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.

[Lazaros Moysis]

The following 2 videos explain very analytically how to plot a bifurcation diagram, and how to interpret it as well. Basically, we are depicting the local maxima for a given state, when solving the system under different parameter values. The number of loxal maxima show us if we have periodic behavior (fixed number of maxima), or chaotic behavior (innumerable number of maxima).

https://www.youtube.com/watch?v=jkrmZB9OM2E

https://www.youtube.com/watch?v=p9mg6TOBMY0

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!