Arnold tongues are regions of parameter space where a driven oscillator locks to rational frequency ratios p/q.
Circle map: θ_{n+1} = θ_n + Ω − K/2π sin(2πθ_n)
At K=0: quasiperiodic (irrational) or periodic (rational). As K→1, tongues widen. At K=1 (critical): devil's staircase — a Cantor function. Above K=1: chaos invades.
Circle map: θ_{n+1} = θ_n + Ω − K/2π sin(2πθ_n)
At K=0: quasiperiodic (irrational) or periodic (rational). As K→1, tongues widen. At K=1 (critical): devil's staircase — a Cantor function. Above K=1: chaos invades.