Circle Map — Arnold Tongues

Explore the circle map θₙ₊₁ = θₙ + Ω − (K/2π) sin(2πθₙ) mod 1. Visualize orbits and mode‑locking regions (Arnold tongues) in (Ω, K).

Top notes — mode locking, devil’s staircase, rotation number.

Side notes — tongues widen with K.

Ω: K: Iter: Speed:

Left: circle with points; Right: θ vs n (unwrapped).

Ω min: Ω max: K max: Grid: Iter:

Side notes — rational p/q regions lock.

Bottom notes — add references or derivation.