How can I add a delay in a function of a function (DDE)?

Hey guys,

I'm new to MATLAB and I'm having some trouble coding the model I need to code. So, I have two delayed differential equations (time delays) but the delays are implicit in my model. What I mean by that is, for exaple, my DDEs look like this:

dy/dt = y(x(t - 1)) + y(t)

instead of:

dy/dt = y(t - 1) + y(t).

I've managed to code the latter with dde23 but I can't figure out how to code the version of the DDE I actually have because I don't know how to delay t in x. With what I've tried I can only get a delay x in y (I'm assuming cause MATLAB didn't recognise x as a function of t as but I don't know how to do that either since I don't have an expression of x(t). I only have dx/dt which in turn depends on the soludion of the system of DDEs).

Any insight will be greatly appreciated. Thank you in advance.

Answer 1

[Steven Lord]

The dde23 function is for solving delay differential equations with constant delays.

The ddesd function is for solving delay differential equations with general delays. There's also ddensd for DDEs of neutral type.

From the description of your problem I think it fits the model of ddesd rather than dde23.

[Sofia Tsangaridou]

I used ddesd with just a random function for x(t) and it worked, thank you!

For my model though, I don't have a definition of x(t), I have a definition of dx/dt, can I still use the same solver?

For example, one of my DDEs is:

dy/dt = y(t)*[x(T(t)) - x(T(t - 1))]

and I know that x(T(t)) = 5 + 6*T(t)/(1+T(t)) BUT I don't have T(t).

I only have:

dT/dt = 100 - 5*T(t) - 4*(u(t) + 7*v(t))*T(t)^2/(4 + T(t)^2)

with du/dt and dv/dt known functions. It's basically a coupled system of 3 ODEs (du,dv,dT) and they all depend on y(t).

I'm a little lost because everything depends on each other.

[Steven Lord]

I think you can use the "DDE With State-Dependent Delays" example in the documentation as a model for your code.

[Sofia Tsangaridou]

Thank you! I’ll try that a look.

[Sofia Tsangaridou]

Hey, so I've seen this example before and I have given it another look just now but I still have the same issue that I don't know how to resolve. If you look at dy/dt above you'll see that yes, y has state-dependent delays because of x but T is also delayed and that delay is a constant time delay (t = time in my model). That is the part I don't know how to tackle and I think ddesd doesn't take into account the delay in T.

Do you know of anything that could help with that? Or did I misunderstand what ddesd does?

[Steven Lord]

Your system of DDEs is where .

The dely function on that page accepts t and Q.

The delays in x will be Z(2, :) and the delays in T will be Z(5, :) in ddefun.