Arnold Tongues

The initial view uses 0 ≤ K ≤ 1, where the map is monotone and the rotation number has a common limit. Larger K is available in the main bounds; use it to explore finite, possibly initial-orbit-dependent winding rates.

The sine-circle-map lift used here is F(x) = x + Ω + K/(2π) sin(2πx), with a plus sign. The paper below uses a minus sign; the two conventions are conjugate by a phase shift of one half. The display wraps phase to one turn while accumulating integer winding, so the sine is evaluated at small arguments. The winding rate itself is not reduced modulo one.

After discarding the chosen burn-in, the calculation measures ρ = [F^burn+iter(x₀) − F^burn(x₀)] / iter. For K ≤ 1, the theoretical finite-iteration error is less than 1/iter, excluding floating-point roundoff. For K > 1, the displayed value describes this initial orbit over the chosen finite time; it can depend on x₀. A unique rotation number or locking is not guaranteed.

See Alsedà and Borros-Cullell (2020), Definition 2.2, Proposition 2.3, Algorithm 1/Lemma 2.8 and Definition 4.6/section 4.2. The nearest reduced fraction uses denominators up to eight and prefers the smaller denominator in a tie. Its absolute deviation is numerical proximity, not proof of synchronization. Dashed slice guides at 0, 1/3, 1/2, 2/3 and 1 provide reference values when visible.

The grid includes both parameter endpoints. K increases upward; Ω increases rightward. Colors use the actual finite minimum and maximum across the computed grid, so colors from different maps need not represent the same rate. Clicking the map or using Probe selects the nearest cached grid point. The slice reads the selected K row without recalculating the map. Narrow plateaus can fall between grid points; longer measurement and finer grids can change the visible detail.

Render starts a background worker. Cancel or any map-setting edit terminates it and removes old output. The PNG contains the complete current heatmap, axes, color scale and selection at the current display size and theme.