Hi @Junaid Ali ,

You asked, “Please guide me how to perform the continuation of this system in MATCONT. “

Please see my response to your comments below.

Here’s how you can set up and perform the continuation analysis using MATCONT. So, I can assume that you have MATCONT already installed and added to your MATLAB path. If not, please click the following links for more information.

https://www.mathworks.com/matlabcentral/answers/1883987-how-to-load-matcont-data-file-and-plot

https://www.mathworks.com/matlabcentral/answers/770418-how-to-plot-bifurcation-diagram-in-matlab

https://dercole.faculty.polimi.it/tds/matcont.html

You have provided the following system of equations as shown in your comments.

function dx = clutchDynamics(t, x, params) % Unpack parameters de = params.de; ds = params.ds; Fn = params.Fn; mukin = params.mukin; mus = params.mus; je = params.je; jd = params.jd; jv = params.jv; ks = params.ks;

% State variables x1 = x(1); x2 = x(2); x3 = x(3); x4 = x(4); x5 = x(5); x6 = x(6); x7 = x(7);

% Define the system of equations dx = zeros(7, 1); dx(1) = x2; dx(2) = -(de + Fn * mukin) * x2 / je + (Fn * mukin * x4) / je - ((x2 - x4) * mukin + mus) / je; dx(3) = x4; dx(4) = (Fn * mukin * x2) / jd - (ds + Fn * mukin) * x4 / jd + ds * x6 / jd + ks * x7 / jd; dx(5) = x6; dx(6) = ds * x4 / jv - ds * x6 / jv - ks * x7 / jv; dx(7) = x6 - x4; end

You also defined the parameters to be used in your system. I would suggest to implement them in a separate MATLAB script or function:

params.de = 0.1; params.ds = 0.1; params.Fn = 8000; params.mukin = linspace(-0.01, 0.01, 100); % Varying mukin for continuation params.mus = 0.136; params.je = 1.57; params.jd = 0.0066; params.jv = 4; params.ks = 56665.5;

To use MATCONT, you need to set up the initial conditions and the continuation parameters. Here’s how you can do it:You need to find an equilibrium point of your system which can be done by solving the system of equations for a specific value of mukin and then initialize MATCONT with the equilibrium point and specify the parameters for continuation. If you are not familiar with MATCONT commands below, I will suggest to review them by clicking the link below.

https://webspace.science.uu.nl/~kouzn101/NBA/ManualMatcontMNov2017.pdf

% Initial conditions (example) x0 = [0; 0; 0; 0; 0; 0; 0]; % Initial guess for the state variables mukin0 = 0; % Initial value of mukin

% Define the continuation parameters options = contset; % Create a default options structure options = contset(options, 'MaxNumPoints', 100); % Maximum number of points options = contset(options, 'Singularities', 1); % Track singularities

% Start continuation [x, v, s] = cont(@clutchDynamics, x0, mukin0, options, params);

Once you have performed the continuation, now you can plot the bifurcation diagram which can be done using MATLAB,

figure; plot(x(1, :), x(2, :)); % Plotting x1 vs x2 as an example xlabel('x1'); ylabel('x2'); title('Bifurcation Diagram of 3-DOF Clutch Dynamics'); grid on;

Now, to analyze co-dimension bifurcations, you may need to vary more than one parameter. My suggestion would be setting up a second parameter in a similar manner to mukin and use MATCONT to track bifurcations in a multi-parameter space.

*Additional Suggestions**

In my opinion, it may be beneficial to perform a sensitivity analysis on the parameters such as mukin, mus to understand their influence on system behavior. Also, depending on your results, consider investigating different types of bifurcations such as saddle-node or Hopf bifurcations, which can provide deeper insights into system stability and a linear stability analysis around the bifurcation points to deduce the nature of the bifurcations.

Hope this should help you get started with your project now.