Radial stability of a circular periodic orbit (${\lambda }_{radial}=-r_{0}g\prime (r_0)$)

For $\dot{r}=r(a-g(r))$, $\dot{\theta }=1$, a positive root $a=g(r_0)$ gives a circular periodic orbit of period $2\pi$. The radial exponent is $-r_{0}g\prime (r_0)$, so $g\prime (r_0)>0$ gives orbital asymptotic stability and $g\prime (r_0)<0$ gives instability. The phase direction is neutral; a periodic orbit is not an asymptotically stable individual point of the flow.