The circle map θₙ₊₁ = θₙ + Ω − (K/2π)sin(2πθₙ) exhibits frequency locking in Arnold tongue regions in (Ω, K) parameter space. At K=1 the tongues touch and form a devil's staircase. Above K=1, chaos appears. Click the parameter map to explore individual orbits.
θₙ₊₁ = θₙ + Ω − K/2π·sin(2πθₙ)
Winding number W = lim(θₙ−θ₀)/n
W=p/q → q:p resonance (tongue)
K<1: quasiperiodic sea + locked tongues
K=1: devil's staircase on critical line
K>1: chaos (overlapping tongues)