I have problems with the following exercise. Prove that the system \begin{align*} \dot{x} &= x(x^2+y^2-2x-3) - y\\ \dot{y} &= y(x^2+y^2-2x-3) + x \end{align*} has a cycle limit.
My attempt: I think I have to apply Poncaire-Bendixon Theorem. First, I changed the system to polar coordinate, then I have
\begin{align*} \dot{r} &= (r^2-2r \cos\theta-3)r\\ \end{align*}
From this point, I noticed that if $r>3$, then $$\dot{r}>0.$$ But for $r<1$ $$\dot{r} < 0.$$ Then, I guess I have a cycle limit in $1<r<3$ for Poncaire-Bendixon theorem. But I think this is only an heuristic way to do the exercise. There exist other approach?
Any help is appreciated.