# Euller method to solve mackey glass system

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L am 28 Apr. 2023
Bearbeitet: Torsten am 2 Mai 2023
In my understanding , dt needs to be very small so that the Euller method works.
I am solving the mackey glass equation
dx(t)/dt = 0.2x(t-tau)/(1+x(t-tau)^10) - 0.1x(t)
If I use dt = 1, the graph is very "mackey-glassy", but for any value different than 1, the system converges to a step function.
What am I doing wrong?
Thanks,
x = make_mg_euller(0.3,17);
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figure;
plot(x)
function [x] = make_mg_euller(x0,tau)
% Euler's Method
% Initial conditions and setup
dt = 0.01; % step size
n = 400000; % simulation steps
x = double(zeros(n,1)); % allocate
x(1) = x0; % initial condition
% The loop to solve the DE
for i = 1:tau-1
x(i+1) = x(i) + double (dt * ( - 0.1 * x(i) ) );
end
for i = tau:n-1
x(i+1) = x(i) + double(dt* ( ( 0.2 * x(i-tau+1) ) / ( 1 + x(i-tau+1)^10 ) - ( 0.1 * x(i) ) )) ;
end
end
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Torsten am 29 Apr. 2023
Bearbeitet: Torsten am 2 Mai 2023
You try to solve a delay differential equation. Use dde23.
If you want to solve it with your own code, note that tau has to be a multiple of dt, and you must access x(i-j) to get x(t-tau) if tau = j*dt. For t <= tau, x must explicitly be prescribed as a "history" function or by an ordinary differential equation without delay terms and an initial condition at t=0.
L am 2 Mai 2023
Thanks for the tip @Torsten.

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### Akzeptierte Antwort

LeoAiE am 28 Apr. 2023
The main issue with your implementation is that you're using the parameter tau as an integer representing the time delay in terms of steps instead of a continuous time value. With this, when you use a different dt, the delay in the equation isn't adjusted properly.
To fix the issue, you need to calculate the number of steps that correspond to the time delay tau based on the dt value. Here's an updated version of your code with the necessary changes:
x = make_mg_euller(0.3,17);
figure;
plot(x)
function [x] = make_mg_euller(x0, tau)
% Euler's Method
% Initial conditions and setup
dt = 0.01; % step size
n = 400000; % simulation steps
x = double(zeros(n, 1)); % allocate
x(1) = x0; % initial condition
% Calculate the number of steps corresponding to the time delay tau
tau_steps = round(tau / dt);
% The loop to solve the DE
for i = 1:tau_steps - 1
x(i + 1) = x(i) + double(dt * (-0.1 * x(i)));
end
for i = tau_steps:n - 1
x(i + 1) = x(i) + double(dt * ((0.2 * x(i - tau_steps + 1)) / (1 + x(i - tau_steps + 1)^10) - (0.1 * x(i))));
end
end
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