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!
