Analyticity and Nonanalyticity of Solutions of Delay-Differential Equations

TL;DR

This paper investigates the conditions under which solutions to delay-differential equations with analytic nonlinearity and variable delays are themselves analytic or nonanalytic at different points, revealing complex local regularity properties.

Contribution

It provides new criteria for determining when solutions are analytic or nonanalytic at specific points, considering the dynamic behavior of the delay function.

Findings

Solutions can be smooth everywhere but nonanalytic at certain points.

Analyticity depends on the dynamic properties of the delay map.

Conditions for both analyticity and nonanalyticity are established.

Abstract

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 $C^\infty$ 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 $t\to \eta(t)$ play a crucial role.