= Delayed logistic Hopf threshold
{title2=$\tau_c=\pi/2,\quad T_c=2\pi$}
A purely imaginary root of $\lambda=-e^{-\lambda\tau}$ has frequency one and first appears at $\tau=\pi/2$. Roots cross with positive real derivative $1/(1+\tau_c^2)$; all roots lie in the left half-plane for smaller positive delay, since a hypothetical root $a+ib$ with $a\geq0$ would require both $|b|\tau\geq\pi/2$ and $|b|\leq1$. A small-amplitude expansion $u=1+a\cos(\omega t)+a^2[\cos(2\omega t)/5+\sin(2\omega t)/10]+\cdots$ yields $\omega=1-3a^2/20+\cdots$ and $\tau=\pi/2+(3\pi-2)a^2/40+\cdots$, verifying a nondegenerate <Hopf bifurcation>. The limiting period is $2\pi=4\tau_c$.
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