Past exam of the topics in applied mathematics course of the University of Oxford / 2007 / B / 5 / c

A Hopf bifurcation occurs when a simple pair of complex conjugate eigenvalues crosses the imaginary axis at nonzero frequency. A supercritical normal form of a dynamical system isWriting $z=r{e}^{i\theta }$ gives $\dot{r}=\mu r-r^3$ and $\dot{\theta }={\omega }_0$. A stable periodic orbit of radius $\sqrt{\mu }$ is born for $\mu >0$ as the equilibrium loses stability. In contrast, a subcritical Hopf has an unstable small periodic orbit on the side where the equilibrium is still stable.