Entrainment Condition

The Entrainment Condition is the criterion for two oscillators to phase-lock: their frequency difference must fall below a critical bandwidth. It marks the step from independent oscillations to a collective synchronized state in the substrate. When the frequencies of two oscillatory modes are close enough, Phase-Locking occurs, and its degree is measured by the Phase-Locking Strength R⊠ = |⟨e^i(φ_i − φ_j)⟩|. The coupled dynamics follow the Kuramoto model through the Phase Coupling Equation dφ_i/dt = ω_i + sum_j K_ij sin(φ_j − φ_i). The critical bandwidth is derived from substrate damping: Δω⨶ = 2πγ⤓ where γ⤓ = ω_0/Q▱, and for a coherent substrate with Q▱ ≈ 10⁶, Δω⨶ ≈ 2πω_0/10⁶. The formal statement is |ω_i − ω_j| < Δω⨶, where ω_i and ω_j are the angular frequencies of oscillators i and j and Δω⨶ is the critical bandwidth. Dimensions as the book gives them: ω_i, ω_j, Δω⨶ [rad/s]; i, j [∅]. The book's check is consistent. It connects to Phase-Locking, Phase-Locking Strength, the Phase Coupling Equation, Synchronization Dynamics and Critical Coupling.