Analyticity and Nonanalyticity of Solutions of Delay-Differential Equations

We consider the equation $\dot x(t)=f(t,x(t),x(η(t)))$ with a variable time-shift $η(t)$. Both the nonlinearity $f$ and the shift function $η$ are given, and are assumed to be analytic (that is, holomorphic) functions of their arguments. Typically the time-shift represents a delay, namely that $η(t)=t-r(t)$ with $r(t)\ge 0$. The main problem considered is to determine when solutions (generally $C^\infty$ 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 $C^\infty$ 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 $t\to η(t)$ play a crucial role.

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