Exercise :
Consider the dynamical system : $$r' = μr(1-g(r^2))$$ $$θ' = p(r^2)$$ where $μ>0$ is a constant and $g,p: \mathbb R_+ \to \mathbb R \space$ are smooth functions with $p(1) > 0$ and $g(1) = 1$. Show the existence of a limit cycle. Is the limit cycle unique ?
Attempt :
So, first of all, we have to check the sign of $r'$. Since $μ>0$, we observe that the points $r=0$ and $r=1$ are stationary points, since $g(1) = 1$. In previous cases, one would have to study the changes of the sign of $r'$ depending on $r$ and would then fix an one-dimension phase portrait with arrows denoting the monotonic domains. If the arrows would "converge" $(\rightarrow \space \leftarrow$) around an equilibria, then this is enough to show that a unique cycle exists.
In this particular case though, I am unable to conduct such a result, since we are not sure of the sign of the expression $(1-g(r^2))$ but we only know that it has a zero point at $r=1$.
I'm pretty sure there's another way around, especially since we're given that $p(1) >0$ which I am not sure where to use yet (I guess it shows a clockwise "movement" of the phase portrait).
I would appreciate any hints or help !