sin(ψ − θ_i), so each oscillator feels the mean field of the population.
Frequently asked questions
Why is the dynamics O(N) if the textbook has a sum over j?
Because Σ_j sin(θ_j − θ_i) = Im(e^{-iθ_i} Σ_j e^{iθ_j}) = N Im(Z e^{-iθ_i}) = N r sin(ψ − θ_i). The entire population enters only through the two numbers r and ψ, so each timestep is linear in N.
Is r = 1 perfect global synchronization?
Usually r ≈ 1 means almost all oscillators are locked into a common rotating cluster; exact r = 1 requires identical phases modulo 2π, which finite spread in ω_i typically prevents unless coupling is very strong relative to disorder.
What does the Lorentzian K_c = 2γ note mean?
For the Kuramoto model in the infinite-N limit with Cauchy-distributed ω of half-width γ, the incoherent state loses stability at K_c = 2γ. Real finite boxes of uniform frequencies behave similarly in spirit but not identically; treat it as intuition, not a fitted critical point here.