Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 85 / 1 / a / Solution

Put $p=\dot{u}$ and extend the system by $\dot{\lambda }=\dot{\kappa }=0$. At the zero equilibrium with zero parameters, the $(u,p)$ block is the nilpotent matrix $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, while the $v,w$ eigenvalues are $-1,-\tau$. Since $\tau >0$, these are stable. The extended centre manifold is a graph $v=h(u,p,\lambda ,\kappa )$, $w=k(u,p,\lambda ,\kappa )$ tangent to the four-dimensional centre space containing the two parameter directions. Its invariance equations areFor the quadratic jets in the ordinary total degree, the coefficient multiplying $h_p,k_p$ has no linear part. Hence $p{\partial }_u{h}_2=-h_2+u^2$ and $p{\partial }_u{k}_2=-\tau k_2+u^2/\tau$. With $h_2=a{u}^2+bup+c{p}^2$, comparison gives $a=1$, $b=-2$, $c=2$. The second graph is obtained in the same way. Terms involving parameters at quadratic order have zero forcing and vanish. Thus the quadratic centre manifold for a pair of linear filters isSubstitution into the centre equations gives the cubic termsThe last displayed cubic term is absent from the proposed second-order approximation, so its omission needs the weighted scaling rather than an ordinary-degree truncation.

Under the hinted scaling $u=\epsilon \tilde{u}$, $\kappa =\epsilon \tilde{\kappa }$, $\lambda ={\epsilon }^2\tilde{\lambda }$, $d/dt=\epsilon d/d\tilde{t}$, one has $p=O({\epsilon }^2)$ and $\dot{p}=O({\epsilon }^3)$. The invariance equations at weighted orders two and three give $v=u^2-2up+O({\epsilon }^4)$ and $w=u^2/{\tau }^2-2up/{\tau }^3+O({\epsilon }^4)$. In particular no additional parameter-dependent weighted-order-three graph terms appear. Therefore the $u{\dot{u}}^2$ term and the omitted manifold corrections affect the reduced acceleration only at order ${\epsilon }^5$, whereas $u^3$ has order ${\epsilon }^3$ and $u^2\dot{u}$ order ${\epsilon }^4$. After division by ${\epsilon }^3$, the slow equation isThis proves why the stated approximation retains the leading nonlinear restoring force and its first nonlinear damping correction. The centre manifold theorem removes the exponentially decaying transverse directions, while the retained generic coefficients determine the nearby reduced bifurcations. It is a local asymptotic description, not an exact global reduction. If either relevant coefficient vanishes, higher-order terms may be needed to resolve that degeneracy; the two extreme $s$ cases in part (b), with $\tau =1$, avoid it.