Arnold Tongues

Mode-Locking in Driven Circle Maps

Circle map:
θ_{n+1} = θ_n + Ω − K/2π·sin(2πθ_n)

Winding number:
W = lim_{n→∞} θ_n/n

Arnold tongue p:q:
W = p/q (mode lock)

Arnold tongue map: scan (Ω,K) space. Colored regions = mode-locked (rational winding number). Click to set point.

Arnold tongues are wedge-shaped regions in (frequency, coupling) parameter space where a driven oscillator locks to a rational ratio p:q with its driver. At K=1 (critical), tongues cover the entire axis (Devil's staircase). They appear in cardiac arrhythmias, Josephson junctions, and laser synchronization.