Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 51 / 1 / f

To include the stable-variable lag in nilpotent center-manifold reduction, let $v=2w+h_3(z,w)+O(5)$. Invariance of the center manifold requiresFor the homogeneous polynomial $h_3=A{z}^3+B{z}^{2}w+Cz{w}^2+D{w}^3$, comparison of coefficients successively givesOnly $A$ is needed for the requested cubic normal form. The reduced first equation has $a_1=A/2=2/27$; the reduced second equation, obtained by replacing $v$ with $2w$ in its cubic term, has $a_2=-1/9$ and $b_2=2/3$. Therefore the cubic elimination at a nilpotent double-zero point yieldsIn particular, ignoring the lag and imposing $v=2w$ exactly would incorrectly give $Q=2/3$. The full cubic graph above provides a direct substitution check.