I am trying to solve numerically the following system of two coupled delay differential equations:

$$\dot x(t)=-\gamma x(t)-\frac{\gamma}{4}e^{i\omega_0\tau_1}y(t-\tau_1)\theta(t-\tau_1)+\frac{\gamma}{4}e^{i\omega_0\tau_2}y(t-\tau_2)\theta(t-\tau_2)+\frac{\gamma}{2}e^{i\omega_0\tau_3}x(t-\tau_3)\theta(t-\tau_3),$$ $$\dot y(t)= -\frac{\gamma}{2}y(t)-\frac{\gamma}{4}e^{i\omega_0\tau_1}x(t-\tau_1)\theta(t-\tau_1)+\frac{\gamma}{4}e^{i\omega_0\tau_2}x(t-\tau_2)\theta(t-\tau_2).$$ where $\tau_1<\tau_2<\tau_3$. The parameters $\gamma, \omega_0$ are constants, and $\theta(t)$ is the Heaviside step function. The history of the system is known for $0\leq t\leq\tau_1$: $$x(t)=e^{-\gamma t}, y(t)=e^{-\gamma t/2}.$$ Here what I tried:

I first solved the system for $0\leq t\leq\tau_2$ using the aforementioned initial history with NDSolve:

\[Gamma] = 1.0;
\[Omega]0 = 2 Pi;
\[Tau]1 = 1.0;
\[Tau]2 = 2.0;
\[Tau]3 = 3.0;

sol1 = NDSolve[{x'[
  t] == - \[Gamma] x[t] - (\[Gamma]/4) E^(I \[Tau]1 \[Omega]0)
    y[t - \[Tau]1], 
y'[t] == - 0.5 \[Gamma] y[t] - (\[Gamma]/4) E^(
   I \[Tau]1 \[Omega]0) x[t - \[Tau]1], 
x[t /; t <= \[Tau]1] == (1.0/Sqrt[2.0]) Exp[-\[Gamma] t], 
y[t /; t <= \[Tau]1] == (1.0/Sqrt[2.0]) Exp[-0.5 \[Gamma] t]}, {x,
 y}, {t, 0, \[Tau]2}];

I get the following solution for $|x(t)|^2$ and $|y(t)|^2$: abs[x]^2,abs[y]^2

The problem arises when I use this first interpolated solution as the initial history to solve for the next interval of time:

sol2 = NDSolve[{x'[
  t] == - \[Gamma] x[t] - (\[Gamma]/4) E^(I \[Tau]1 \[Omega]0)
    y[t - \[Tau]1] + (\[Gamma]/4) E^(I \[Tau]2 \[Omega]0)
    y[t - \[Tau]2], 
y'[t] == - 0.5 \[Gamma] y[t] - (\[Gamma]/4) E^(
   I \[Tau]1 \[Omega]0) x[t - \[Tau]1] + (\[Gamma]/4) E^(
   I \[Tau]2 \[Omega]0) x[t - \[Tau]2], 
x[t /; t <= \[Tau]2] == Evaluate[x[t] /. sol1], 
y[t /; t <= \[Tau]2] == Evaluate[y[t] /. sol1]}, {x, y}, {t, 
0, \[Tau]3}]; 

This time I get the following messages: enter image description here enter image description here

It seems that the second NDSolve (sol2) does not allow the interpolation of the first result as initial history. Any suggestion? Thank you in advance.


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