# Centre Manifold

The centre manifold of the system
\[ \dot{x} = xy,\qquad \dot{y}=-y-x^2 \]
is approximately
\[
y = -x^2 - 2x^4 - 12x^6-112x^8-\cdots
\]
near the origin (derived in the notes, and shown as the solid blue curve below).
In general, the points converge to the centre manifold *quickly*, and the stability
of the system is determined by the (reduced) *slow* dynamics on the centre manifold.