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

At $r=1$, the Jacobian at the origin isIts eigenvalues are $0,-2,-1$. The zero eigenvalue makes the equilibrium nonhyperbolic.

Eigenvectors for $0,-2,-1$ are respectivelyHence

With $x=v+w$ and $y=v-w$, the equations at $r=1$ becomeThe symmetry $(v,w,z)\mapsto (-v,-w,z)$ suggestsSubstitute these into the invariance equationsComparison at orders $v^2,v^3,v^4$ givesThus the center manifold of the 2024 Cambridge cubic system is

Substitution of the center-manifold series into the $v$ equation gives at cubic orderKeeping the $v^5$ terms from $3a{v}^{2}w$, $zw$, and the quartic term in $z$ gives

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