Coupled Oscillators — Synchronization Onset

Frequency entrainment, Arnold tongue, phase portrait in (Δω, K) space
Live Oscillators
Arnold Tongue
Phase Difference
ω₁=3.0 | ω₂=3.5 | Δω=0.5 | K=1.0 | Phase diff=° | Synced:
ω₁: 3.0
ω₂: 3.5
K (coupling): 1.0
Two coupled phase oscillators: dθᵢ/dt = ωᵢ + K·sin(θⱼ−θᵢ). Synchronization (phase locking) occurs when the coupling K exceeds the frequency detuning: K > |Δω|/2. In the (Δω, K) plane, the region of synchronization is the Arnold tongue — a wedge-shaped region. Inside the tongue, the phase difference locks to φ* = arcsin(Δω/2K). Outside, phases slip continuously. This is the 1:1 locking; higher-order tongues exist at rational frequency ratios.