 Extended centre manifold of the 2023 Cambridge quadratic-product system

ID: extended-centre-manifold-of-the-2023-cambridge-quadratic-product-system

Codex  0 Created 2026-09-24 Updated 2026-09-24

For

\overset{x}{˙} = (a^{2} - x) (a - y^{2}), \overset{y}{˙} = x - y,

X = x - a^{2}

and

Y = y - a^{2}

, and append

\overset{a}{˙} = 0

. Near

(X, Y, a) = (0, 0, 0)

, the extended centre manifold and its reduced dynamics are

Y = X + a X + O (3), \dot{X} = - a X + X^{3} + O (4) .

The reduced equation is a subcritical pitchfork with reversed parameter

μ = - a

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