# How to use Lyapunov Exponent to evaluate the performance of a chaotic map

Any simple Mathematica code for Lyapunov exponent. Considering the figure below which is a chaotic map using Sine Map, Tent Map and Henon Map . How to use Lyapunov Exponent to evaluate the performance of this chaotic map. The equations are as follows enter image description here. The STH (Sine Map, Tent Map and Henon Map) that I did is as

sth0 = 0.929688;
sth1 = 0.488281;
r = 0.5;
u = RandomReal[{0.87, 1}];
β = RandomReal[{0, 1}];
x0 = RandomReal[{0, 1}];
sth[0] = Abs[u - 10*Sin[Pi*sth1]^2 + β*r*Abs[1 - 2*sth0]];

sth[m_] := sth[m] = Abs[u - 10*Sin[Pi*sth[m - 1]]^2 +
β*r*Abs[1 - 2*sth[m - 1]]];
Table[sth[m], {m, 0, Ceiling[512*512*(3/4)]}];


I need to plot the Lyapunov exponent for this STH to evaluate its performance. I found similar problems here and here but can't apply them to my solution.

• Can you clarify $HM$; is $STH(n+1)=HM(n+1)=1-ur\cdot\sin^2(\pi STH(n))+\beta r\cdot|(1-2STH(n-1)|$? Also, are you interested in computing $\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1}\ln|STH'(x_i)|$ for many $x_0$'s (from en.wikipedia.org/wiki/Lyapunov_exponent)? Is numerical approximation OK?
• Look at the code from mathematica.stackexchange.com/questions/81230/…. How do you compute Lyapunov exponents for double recurrences (i.e. $f'(x_i,x_{i-1})$)? Do you want a plot with $u$ varying or $r$ varying or $\beta$ varying? For what fixed $x_0$ starting point? Once these details are hashed out, the code can be adapted.