Comparing Two Codes for Plotting Maximal Lyapunov Exponent: Which One is Correct?

Hello MATLAB community,

I have been working on calculating and plotting the Maximal Lyapunov Exponent for the Logistic map. I came across two different implementations, and I am unsure which one correctly computes the Lyapunov exponent.

Code 1:

% Parameters for the Logistic Map
r = 3.5;       % Parameter 'r'
x0 = 0.1;      % Initial condition
v0 = 1;        % Initial tangent vector
N = 1000;      % Number of iterations to compute the exponent
format long
% Initialize variables
x = x0;
v = v0;
lyapunov = 0;
% Iteration loop
for i = 1:N
    % Logistic map equation
    xn = r * x * (1 - x);
    % Jacobian of the Logistic map
    J = r * (1 - 2 * x);
    % Update tangent vector
    v = J * v;
    % Compute norm of the tangent vector
    norm_v = abs(v);
    % Renormalize the tangent vector
    v = v / norm_v;
    % Accumulate Lyapunov sum
    lyapunov = lyapunov + log(norm_v)
    % Update state
    x = xn;
end
% Compute the maximal Lyapunov exponent
lyapunov =lyapunov / N;
format long
% Display result
fprintf('Maximal Lyapunov Exponent: %.6f\n', lyapunov);

Code 2:

function LE = LEofLogisticMap( rStart, rEnd, rStep )
rValues = rStart:rStep:rEnd;  
nPoints = length( rValues );
nIterations = 1000; % number of iterations
LE      = zeros( 1, nPoints );
x       = zeros( 1, nIterations + 1 );
x( 1 ) = 0.1;                
for k = 1:nPoints
    sum = 0;
    for i = 1:nIterations
        x( i + 1 ) = rValues( k )*x( i )*( 1 - x( i ) );
        sum = sum + log( abs( rValues( k ) - 2*rValues( k )*x( i ) ) ) 
    end
    LE( k ) = sum / nIterations;     
end
rStart = 3.5;    % Start value of r
rEnd = 4.0;      % End value of r
rStep = 0.01;    % Step size
format long
LE = LEofLogisticMap(rStart, rEnd, rStep)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Question:

Both codes compute a Lyapunov exponent using similar steps . However:

1. In the first code, the tangent vector is renormalized at each step. Is this approach correct?
2. Do these codes correctly compute the maximal Lyapunov exponent for their respective map?
3. Is there a universal approach or standard practice that I should follow for these types of calculations?
4. Are there any potential pitfalls in the logic of either code that I should address?

Thank you for your guidance!

[Ahmed Yousef]

@Umar, could you please help clarify this point?

[Ahmed Yousef]

@Walter Roberson, could you please help clarify this point?

[Walter Roberson]

@Ahmed Yousef

Sorry, I have no information about Maximal Lyapunov Exponents

[Ahmed Yousef]

Thank you very much

[Ahmed Yousef]

@Sam Chak, could you please help clarify this point?

[Torsten]

https://www.youtube.com/watch?v=Xv1_7kFuri4

This computes Lyapunov exponents for given r-values. What is the definition of "Maximal Lyapunov Exponent" ?