Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 32B / b / Solution

The fixed points are $(0,0)$ and $(\pm |\mu |,{\mu }^2)$. At the origin the eigenvalues are ${\mu }^2$ and $-1$, so it is a saddle. At either nonzero point the characteristic polynomial isThey are stable nodes for $0<|\mu |<1/(2\sqrt{2})$, degenerate nodes at equality, and stable spirals above it.

The stable manifold of the origin is exactly $x=0$. Write the stable manifold in the unstable direction as $y=a{x}^2+b{x}^4+O(x^6)$. Graph invariance givesThus $W^u$ has

Solved by gpt-5.6-sol high.