Given the system:
$$ \dot{r} = -\mu r + r^3, \\ \dot{\theta} = r $$
There is clearly one single node at $r=0$.
The Jacobian is then: $$ \begin{pmatrix} -\mu + 3r^2 & 0 \\ 1 & 0 \end{pmatrix}$$ Setting $r=0$ and finding the eigenvalues I get: $\lambda = 0 , \lambda = -\mu $. The problem statement says "show that a subcritical Hopf bifurcation occurs at the parameter value $\mu = 0$ ". I don't see how a Hopf bifurcation appears here when all my eigenvalues are all real and I am failing to interpret $\lambda = 0$
Hint: $-\mu r + r^3=r(-\mu + r^2)$. For the stability of the orbits look at the sign of $-\mu+r^2$.
Don't use the Jacobian, no need for it. Better drawing the orbits based on the former identity.