Chaos Lab

Discrete map · 2D · dim 2

Arnold cat map

Toral automorphism with eigenvalues (3 ± √5)/2. Exponential stretching along the unstable eigendirection, then folding by mod 1. Mixes phase space exponentially; popular in image scrambling.

Three iterations of the Arnold cat map on a 32×32 quadrant pattern: mixing in action.

Equations

(x, y) ↦ ((2x + y) mod 1, (x + y) mod 1)

At a glance

Parameters(no free parameter)
Chaotic foralways
Lyapunov exponentλ = ln((3 + √5)/2) ≈ 0.962
HistoryVladimir Arnold's 1960s illustration of hyperbolic mixing.

Try it

Open the interactive playground at /tools/arnold-cat.

See also

Quick quiz

Test yourself on arnold-cat

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

  1. Q1.The Arnold cat map's matrix [[1,1],[1,2]] has eigenvalues whose larger one is:

  2. Q2.On a finite N×N grid the cat map has:

  3. Q3.Geometrically each step does what?

  4. Q4.The cat map is famous for:

  5. Q5.Arnold demonstrated the map by mixing:

  6. Q6.The map is an automorphism of:

  7. Q7.Cat-map image scrambling has the weakness that:

  8. Q8.K-S entropy of the cat map equals:

0 of 8 answered