Critical Coupling

Critical Coupling is the threshold coupling strength at which a set of coupled oscillators passes into synchronization. The book treats substrate oscillators with the Kuramoto model, in which each oscillator's phase is pulled toward the others through the Phase Coupling Equation. Below the critical coupling the oscillators keep their own frequencies; above it they lock together, and above the critical value the synchronization measure r grows from 0, reaching 1 as K → ∞, while the phase coherence |⟨e^i(φ_j − φ_k)⟩| ≈ r² rises toward 1 and the coherence factor of the combined form-emergence criterion Θ reaches 1 once r² ≥ C⨶, so critical coupling marks where collective behavior sets in. The book lists critical coupling thresholds among its predictions, detectable through synchronization measurements in oscillator networks. The formal statement is K⨶ = 2*Δω⥣/(π*N), where K⨶ is the Critical Coupling for N coupled oscillators, the coupling per pair at the onset of synchronization, Δω⥣ the frequency spread (the maximum frequency difference), over which evenly spread natural frequencies have peak density g(0) = 1/Δω⥣, and N the number of oscillators. Dimensions from the book: K⨶ and Δω⥣ [rad/s], π and N [∅]. It follows from the Kuramoto onset K_c = 2/(π*g(0)), which the Chapter 7.6 predictions state for coupled oscillator arrays, with K_c = N × K⨶. It connects to the Phase Coupling Equation, Phase-Locking Strength, Synchronization Dynamics, Combined Threshold Function, Phase Coherence Condition and Thresholds of Recursion.