Poincaré Recurrences


The purpose of this page is to distinguish the role of chaotic and periodic orbits in visual illustrations of Poincaré recurrences and to estimate the dependence of the recurrence time in each case.


A beautiful illustration of the Poincaré recurrence theorem was provided by Crutchfield et al. It consists in dynamically evolving (the piexels of) a picture according to a deterministic procedure that corresponds to a Hamiltonian system. A recurrence occurs when the origin image is again obtained. Usually a chaotic system with strong mixing properties is used to illustrate this effect, such as the cat map defined as

$$y_{n+1}=x_n+2y_n\text{ mod }1$$

$$x_{n+1}=x_n+ y_n\text{ mod }1$$

This illustration was repeated in different contexts and the extremely small value of the recurrence time is usually a matter of surprise. We describe below the procedure in detail and we discuss the dependence of the typical recurrence time on the resolution of the picture and on the choice of initial conditions.

Iteration method

A color is attributed to each of the N x N pixels of the picture, defined in a square grid. The color of each pixel (i,j) at iteration time n' is obtained by the following procedure:

  1. One initial condition $(x_0^i,y_0^i)$ is chosen inside each pixel (two different possibilities are mentioned below);
  2. The inverse cat map is iterated n' times: $(x_0^i,y_0^i)\Rightarrow(x_{-n'}^i,y_{-n'}^i)$;
  3. The color of the pixel of the original image to which $(x_{-n'}^i,y_{-n'}^i)$ belongs is atributted to the pixel at position (i,j) at time n'.

Recurrence time and initial Conditions

Consider the following two kinds of initial conditions inside each pixel:

  • A natural choice is to take the point at left/low edge of the pixel (or, similarly, at the center of the pixel): $(x_0^i,y_0^i)=(i/N,j/N)$ . In this case trajectories are restricted to rational numbers and are periodic orbits of the cat map. Recurrence will occur after T iterations, where T is the smallest common period of all N x N periodic orbits. This can be related to the divisibility properties of Fibonnacci numbers, and the period T is given by the smallest positive integer such that $u_{2T}=0\text{ mod }N$ and $u_{2T-1}=1\text{ mod }N$  where $u_{i}$.is the i-th Fibonacci number Dyson and Falk. In this paper it's shown that
    $$\frac{\log N}{\gamma}<T<3T$$
    where $\gamma = \left(\sqrt{5}+1\right)/2$ is the golden ratio. In the left picture below we used N = 256 and found T = 192.
  • From the point of view of the cat map defined in R x R, the initial conditions above are non-typical since with probability one initial conditions are on irrational numbers and trajectories are chaotic. Our second choice consist of typical initial conditions, which are taken randomly inside each pixel (Notice that the dynamics is deterministic and randomness is used only on the choice of initial conditions). This was used in the right picture below. It resembles the left picture for small times, but no recurrence is observerd at time comparable to N. The reason is that the mean recurrence time is given by $\langle T\rangle=N^{2N^2}$

For N = 256 used below we find $\langle T\rangle=6.74\times 10^{315652}$. Since computers work only on rational numbers finally even the second choice of initial conditions will lead to a precise periodicity. The period in this case can be estimated through the formulas of the first choice of initial conditions with an effective $N\approx10^{16}$.


The comparison between the scaling of the recurrence time with N performed above show that Poincaré recurrences in chaotic systems occur only after a very long time. This is in agreement with the usual assumptions of statistical physics when describing thermodynamical systems. Short time recurrences as those observed by Crutchfield et al. (as well as by other authors) are related to the particular non-generic choice of initial conditions in a chaotic system.


Chaos Crutchfield et al., Scientific American, DEC 1986.

Period of a discrete cat mapping , Freeman J. Dyson and Harold Falk, American Mathematical Monthly 99 603-614 (1992). Full text, permission needed.