On periodic solutions and global dynamics in a periodic differential delay equation
MFO Scientific ProgramResearch in Pairs 2013
Ivanov, Anatoli F.
Trofimchuk, Sergei I.
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, $x \cdot f(x) < 0 x \neq 0, 0\leq a(t)$ is continuous $\omega$-periodic, $[\cdot]$ is the integer part function, and the integer $K \geq 0$ is the delay. The case of integer period $\omega$ 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 $(K = 0)$.