CDS - Hopf Bifurcation General Formula

The basic example of the Hopf Bifurcation:$\{\begin{array}{l}\dot{x}=\beta x-y+\sigma x(x^2+y^2)\\ \dot{y}=x+\beta y+\sigma y(x^2+y^2)\end{array}$Where $\beta$ and $\sigma$ are parameters and we have that:

• $\sigma =-1$, we will have the SUPERcritical hopf bifurcation.
• $\sigma =+1$, we will have the SUBcritical hopf bifurcation.