Winding ρ: —
State: —
Ω: —
About: The circle map θ_{n+1} = θ_n + Ω + (K/2π)sin(2πθ_n) models periodically forced oscillators. At K=0 the motion is rigid rotation; at K=1 the map becomes critical. Arnold tongues are regions of parameter space (Ω, K) where the winding number ρ = lim(θ_n/n) locks to a rational value p/q — the system synchronizes. Between tongues lies a Cantor set of quasiperiodic (irrational ρ) motion. The devil's staircase structure of ρ(Ω) at K=1 is a canonical example of fractal dimension.