Circle map parameter space: rational winding numbers form wedge-shaped "tongues" — regions of frequency locking between coupled oscillators.
Click anywhere — Ω (x-axis, 0→1) vs K (y-axis, 0→1). Color = winding number.
Circle map orbit for selected (Ω, K)
Click the parameter space to probe a point.
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Circle map: θ_{n+1} = θ_n + Ω + (K/2π)·sin(2πθ_n). The winding number W = lim_{n→∞} θ_n/n measures the average rotation rate. For rational W = p/q, the map is mode-locked (frequency synchronized). Arnold tongues are the regions in (Ω, K) space with rational winding numbers — they widen as K increases. At K=1 (critical line), tongues form a devil's staircase with measure 1. For K>1, the map is non-invertible and chaos emerges.