Smallest Alignment Index (SALI)

Can any one help me to get the matlab code to compute the Smallest Alignment Index(SALI(t),as function of time) of the orbits in a phase space of a time series signal?

Answer 1

[Anshuman]

Here is a basic MATLAB code to compute the SALI as a function of time for orbits in a phase space of a given time series signal:

function sali = compute_sali(time_series, dt, t_max)
    % Function to compute SALI(t) for a given time series signal.
    % time_series: The time series data.
    % dt: Time step.
    % t_max: Maximum time.

    % Initialize variables
    num_points = length(time_series);
    t = 0:dt:t_max;
    num_steps = length(t);
    sali = zeros(1, num_steps);

    % Initialize two deviation vectors
    d1 = rand(num_points, 1);
    d2 = rand(num_points, 1);

    % Normalize the deviation vectors
    d1 = d1 / norm(d1);
    d2 = d2 / norm(d2);

    % Time evolution loop
    for i = 1:num_steps
        % Compute the next state of the system (simple Euler integration as example)
        next_state = time_series + dt * time_series_derivative(time_series);
        
        % Compute the next state of the deviation vectors
        d1_next = d1 + dt * deviation_derivative(d1, time_series);
        d2_next = d2 + dt * deviation_derivative(d2, time_series);

        % Normalize the deviation vectors
        d1_next = d1_next / norm(d1_next);
        d2_next = d2_next / norm(d2_next);

        % Compute SALI
        sali(i) = min(norm(d1_next + d2_next), norm(d1_next - d2_next));

        % Update the state and deviation vectors
        time_series = next_state;
        d1 = d1_next;
        d2 = d2_next;
    end
end

function dxdt = time_series_derivative(x)
    % Placeholder function for the derivative of the time series.
    % Replace this with the actual derivative calculation.
    dxdt = -x; % Example: simple linear system
end

function ddt = deviation_derivative(d, x)
    % Placeholder function for the derivative of the deviation vector.
    % Replace this with the actual deviation derivative calculation.
    ddt = -d; % Example: simple linear system
end

The provided code uses simple linear systems as placeholders. You need to replace these with the actual dynamics of your specific time series.

Hope it helps!