2013/05/02 by John Mallet-Paret, Mallet-Paret, John, Roger D. Nussbaum +1 · 1 citation
Mathematics · #30B10 #34K06 #34K13 #34K99 #37E10 #40A05 (Secondary) #45C05 (Primary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:30B10 #msc:34K06 #msc:34K13 #msc:34K99 #msc:37E10 #msc:40A05 #msc:45C05
paper · pdf · doi:10.48550/arxiv.1305.0579
This is identical to the earlier version 1305.0579v1 except for author email addresses added to the cover page
arxiv created 2013/05/09 · arxiv updated 2013/05/10
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.