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');
