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Consider the simple delayed differential equation:

$X'(t) = -a X(t) + a X(t - d)$

where $d$ and $a$ are positive constants. I'm interested in the possible steady-state (stationary) solutions of this and other similar equations as $t \rightarrow \infty$.

The steady-state seems to vary depending on the initial conditions, for example, I am interested in the case with initial data given by $X(t) = 0$ for $t \in [-d, 0)$ and $X(0) = 1$. Numerical integration with Matlab's dde23 routine gives the steady-state in the case at around 0.33. My question is whether there is another way (other than numerical integration for a long time period) to find such steady-states analogous to finding fixed points of ODEs. If you try and do that here, by assuming that the terms $X(t)$ and $X(t - d)$ become equal as $t \rightarrow \infty$, and that $X'(t) = 0$, you get the useless relation:

$0 = -a X^* + a X^*$

which, of course, tells you nothing about possible fixed points $X^*$.

Any ideas?


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