Bifurcation of 3D system

I am working with a system of differential equations with three variable.Now, I need to check bifurcation of the system. I need a sample code for bifurcation of 3D system.

[Ashutosh Thakur]

Hi Sarowar,

You can refer to the following MATLAB Answers regarding the generation of code for bifurcation or 3D system:

• https://www.mathworks.com/matlabcentral/answers/20382-bifurcation-diagram.
• https://www.mathworks.com/matlabcentral/answers/319767-how-to-plot-a-bifurcation-diagram-for-differential-equation-is-there-any-formula-to-plot-the-bifurc.

Answer 1

[Anshuman]

Hello,

Typically for bifurcation analysis, tools like MATCONT are used. Below is a sample MATLAB script for a simple 3D system. This example assumes you have a system of differential equations and you want to perform bifurcation analysis by varying a parameter r. Here I am taking Lorenz system for using as an example.

% Define the system of differential equations
function dxdt = mySystem(t, x, r)
    % Example system: Lorenz system
    sigma = 10;
    beta = 8/3;
    dxdt = zeros(3,1);
    dxdt(1) = sigma * (x(2) - x(1));
    dxdt(2) = r * x(1) - x(2) - x(1) * x(3);
    dxdt(3) = x(1) * x(2) - beta * x(3);
end

% Initial conditions and parameter
x0 = [1; 1; 1]; % Initial conditions for x, y, z
r = 28; % Initial parameter value

% Time span for the simulation
tspan = [0 100];

% Solve the system using ODE45 or any suitable solver
[t, x] = ode45(@(t, x) mySystem(t, x, r), tspan, x0);

% Plot the results
figure;
plot3(x(:,1), x(:,2), x(:,3));
xlabel('x');
ylabel('y');
zlabel('z');
title('3D System Dynamics');
grid on;

% For bifurcation analysis, you would typically use a tool like MATCONT
% Here, a simple parameter sweep is done
r_values = linspace(0, 50, 100); % Range of r for bifurcation analysis
bifurcation_points = [];

for r = r_values
    [t, x] = ode45(@(t, x) mySystem(t, x, r), tspan, x0);
    % Analyze the steady-state behavior, fixed points, etc.
    % Here, you could check for changes in stability or periodicity
end

Hope it helps!