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

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    $\begingroup$ 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? $\endgroup$
    – Adam
    Mar 21, 2021 at 9:12
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    $\begingroup$ 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. $\endgroup$
    – Adam
    Mar 21, 2021 at 10:31
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    $\begingroup$ People here generally like users to post code as Mathematica code instead of just images or TeX, so they can copy-paste it. And they like it to appear in the question instead of the comments. It makes it convenient for them and more likely you will get someone to help you. You may find the meta Q&A, How to copy code from Mathematica so it looks good on this site, helpful $\endgroup$
    – Michael E2
    Mar 21, 2021 at 14:26
  • $\begingroup$ @Adam I have modified in the post $\endgroup$
    – mtl Kh
    Mar 21, 2021 at 15:05

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