Discrete map · 1D · dim 1
Circle map
Arnold's standard form for the iterates of a circle homeomorphism with a sinusoidal nonlinearity. Famous for Arnold tongues (resonance regions) and devil-staircase rotation number.
Equations
θ_{n+1} = θ_n + Ω − (K/2π) sin(2π θ_n) mod 1At a glance
| Parameters | Ω ∈ [0, 1], K ≥ 0 |
|---|---|
| Chaotic for | K > 1 (above the critical curve, tongues overlap) |
| History | Arnold (1965); central to Kolmogorov-Arnold-Moser (KAM) studies. |
Try it
Open the interactive playground at /tools/cobweb.