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Hi @Le,

I do agree with @Torsten’s comments about, “ use MATLAB's "ddesd" to solve this system of delay differential equations”.

Brief summary: The script deli.m sets up a system of delay differential equations with defined functions for system equations, initial conditions, delays, parameters, and solver options. By executing the script and calling the ddesd solver, the delay differential equations are solved numerically. The solution is then extracted and plotted to visualize the behavior of the system over time.

function dxdt = ddex1de(t, x, Z, A, Ah, Ad)

    % Define the system of delay differential equations

    d1 = 1.5 + 0.5 * sin(1.5*t);

    d2 = 2 + cos(2*t);

    x1 = Z(:,1);

    x2 = Z(:,2);

    % trapz function

    dxdt = [A*x1 + Ah*x1 + Ad*trapz(t, x1) - x1; A*x2 + Ah*x2 + Ad*trapz(t, x2) - x2];

end

function Z = ddex1hist(t)

    % Define the initial conditions for x1 and x2

    Z = [0.05*sin(t); 0.05*cos(t)];

end

function d = ddex1delays(t)

    % Define the delay functions d1 and d2

    d = [1.5 + 0.5*sin(1.5*t); 2 + cos(2*t)];

end

% Define the parameters

A = 1.0;

Ah = 0.5;

Ad = 0.2;

% Define the options for the solver

options = ddeset('RelTol',1e-6,'AbsTol',1e-6);

% Solve the delay differential equations sol = ddesd(@ddex1de, @ddex1delays, @ddex1hist, [0, 5], options);

% Extract and plot the solution t = linspace(0, 5, 1000); y = deval(sol, t); plot(t, y(1,:), t, y(2,:)); legend('x1(t)', 'x2(t)'); xlabel('Time'); ylabel('Value'); title('Solution of Delay Differential Equations');

Please see attached plots and script.

Hi @Le,

Without alterations to code, I received the following results. In your code snippet, you have three functions related to Delay Differential Equations (DDEs) which are essential for solving DDE problems. From analysis of code, the ddex1delays function defines the delays in the DDE system. It returns a vector d representing the delays at time t with state variables y, The history function provides initial conditions or past values for the DDE system. It influences the system's behavior by incorporating past information into the current state and the last one ddex1de function defines the DDE system's differential equations. It computes the derivatives of the state variables y at time t with the history matrix Z. In nutshell, enabling the simulation and analysis of systems with delayed feedback. So, @Torsten is expert in field of dynamics and I would like his feedback if he can elaborate little bit about your recent comments, “The above code comes from the differential equation below. Can you help me check if I wrote the code correctly?”, and so far I haven’t see his feedback on it. However, my response to your comments “The above code comes from the differential equation below. Can you help me check if I wrote the code correctly?” is mentioned below.

Delay Functions:

The delay functions in the code are defined as:

        * d1(t) = t - 1.5 - 0.5sin(1.5t)
        * d2(t) = t - 2 - cos(2*t)

These functions correspond to the model's delay functions:

        * d11(t) = 1.5 + 0.5sin(1.5t)
        * d2(t) = 2 + cos(2*t)
        * My conclusion: The code implementation matches the specified delay functions.

Initial Conditions:

The initial conditions in the model are:

        * x1(t) = 0.05*sin(t)
        * x2(t) = 0.05*cos(t)

My conclusion: The code does not explicitly define these initial conditions but uses them in the history function ddex1hist(t). The history function ddex1hist(t) in the code corresponds to the initial conditions specified in the model.

Differential Equations:

The differential equations in the code are defined by the function ddex1de(t, y, Z). These equations involve the system dynamics and interactions based on the delayed states.

My conclusion: The equations in the code should be consistent with the mathematical model's system dynamics, considering the delays and interactions between variables. Hope, this helps.