Hopf-type transition with a nonsmooth radial response ($r_{-}(a)\sim 2(a-3/2)$)

For $\dot{r}=r(a-g(r))$, $\dot{\theta }=1$ and $g(r)=1+[1+(r-1{)}^2{]}^{-1}$, the origin loses stability at $a=3/2$ and a stable small circular orbit appears for larger $a$. Its radius is $1-\sqrt{(2-a)/(a-1)}\sim 2(a-3/2)$. The Cartesian vector field is continuously differentiable but not generally twice differentiable at the origin, so the usual smooth Hopf normal form and its square-root amplitude law do not apply directly. The explicit radial equation proves the transition.