= 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> $\begin{pmatrix}0&1\\0&0\end{pmatrix}$, 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 are
$$
p h_u+(-\lambda u+\kappa p-uh+suk)h_p=-h+u^2,
$$
$$
p k_u+(-\lambda u+\kappa p-uh+suk)k_p=-\tau k+u^2/\tau.
$$
For 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 $h_2=a u^2+bup+cp^2$, 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> is
$$
\boxed{v=u^2-2up+2p^2+O(3),\qquad
w=\frac{u^2}{\tau^2}-\frac{2up}{\tau^3}+\frac{2p^2}{\tau^4}+O(3).}
$$
Substitution into the centre equations gives the cubic terms
$$
\ddot u+\lambda u=\kappa\dot u+\left(\frac{s}{\tau^2}-1\right)u^3
+2\left(1-\frac{s}{\tau^3}\right)u^2\dot u
+2\left(\frac{s}{\tau^4}-1\right)u\dot u^2+O(4).
$$
The 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 $u=\varepsilon\tilde u$, $\kappa=\varepsilon\tilde\kappa$, $\lambda=\varepsilon^2\tilde\lambda$, $d/dt=\varepsilon d/d\tilde t$, one has $p=O(\varepsilon^2)$ and $\dot p=O(\varepsilon^3)$. The invariance equations at weighted orders two and three give $v=u^2-2up+O(\varepsilon^4)$ and $w=u^2/\tau^2-2up/\tau^3+O(\varepsilon^4)$. 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 $\varepsilon^5$, whereas $u^3$ has order $\varepsilon^3$ and $u^2\dot u$ order $\varepsilon^4$. After division by $\varepsilon^3$, the slow equation is
$$
\tilde u''+\tilde\lambda\tilde u
=\tilde\kappa\tilde u'+\left(\frac{s}{\tau^2}-1\right)\tilde u^3
+2\varepsilon\left(1-\frac{s}{\tau^3}\right)\tilde u^2\tilde u'+O(\varepsilon^2).
$$
This 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.
Back to article page
