Quadratic centre manifold for a pair of linear filters ($v=u^2-2u\dot{u}+2{\dot{u}}^2,w=u^2/{\tau }^2-2u\dot{u}/{\tau }^3+2{\dot{u}}^2/{\tau }^4$)

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 $u=O(\epsilon )$, $p=O({\epsilon }^2)$, the $p^2$ graph term affects an $uh$ coupling only at order ${\epsilon }^5$.

// Target: dynamical-systems.bigb