Phase-Locking

Phase-Locking is the synchronization of oscillators that occurs when their frequency differences fall below a critical bandwidth, after which they oscillate with a fixed phase relation and behave collectively. In Chapter 7 it is the result of the Entrainment Condition: oscillators whose frequencies are close enough lock together, the degree of locking is measured by the Phase-Locking Strength, and the coupled dynamics follow the Kuramoto-model Phase Coupling Equation. The current definition sets the critical bandwidth by substrate damping, Δω⨶ = 2πγ⤓. In Chapter 9 the book carries the idea into navigation through the substrate: among its testable predictions, superconducting circuits held in coherent phase control at cryogenic temperatures should demonstrate controlled navigation by substrate phase manipulation, alongside gravitational wave phase modulation and phase attractor convergence. The condition is |ω_i − ω_j| < Δω⨶ = 2πγ⤓, where ω_i and ω_j are the angular frequencies of two oscillators, Δω⨶ the critical bandwidth and γ⤓ the substrate damping rate. It connects to the Entrainment Condition, Phase-Locking Strength, Phase Coupling Equation, Phase-Locked Domains and Synchronization Dynamics.