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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.