Exercise :
Translate the dynamical system given in polar coordinates to cartesian coordinates : $$r' = r(μ-r^2)$$ $$θ' = ρ(r^2)$$ where $μ>0$ is a constant and $p: \mathbb R_+ \to \mathbb R$ smooth function with $ρ(μ) > 0$. Show that there exists a unique limit cycle. Would we have the same result if it was $ρ(μ) = 0$ ?
Attempt :
Using the relations :
$$x = r\cos(θ) \Rightarrow x' = r'\cos(θ) - r\sin(θ)θ'$$ $$y = r\sin(θ) \Rightarrow y' = r'\sin(θ) + r\cos(θ)θ'$$
I derived the equations for the given system in cartesian coordinates, as :
$$x' = μx - yρ(x^2 + y^2) - x(x^2+y^2)$$ $$y' = μy - xρ(x^2 + y^2) - y(x^2+y^2)$$
Now, regarding the limit cycle, if :
$$f(x,y) = \begin{bmatrix} μx - yρ(x^2 + y^2) - x(x^2+y^2) \\ μy - xρ(x^2+y^2) - y(x^2+y^2)\end{bmatrix}$$
How would I proceed about the limit cycle part of the problem though ? I know that one approach could be showing that $\text{div}(f) \neq 0$ for a simply connected set, but I'm not sure if this is correct and it seems that it wouldn't be leading anywhere since there is an unknown $ρ$ function there.
This is the first problem I'm trying to solve regarding that part of dynamical systems so I'd appreciate any help.
What would I need to do to show the existence of a unique limit cycle ?