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 $t\to\infty$.
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 $t\to\infty$, 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 $X^*$.