We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawt...

We prove oscillation theorems for the nonlinear delay differential equation

In this article, we analyze the first order linear delay differential equation
 \begin{equation*}
 x^{\prime }(t)+p(t)x\left( \tau (t)\right) =0,\text{ }t\geq t_{0},
 \end{equation*}
 where $p,$ $\tau \in C\left( [t_{0},\infty ),\mathbb{R}^{+}\right) $ and $%
 (t)\leq t,\ \lim_{t\rightarrow \infty }\tau (t)=\infty $. Under assumption that (t)$ is not necessarily monoton...