File:LogisticTopMixing1-6.gif

English: Six iterations of a set of states $[x,y]$ passed through the logistic map. (a) the blue plot (legend 1) shows the first iterate (initial condition), which essentially forms a circle. Plots in (a), (b) and (c), show the first to the sixth iteration of the circular initial conditions. It can be seen that \emph{mixing} occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogenous and irreversible. The logistic map has equation $x_{k+1} = 4 x_k (1 - x_k )$. In order to expand the state-space of the logistic map into two dimensions, a second state, $y$, was created as $y_{k+1} = x_k + y_k $, if $x_k + y_k <1$ and $y_{k+1} = x_k + y_k -1$ otherwise.