 Hopf normal form

The complex normal form near a simple Hopf bifurcation is $\dot{z}=(\mu +i\omega )z+\ell z|z{|}^2+\cdots$. The sign of $\mathrm{Re}\ell$ determines the critical cubic radial drift: negative gives supercritical saturation and decay at the critical parameter, while positive gives destabilizing drift.