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Kuramoto Oscillators — Synchronization

Coupled oscillators with random frequencies spontaneously synchronize above a critical coupling strength

Phase circle (dot = oscillator phase)
Order parameter r(t)
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Current order parameter r = |⟨e^{iθ}⟩|
The Kuramoto model: dθᵢ/dt = ωᵢ + (K/N)Σⱼ sin(θⱼ − θᵢ). Natural frequencies ωᵢ are drawn from a Lorentzian distribution with half-width γ = 1. The critical coupling is Kc = 2γ/πg(0) = 2γ·π/(π·π) = 2γ = 2. Below Kc: r → 0 (incoherence). Above Kc: r → √(1 − Kc/K) (partial sync).