Characteristic equation of a delay differential equation

Substitution of $n(t)=e^{\lambda t}$ into $n^{\prime }(t)=rn(t-\tau )$ gives the transcendental characteristic equation

Purely imaginary characteristic roots produce oscillatory solutions. In particular, $r\tau =-\pi /2$ gives $\lambda =\pm i\pi /(2\tau )$ and the real solution $\cos (\pi t/(2\tau ))$.

Forlinearization about $x=1$ gives $u^{\prime }=\alpha [u(t)-2u(t-T)]$. The equilibrium is stable forand loses stability through imaginary roots $\lambda =\pm i\sqrt{3}\alpha$ at equality.

For $x^{\prime }(t)=\alpha x(t-T)[1-x(t)]$, linearization about $x=1$ is $u^{\prime }=-\alpha u$ because the delayed perturbation appears only in a quadratic term. The equilibrium is therefore locally asymptotically stable for every delay $T\ge 0$.