= First Hopf boundary of scalar delayed negative feedback
{title2=$q'=-\beta q-\gamma q(T-1)$}
For nonnegative instantaneous damping $\beta$ and positive delayed gain $\gamma$, <characteristic> roots obey $\lambda+\beta+\gamma e^{-\lambda}=0$. There is no nonnegative real root. The first imaginary crossing has $\omega\in[\pi/2,\pi)$ and $\beta=-\omega\cot\omega$, $\gamma=\omega/\sin\omega$. In particular stability is retained below this first boundary and $\gamma=\pi/2$ is exact only at $\beta=0$. For small positive $\beta$, the critical gain is $\pi/2+2\beta/\pi+O(\beta^2)$ and the scaled <period> tends to four. If the <time> delay is dimensional $\tau$, that <period> is multiplied by $\tau$.
Back to article page
