Solution

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

For the system in the 2024 Part II Paper 2 question at $r=1$, with
$v=(x+y)/2$ and $w=(x-y)/2$, the center manifold is
$$
w=\frac{a+1}{4}v^3+O(v^5),
\qquad
z=v^2+(1-a)v^4+O(v^6).
$$
Its reduced equation is
$$
\dot v=\frac{a-1}{2}v^3+
\frac{(3a-1)(a+3)}8v^5+O(v^7).
$$


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 Incoming links: Center manifold of the 2024 Cambridge cubic system