visit matlab.com

Derbouche Assia Certainly! Calculating the Lyapunov exponents for a 3D dynamic system can be a complex task, but I can provide you with a basic outline of the steps and some MATLAB code to get you started. Please note that this is a simplified example, and the implementation may vary depending on the specific system you're working with.

Here's a general outline of the process:

1. Define your 3D dynamic system and its equations of motion.

2. Choose an initial condition for the system.

3. Perturb the initial condition slightly to create a nearby trajectory.

4. Integrate the perturbed trajectory and the unperturbed trajectory forward in time.

5. Calculate the difference between the two trajectories at each time step.

6. Compute the Lyapunov exponents based on the rate of divergence or convergence of nearby trajectories.

Here's a MATLAB code snippet for a simple 3D dynamic system (Lorenz system) to compute the first Lyapunov exponent:

```matlab

% Define the Lorenz system equations

sigma = 10;

rho = 28;

beta = 8/3;

% Define the time span and initial conditions

tspan = 0:0.01:10;

x0 = [1; 0; 0]; % Initial condition

% Initialize Lyapunov exponent

lambda1 = 0;

% Perturbation magnitude

epsilon = 1e-6;

for t = tspan

% Solve the unperturbed system

[~, x] = ode45(@(t, x) lorenz(t, x, sigma, rho, beta), [t, t + 0.01], x0);

x_unperturbed = x(end, :);

% Perturb the initial condition

x_perturbed = x0 + epsilon * randn(3, 1);

% Solve the perturbed system

[~, x] = ode45(@(t, x) lorenz(t, x, sigma, rho, beta), [t, t + 0.01], x_perturbed);

x_perturbed = x(end, :);

% Calculate the difference and normalize

dx = x_perturbed - x_unperturbed;

dx_norm = norm(dx);

% Update the Lyapunov exponent

lambda1 = lambda1 + log(dx_norm / epsilon);

% Update the initial condition for the next iteration

x0 = x_unperturbed;

end

% Calculate the average Lyapunov exponent

lambda1 = lambda1 / length(tspan);

% Display the Lyapunov exponent

disp(['The first Lyapunov exponent is: ', num2str(lambda1)]);

% Lorenz system equations

function dxdt = lorenz(t, x, sigma, rho, beta)

dxdt = [sigma * (x(2) - x(1));

rho * x(1) - x(2) - x(1) * x(3);

x(1) * x(2) - beta * x(3)];

end

```

This code calculates the first Lyapunov exponent for the Lorenz system. You can adapt it for your specific 3D dynamic system by modifying the equations of motion and initial conditions. Additionally, you can extend the code to compute multiple Lyapunov exponents if needed.

Please just check the answers in the previous years similar questions - good luck.

from 2022:

https://www.researchgate.net/post/Can_somebody_help_me_to_find_the_lyapunov_exponent_of_3d_discrete_system_in_matlab_or_python

from 2021:

https://www.researchgate.net/post/Can_anyone_provide_me_with_a_matlab_code_for_calculating_the_lyapunov_exponents

etc.