How to plot 2d and 3D phase portrait as in the image attachment for chaotic visualization? Also see my code.

function chaotic

% Define the parameters

k = 1;

lambda_1 = 1;

k_1 = 1;

k_2 = 2;

k_3 = 1;

v_1 = 0.4;

rho = 2;

omega = 1;

mu = 1;

% Calculate W_1 and W_2 based on the given parameters

W_1 = 6 * (k * lambda_1 + rho) / (3 * k * k_3 + 3 * k_2);

W_2 = (k^3 * k_3 + 3*k^2*k_2 + 6*(k*k_1 - v_1)) / (3 * k * k_3 + 3 * k_2);

% Define the time span

tspan = [0 100];

% Generate a grid of initial conditions

[G0, P0] = meshgrid(linspace(0, 0.2, 10), linspace(0, 0.2, 10));

initial_conditions = [G0(:), P0(:)];

% Prepare figures

figure('Name', '2D Phase Portrait');

figure('Name', '3D Phase Portrait');

% Loop over the initial conditions

for i = 1:size(initial_conditions, 1)

% Solve the ODE

[T, Y] = ode45(@(t, y) odesys(t, y, W_1, W_2, omega, mu), tspan, initial_conditions(i, :));

% Plot the 2D phase portrait

figure(findobj('Name', '2D Phase Portrait'));

plot(Y(:, 1), Y(:, 2));

hold on;

% Plot the 3D phase portrait

figure(findobj('Name', '3D Phase Portrait'));

plot3(T, Y(:, 1), Y(:, 2));

hold on;

end

% Finalize the 2D figure

figure(findobj('Name', '2D Phase Portrait'));

xlabel('G');

ylabel('P');

title('2D Phase Portrait');

grid on;

hold off;

% Finalize the 3D figure

figure(findobj('Name', '3D Phase Portrait'));

xlabel('Time');

ylabel('G');

zlabel('P');

title('3D Phase Portrait');

grid on;

hold off;

end

% Nested function for the system of ODEs

function dydt = odesys(t, y, W_1, W_2, omega, mu)

G = y(1);

P = y(2);

dGdt = P;

dPdt = W_1 * G^3 + W_2 * G + omega * cos(mu * t)^2;

dydt = [dGdt; dPdt];

end

Answer 1

[sanidhyak]

Hi @Farooq Aamir,

I understand that you are trying to plot 2D and 3D phase portraits for chaotic visualization using MATLAB. When running the code you shared, I too observed that while the ODE integration works fine, the visualization can be improved to properly display chaotic attractors similar to what you have shared in the figure.

Actually the current implementation loops over a grid of initial conditions and directly plots the results, which can make the plots look cluttered. For a clear chaotic phase portrait, you should use a single trajectory over a longer simulation time.

Kindly refer to the following corrected code for the same:

function chaotic_phase
% Parameters
k = 1; lambda_1 = 1; k_1 = 1; k_2 = 2; k_3 = 1;
v_1 = 0.4; rho = 2; omega = 1; mu = 1;

% Calculate W1 and W2
W1 = 6*(k*lambda_1 + rho) / (3*k*k_3 + 3*k_2);
W2 = (k^3*k_3 + 3*k^2*k_2 + 6*(k*k_1 - v_1)) / (3*k*k_3 + 3*k_2);

% Time span
tspan = [0 200]; 

% Initial condition
y0 = [0.1, 0.1];

% Solve ODE
[T, Y] = ode45(@(t,y) odesys(t,y,W1,W2,omega,mu), tspan, y0);

% 2D Phase Portrait
figure;
plot(Y(:,1), Y(:,2), 'g');
xlabel('G'); ylabel('P');
title('2D Chaotic Phase Portrait');
set(gca,'color','k'); grid on; axis tight;

% 3D Phase Portrait
figure;
plot3(Y(:,1), Y(:,2), T, 'g');
xlabel('G'); ylabel('P'); zlabel('t');
title('3D Chaotic Phase Portrait');
set(gca,'color','k'); grid on; axis tight;
end

% System of ODEs
function dydt = odesys(t,y,W1,W2,omega,mu)
G = y(1);
P = y(2);
dGdt = P;
dPdt = W1*G^3 + W2*G + omega*(cos(mu*t))^2;
dydt = [dGdt; dPdt];
end

Here, we have done the following changes:

• Used a single initial condition.
• Increased the simulation time (“tspan = [0 200]”) to observe chaotic dynamics.
• Added black background with green traces (same as that you shared).
• Plotted “P vs G” (2D) and “(G, P, t)” in 3D for clearer chaotic attractor visualization.

For more details on phase portrait plotting, you may refer to the MATLAB documentation:

• https://www.mathworks.com/help/matlab/ref/plot3.html?utm_source=chatgpt.com

I hope this helps!