ANALYTICITY AND NONANALYTICITY OF SOLUTIONS OF DELAY-DIFFERENTIAL EQUATIONS

摘要

We consider the equation 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) ≥ 0. The main problem considered is to determine when solutions (generally C~∞ 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~∞ 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 → η(t) play a crucial role.