= Hopf-type transition with a nonsmooth radial response
{c}
{title2=$r_-(a)\sim2(a-3/2)$}
For $\dot r=r(a-g(r))$, $\dot\theta=1$ and $g(r)=1+[1+(r-1)^2]^{-1}$, the origin loses <Lyapunov stability>[stability] at $a=3/2$ and a <Lyapunov stability>[stable] small <limit cycle>[circular orbit] appears for larger $a$. Its radius is $1-\sqrt{(2-a)/(a-1)}\sim2(a-3/2)$. The <Cartesian coordinate system>[Cartesian] <vector field> is continuously differentiable but not generally twice differentiable at the origin, so the usual <Hopf bifurcation>[smooth Hopf normal form] and its square-root amplitude law do not apply directly. The explicit radial equation proves the transition.
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