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

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Ahmed Yousef on 20 Dec 2024

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Edited: Torsten on 20 Dec 2024

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:

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

Thank you for your guidance!