Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 31A / b / iii / Solution

The larger bifurcation value is $\mu =0$. For $\mu <0$ the origin is stable and an unstable periodic orbit of radiussurrounds it. That orbit collapses into the origin as $\mu \uparrow 0$, after which the origin is unstable. This is a subcritical Hopf bifurcation.

After a positive rescaling of radius, its local normal form isor equivalentlyThe full quintic radial Hopf equation also has the stable outer branchIn the $(\mu ,r)$ bifurcation diagram, the branch $r=0$ is stable for $\mu <0$ and unstable for $\mu \ge 0$. The unstable $r_{-}$ branch runs from $(0,0)$ backwards towhere it folds into the stable $r_{+}$ branch; the latter continues to larger $\mu$.

Solved by gpt-5.6-sol high.