= Quadratic centre manifold for a pair of linear filters
{title2=$v=u^2-2u\dot u+2\dot u^2,\quad w=u^2/\tau^2-2u\dot u/\tau^3+2\dot u^2/\tau^4$}
At a double-zero <eigenvalue> with centre equations $\dot u=p$, $\dot p=0$ to linear order, stable filters $\dot v=-v+u^2$ and $\dot w=-\tau w+u^2/\tau$ have the displayed quadratic <centre manifold> graphs. Solving $p\partial_u h=-\alpha h+u^2/\alpha$ gives $h=u^2/\alpha^2-2up/\alpha^3+2p^2/\alpha^4$. Under the weighted scaling $u=O(\varepsilon)$, $p=O(\varepsilon^2)$, the $p^2$ graph term affects an $uh$ coupling only at order $\varepsilon^5$.
// Target: dynamical-systems.bigb
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