I want to extract outer points of my plot(Chaotic Data Points) and plot it with the extracted outer points.

alpha = 0;

% Initial conditions
x = 1;
y = 1;
z = 1;

% Time step
dt = 0.01;

% Arrays to store the trajectory points
x_traj = [];
y_traj = [];
z_traj = [];

% Loop over time
for t = 0:dt:300
    
    % Calculate x', y', and z' for the current alpha
    x_prime = (25*alpha + 10)*(y - x);
    y_prime = (28 - 35*alpha)*x - x*z + (29*alpha - 1)*y;
    z_prime = x*y - ((alpha+8)/3)*z;
    
    % Update x, y, and z
    x = x + dt*x_prime;
    y = y + dt*y_prime;
    z = z + dt*z_prime;
    
    % Store the trajectory points
    x_traj = [x_traj, x];
    y_traj = [y_traj, y];
    z_traj = [z_traj, z];
    
end
% Plot x and y
figure;
plot(x_traj, y_traj);      
xlabel('x')                          
ylabel('y');
title('Phase Portrait: x and y');
grid on;

I have written this code and from this I have got the following plot

;

There are a lot of self intersecting points in this plot.So I want a exact shape but without using the selfintersecting points.Then I used Convhull command

convhull_indices = convhull(x_traj, y_traj);

plot(x_traj(convhull_indices), z_traj(convhull_indices), 'r');

and I got this plot

But this is not the excat representation of the previous shape

I am actually looking something like this.Which command or procedure I should use.

Answer 1

[Mathieu NOE]

hello

maybe this ?

I needed to use a refine time increment dt (1/10th of the original value) so that the outer points are close enough to make the boundary smooth and not with zigzags between outer and more inner points

hope it helps

alpha = 0;
% Initial conditions
x = 1;
y = 1;
z = 1;
% Time step
dt = 0.001;
% Arrays to store the trajectory points
x_traj = [];
y_traj = [];
z_traj = [];
% Loop over time
for t = 0:dt:300
    % Calculate x', y', and z' for the current alpha
    x_prime = (25*alpha + 10)*(y - x);
    y_prime = (28 - 35*alpha)*x - x*z + (29*alpha - 1)*y;
    z_prime = x*y - ((alpha+8)/3)*z;
    % Update x, y, and z
    x = x + dt*x_prime;
    y = y + dt*y_prime;
    z = z + dt*z_prime;
    % Store the trajectory points
    x_traj = [x_traj; x];
    y_traj = [y_traj; y];
    z_traj = [z_traj; z];
end

% Plot x and y
figure;
plot(x_traj, y_traj);      
hold on 
xlabel('x')                          
 ylabel('y');
title('Phase Portrait: x and y');
grid on;

 
% k = boundary(___,s) specifies shrink factor s using any of the previous syntaxes.
% s is a scalar between 0 and 1. Setting s to 0 gives the convex hull,
% and setting s to 1 gives a compact boundary that envelops the points.
% The default shrink factor is 0.5.
k = boundary(x_traj, y_traj,0.95);
 plot(x_traj(k), y_traj(k), 'm');

[Mathieu NOE]

this is an alternative that does not need dt increment refined as before

I simply used this fex submission :

Fast and Robust Curve Intersections - File Exchange - MATLAB Central (mathworks.com)

this is the result obtained :

alpha = 0;
% Initial conditions
x = 1;
y = 1;
z = 1;
% Time step
dt = 0.01;
% Arrays to store the trajectory points
x_traj = [];
y_traj = [];
z_traj = [];
% Loop over time
for t = 0:dt:100
    % Calculate x', y', and z' for the current alpha
    x_prime = (25*alpha + 10)*(y - x);
    y_prime = (28 - 35*alpha)*x - x*z + (29*alpha - 1)*y;
    z_prime = x*y - ((alpha+8)/3)*z;
    % Update x, y, and z
    x = x + dt*x_prime;
    y = y + dt*y_prime;
    z = z + dt*z_prime;
    % Store the trajectory points
    x_traj = [x_traj; x];
    y_traj = [y_traj; y];
    z_traj = [z_traj; z];
end

% Plot x and y
figure;
plot(x_traj, y_traj);      
hold on 
xlabel('x')                          
ylabel('y');
title('Phase Portrait: x and y');
grid on;


% home made outer boudary selection 
% obtained by looking at top / bottom y points (intersection with a
% vertical line scrolling from left to right)

xx = linspace(min(x_traj),max(x_traj),100);
x_store = [];
y_store = [];
yrange = ceil(max(abs(y_traj)));    

for ci =1:numel(xx)
    [x0,y0,iout,jout] = intersections(x_traj,y_traj,[xx(ci) xx(ci)],[-yrange yrange]); % FEX : https://fr.mathworks.com/matlabcentral/fileexchange/11837-fast-and-robust-curve-intersections?s_tid=srchtitle
    
    [y0max,~] = max(y0);
    [y0min,~] = min(y0);
    
    x_store = [x_store;xx(ci);xx(ci)];
    y_store = [y_store;y0min;y0max];    
    
end

plot(x_store, y_store, '*r');

[Mashrur Zawad]

When I was trying to run the code.I am getting this error. Unrecognized function or variable 'intersections'.

Error in UnifiedChaos (line 606)

[x0,y0,iout,jout] = intersections(x_traj,y_traj,[xx(ci) xx(ci)],[-yrange yrange]); % FEX : https://fr.mathworks.com/matlabcentral/fileexchange/11837-fast-and-robust-curve-intersections?s_tid=srchtitle.

Any idea why I am getting this.

[Mathieu NOE]

sure

I was saying that you need to download the function 'intersections.m'. at this page : Fast and Robust Curve Intersections - File Exchange - MATLAB Central (mathworks.com)

if you prefer , it's also in the attachment here

store it in your matlab path

[Mashrur Zawad]

Thanks a lot

[Mathieu NOE]

as always, my pleasure

if my submission has helped you, do you mind accepting it ? tx