Phase-Locking Strength

The Phase-Locking Strength is the order parameter that measures how synchronized two oscillatory modes are, approaching 1 when their phases are fully aligned. It quantifies the transition from random phase differences to synchronized behavior. When the phase difference between two oscillators wanders freely, the averaged complex phase factor cancels toward zero; when the oscillators are locked, it stays fixed and the magnitude approaches unity. The book pairs it with the Entrainment Condition, which says when locking can occur, and with the Kuramoto-model Phase Coupling Equation dφ_i/dt = ω_i + sum_j K_ij sin(φ_j − φ_i), which describes how coupling drives oscillators toward the synchronized state, with critical coupling K⨶ = 2*Δω⥣/(π*N) for N oscillators. The formal statement is R⊠ = |⟨e^i(φ_i − φ_j)⟩|, where R⊠ is the Phase-Locking Strength, | | the magnitude, ⟨ ⟩ the ensemble average and φ_i, φ_j the phases of the ith and jth oscillators. All quantities are dimensionless [∅]; the book's check is consistent. It connects to Phase-Locking, the Entrainment Condition, Phase Coupling Equation, Critical Coupling and Synchronization Dynamics.