basin for attraction of multiple attractor

how can basin of attraction of multiple attractor for a three dim system can plot as i am just able to plot only for one attractor and don't have an idea how to plotbasin of attraction for multiple attractor.

[Sam Chak]

Can you show the MATLAB code of how you plotted the basin of attraction for a single attractor?

[Ajay]

% Lorenz system parameters
clear all;
clc;
alpha = 10;
row= 28;
beta = 8/3;
% Define the Lorenz system equations
lorenz = @(t, xyz) [alpha * (xyz(2) - xyz(1));
    xyz(1) * (row - xyz(3)) - xyz(2);
    xyz(1) * xyz(2) - beta * xyz(3)];
% Define the attractor coordinates (you can adjust )
attractor = [0, 0, 0];
% Define the time span for simulation
tspan = [0, 5];
% Number of points for the grid
num_points = 20;
% Create a grid of initial conditions for x, y, and z
x_range = linspace(-30, 30, num_points);
y_range = linspace(-30, 30, num_points);
z_range = linspace(0, 50, num_points);
[X, Y, Z] = meshgrid(x_range, y_range, z_range);
% Create a colormap for plotting
colormap([1 0 0; 0 0 1]);  % Red and blue
% Initialize a matrix to store basin of attraction information
basin = zeros(size(X));
% Loop through all initial conditions
for i = 1:numel(X)
    x0 = X(i);
    y0 = Y(i);
    z0 = Z(i);
    
    % Simulate the Lorenz system from (x0, y0, z0)
    [~, sol] = ode45(lorenz, tspan, [x0, y0, z0]);
    
    % Check if the trajectory converges to the attractor
    if norm(sol(end, :) - attractor) < 1.0
        % Mark this point as part of the basin of attraction
        basin(i) = 1;
    end
end
% Plot the basin of attraction
figure;
slice(X, Y, Z, basin, [], [], []);
xlabel('X');
ylabel('Y');
zlabel('Z');
title('Basin of Attraction for Lorenz Attractor');
colorbar;
axis tight;

[Kalyanashis]

This program will give a blank figure

Answer 1

[Varun]

Hi Ajay,

I understand that you are currently able to plot basin of attraction for one attractor, but you want to plot basin of attraction for multiple attractions.

To achieve this, you can modify the existing code by introducing an array of attractor coordinates. You define an array “attractor” that holds the coordinates of all the attractors you want to consider. .

Then, during the loop that checks if the trajectory converges to an attractor, you iterate over each attractor in the attractors array and break the loop as soon as a match is found. The value assigned to “basin(i)” represents the index of the attractor that the point converges to.

Please refer the updated code:

% Lorenz system parameters
clear all;
clc;
alpha = 10;
row= 28;
beta = 8/3;

% Define the Lorenz system equations
lorenz = @(t, xyz) [alpha * (xyz(2) - xyz(1));
                   xyz(1) * (row - xyz(3)) - xyz(2);
                   xyz(1) * xyz(2) - beta * xyz(3)];

% Define the attractor coordinates
attractors = [  % Add the coordinates of your attractors here
    0, 0, 0;
    10, -10, 20;
    -10, 10, 30
];

% Define the time span for simulation
tspan = [0, 5];

% Number of points for the grid
num_points = 20;

% Create a grid of initial conditions for x, y, and z
x_range = linspace(-30, 30, num_points);
y_range = linspace(-30, 30, num_points);
z_range = linspace(0, 50, num_points);
[X, Y, Z] = meshgrid(x_range, y_range, z_range);

% Create a colormap for plotting
colormap([1 0 0; 0 0 1]);  % Red and blue

% Initialize a matrix to store basin of attraction information
basin = zeros(size(X));

% Loop through all initial conditions
for i = 1:numel(X)
    x0 = X(i);
    y0 = Y(i);
    z0 = Z(i);
    
    % Simulate the Lorenz system from (x0, y0, z0)
    [~, sol] = ode45(lorenz, tspan, [x0, y0, z0]);
    
    % Check if the trajectory converges to any of the attractors
    for j = 1:size(attractors, 1)
        if norm(sol(end, :) - attractors(j, :)) < 1.0
            % Mark this point as part of the basin of attraction
            basin(i) = j;  % Use the index of the attractor as the value
            break;  % Break the loop if a match is found
        end
    end
end

% Plot the basin of attraction
figure;
slice(X, Y, Z, basin, [], [], []);
xlabel('X');
ylabel('Y');
zlabel('Z');
title('Basin of Attraction for Lorenz Attractors');
colorbar;
axis tight;

Hope this helps.

[Kalyanashis]

But, how will it produce basin of attraction? Its a blank figure.

[Kalyanashis]

Kindly give the answer.

[Ajay]

https://in.mathworks.com/matlabcentral/fileexchange/43564-basins-of-attraction

You can use this program to solve your problem.

[Kalyanashis]

But, in that program that is only a system of non-linear equations but not a system of nonlinear ODEs. I want to know the error of your program that you have written for multiple attractors. Why does it give a blank figure? Kindly solve my problem.

[Ajay]

Actually, i use above program for my continous model. To show program, i just express my program for random function. This program is completely fine for continous function.

[Kalyanashis]

[Kalyanashis]

This is the corresponding figure of your code