Analyticity and Nonanalyticity of Solutions of Delay-Differential Equations

We consider the equation $$\dot{x}(t)=f(t,x(t),x(\eta (t)))$$ with a variable time-shift $\eta (t)$. Both the nonlinearity $f$ and the shift function $\eta$ are given, and are assumed to be analytic (that is, holomorphic) functions of their arguments. Typically the time-shift represents a delay, namely that $\eta (t)=t-r(t)$ with $r(t)\ge 0$. The main problem considered is to determine when solutions (generally $C^{\mathrm{\infty }}$ and often periodic solutions) of the differential equation are analytic functions of $t$; and more precisely, to determine for a given solution at which values of $t$ it is analytic, and at which values it is not analytic. Both sufficient conditions for analyticity, and also for nonanalyticity, at certain values of $t$ are obtained. It is shown that for some equations there exists a solution which is $C^{\mathrm{\infty }}$ everywhere, and is analytic at certain values of $t$ but is not analytic at other values of $t$. Throughout our analysis, the dynamic properties of the map $t\to \eta (t)$ play a crucial role.