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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 - 10Sin[Pisth[m - 1]]^2 + 
  βrAbs[1 - 2sth[m - 1]]]; 
Table[sth[m], {m, 0, Ceiling[512512*(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.

mtl Kh
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    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 https://en.wikipedia.org/wiki/Lyapunov_exponent)? Is numerical approximation OK? – Adam Mar 21 '21 at 09:12
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    Look at the code from https://mathematica.stackexchange.com/questions/81230/discrete-time-lyapunov-exponent-plot. 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. – Adam Mar 21 '21 at 10:31
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    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 – Michael E2 Mar 21 '21 at 14:26
  • @Adam I have modified in the post – mtl Kh Mar 21 '21 at 15:05

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