File:Markus-Ljapunow Exponent 2.jpg

English: The picture was constructed with an iterative equation of the form ${\displaystyle x_{n+1}=f(r,x_{n}),n=1,2,3...,}$ where ${\displaystyle r}$ is a parameter that varies periodically between two values A and B. An exponent on A or B indicates a consecutive repetion of A or B, as given by the exponent. Example: A with exponent 4 means to be an r-sequence AAAA. The parameter A corresponds to the x-axis of the picture and B to the y-axis. ${\displaystyle f}$ is given by te following formula:

${\displaystyle x_{n+1}=rx_{n}(1-x_{n})}$
${\displaystyle A}$ ${\displaystyle versus}$ ${\displaystyle B,}$ ${\displaystyle r:B^{7}A^{2}B^{9}(BA)^{9}A^{7}B^{2}A^{7}...}$

The exponential growth of a small perturbation is determined for each point in the picture, i.e. for each pair (A,B). The value of the resulting exponent (Markus-Lyapunov exponent) is show with a color.