The circle map models a driven oscillator. Mode locking occurs when the winding number locks to a rational p/q: the system cycles in q periods of the driver for every p cycles of itself.
At K=0, the winding number equals Ω (quasiperiodic for irrational Ω). At K>0, Arnold tongues open around each rational Ω=p/q. The most prominent tongue is W=1/2 (period doubling). At K=1 (critical), the tongues cover a Cantor set — devil's staircase. Above K=1, the map is non-invertible and chaos emerges.
Applications: cardiac arrhythmias, Josephson junctions, mode-locked lasers, planetary orbital resonances.