File:Markus-Ljapunow Exponent 7.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:

${\displaystyle x_{n+1}={\begin{cases}b\sin ^{2}(x_{n}+r)+\alpha r^{k}&{\text{if }}[x_{n}]mod(\pi )<{\frac {\pi }{2}}\\b\sin ^{2}(x_{n}+r)+\beta r^{k}&{\text{else }}\end{cases}}}$
${\displaystyle r:A^{3}B^{3}}$ ${\displaystyle A^{3}B^{3}...,b=2.3,\beta =0.2}$

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.