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I have a delay differential system: $$\dot{x}(t) = -2x(t) - x(t-\tau)$$ $$\dot{y}(t) = -0.9y(t) - x(t-\tau) - y(t-\tau)$$ where $\tau$ is just some constant.

Is there a way to transform this system into a regular ODE system?

Like let $u=x(t-\tau)$ and $v=y(t-\tau)$. Then $\dot{u}=\dot{x}$ and $\dot{v}=\dot{y}$. But after this, I'm confused as to where to go from here since I still have $x(t)$ and $y(t)$ in the system.

Desperate Fluffy
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  • Usually the answer to this is no. In some sense a DDE is sort of like an evolution PDE, except without spatial derivatives. To be more precise, the "state" of the system at time $t$ (i.e. the information that we need to specify an initial value problem) is not a single vector $x(t)$ but a vector-valued function $x_t$ with domain $[-\tau,0]$. Then we identify $x_t(s)$ with $x(t-s)$. So for instance we can't make an initial value problem for your DDE by just specifying $x(0),y(0),x(-\tau),y(-\tau)$, we need to know the values on all of $[-\tau,0]$ to proceed. – Ian Oct 01 '15 at 15:53
  • Sorry, typo: if I take the domain of the "state" function to be $[-\tau,0]$ then $x_t(s)$ is identified with $x(t+s)$, not $x(t-s)$. – Ian Oct 01 '15 at 21:41

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