Stability threshold for delayed quadratic crowding
= Stability threshold for delayed quadratic crowding
For
$$
x'(t)=\alpha\left[x(t)-x(t-T)^2\right],
$$
linearization about $x=1$ gives $u'=\alpha[u(t)-2u(t-T)]$. The equilibrium is stable for
$$
0\leq T<\frac{\pi}{3\sqrt3\alpha}
$$
and loses stability through imaginary roots $\lambda=\mathord\pm i\sqrt3\alpha$ at equality.