Yoshiki Kuramoto (1975) showed that N coupled phase oscillators with distributed natural frequencies ωᵢ ~ N(ω̄, σ²) undergo a phase transition at K_c = 2/(πg(ω̄)). Below K_c, incoherence. Above K_c, a synchronized cluster forms (order parameter r → 1).
Male Pteroptyx malaccae fireflies in Southeast Asia synchronize their flashes to femtosecond precision across riverside trees, creating spectacular displays. Each firefly adjusts its rhythm by perceiving neighbors' flashes — a real-world Kuramoto oscillator network.