Chaos Lab

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.

Devil's staircase: rotation number ρ vs Ω at K = 1. Plateaux are Arnold tongues (mode locking).

Equations

θ_{n+1} = θ_n + Ω − (K/2π) sin(2π θ_n) mod 1

At a glance

ParametersΩ ∈ [0, 1], K ≥ 0
Chaotic forK > 1 (above the critical curve, tongues overlap)
HistoryArnold (1965); central to Kolmogorov-Arnold-Moser (KAM) studies.

Try it

Open the interactive playground at /tools/cobweb.

See also

Quick quiz

Test yourself on circle

8 multiple-choice questions. Pick an answer for each, then submit to see explanations.

  1. Q1.Arnold tongues in the circle map appear because:

  2. Q2.Below the critical K = 1, the circle map is:

  3. Q3.The standard form of the circle map is:

  4. Q4.The rotation number ρ is:

  5. Q5.The devil's staircase is the plot of:

  6. Q6.The total measure of the rational plateaux on the critical curve K = 1 is:

  7. Q7.Arnold tongues are named after:

  8. Q8.Real-world phenomena modelled by the circle map include:

0 of 8 answered