Consider the system:
$\dot{x} = \mu x-y-xy^2-x^3$
$\dot{y} = x+\mu y - x^2y-y^3$
I have shown that a Hopf bifurcaiton takes place at the origin $(0,0)$ as a stable spiral becomes an unstable spiral as $\mu$ passes through $0$. However I am not sure how to classify its type.
I have converted the system into polar coordinates and found $\dot{r} = \mu r -r^3\cos^4\theta-r^3\sin^4\theta$.
If $\theta = \pi/4$ then the system has a stable limit cycle, but this is the minimum $r$ can grow.
I dont think this classifies it as a super critical bifurcation through...