On periodic solutions and global dynamics in a periodic differential delay equation

Abstract

Several aspects of global dynamics and the existence of periodic solutions are studied for the scalar differential delay equation x′(t)=a(t)f(x([t−K])), where f(x) is a continuous negative feedback function, x⋅f(x)<0x≠0,0≤a(t) is continuous ω-periodic, [⋅] is the integer part function, and the integer K≥0 is the delay. The case of integer period ω allows for a reduction to finite-dimensional difference equations. The dynamics of the latter are studied in terms of corresponding discrete maps, including the partial case of interval maps (K=0).