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

Put $r^2=x^2+y^2$. The system can be writtenIt therefore vanishes at the origin. The linearization there has eigenvaluesHence the origin is an asymptotically stable focus for $\mu <0$ and an unstable focus for $\mu >0$.

When $\mu =0$ the linearization alone is inconclusive. Direct radial reduction giveswhich is positive for every sufficiently small $r>0$. The origin is therefore unstable at $\mu =0$ as well, although trajectories leave it only at cubic order.