Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 7A / ii / Solution

The only equilibrium is $(0,0)$. Its linearization iswhose eigenvalues are $\pm i$, so the linearized system has a centre.

The exact radial identity from part (i) givesThus the origin is asymptotically stable for $k<0$, neutrally stable for $k=0$, and unstable for $k>0$. Moreover,so trajectories rotate clockwise.

Solved by gpt-5.6-sol high.