Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 33A / b / v / Solution

The transverse eigenvalues $-2$ and $-1$ are stable, so the center manifold reduction determines local asymptotic stability. If $a<1$, the leading center term has negative coefficient and $v$ flows toward zero from both sides. If $a>1$, it flows away. At $a=1$ the cubic term vanishes but the quintic coefficient isso the origin is again unstable. Therefore