English: The circle map is a chaotic map showing a number of interesting chaotic behaviors.

This figure shows the average Poincaré recurrence time for the iterated circle map modulo 1. The circle map is given by iterating on the map

${\displaystyle {\theta }_{n+1}={\theta }_n+\mathrm{\Omega }-\frac{K}{2\pi }\sin (2\pi {\theta }_n)}$

Given a fixed value of K, Ω, ${\displaystyle {\theta }_n}$ and ε, the recurrence time N is given by the smallest integer N such that

${\displaystyle !({\theta }_{n+N}-{\theta }_n)!<ϵ}$

is satisfied. This figure shows the average values of N for a variety of random starting values ${\displaystyle {\theta }_n}$, and an ε = 0.0001. Black indicates values of N of less than about 10, blue values of about 50, green shows values of about 140 and red shows values greater than 250.

The frequency Ω runs from 0 to 1 along the horizontal axis, and the coupling constant K runs from 0 at the bottom to 4π at the top of the image. The upper limit of the red area at the bottom occurs at K=1.