= Solution
Set $A=s-1\ne0$ and rescale $q=\sqrt{|A|}u$, $p=\dot q$. The <reflection-symmetric cubic double-zero normal form> becomes
$$
\dot q=p,\qquad \dot p=-\lambda q+\kappa p+\sigma q^3-2\sigma q^2p,
\qquad \sigma=\operatorname{sgn}(A).
$$
It is invariant under $(q,p)\mapsto(-q,-p)$. Thus the large-$s$ case has $\sigma=1$ and the small-$s$ case $\sigma=-1$. At the origin the <Jacobian matrix> has determinant $\lambda$ and trace $\kappa$: it is a saddle for $\lambda<0$, stable for $\lambda>0,\kappa<0$, and unstable for $\lambda>0,\kappa>0$. In the latter two cases the discriminant $\kappa^2-4\lambda$ distinguishes nodes from foci; its zero is not an <equilibrium>-creation <bifurcation>.
The nonzero <equilibria> satisfy $p=0$, $q^2=\lambda/\sigma$. At either of them the determinant is $-2\lambda$ and the trace is $\kappa-2\lambda$. Therefore \b[for $s\gg1$, a pair of saddles exists when $\lambda>0$], while \b[for $s\ll1$, a pair of attracting <equilibria> exists when $\lambda<0,\kappa<2\lambda$, and a repelling pair when $\lambda<0,\kappa>2\lambda$]. The repelling or attracting points may be nodes or foci. Their origin collision occurs on the <pitchfork bifurcation> line $\lambda=0$, away from $\kappa=0$. For fixed $\kappa\ne0$, the scalar centre equation there begins
$$
\dot q=\frac\lambda\kappa q-\frac\sigma\kappa q^3+\cdots.
$$
For $\kappa<0$, the large-$s$ case is a subcritical pitchfork and the small-$s$ case a supercritical pitchfork. For $\kappa>0$, the scalar criticality reverses, but there is already an unstable transverse <eigenvalue>, so the full-plane branches have the stability just listed.
At the origin, $\kappa=0$, $\lambda>0$ is a <Hopf bifurcation> with frequency $\sqrt\lambda$. Averaging the leading weakly nonlinear oscillator gives the amplitude equation
$$
\dot r=\frac\kappa2r-\frac\sigma4r^3+\cdots.
$$
Consequently
$$
\boxed{\sigma=1:\ \text{stable cycle for }\kappa>0,\ r^2\sim2\kappa;\qquad
\sigma=-1:\ \text{unstable cycle for }\kappa<0,\ r^2\sim-2\kappa.}
$$
These are respectively a <supercritical Hopf bifurcation> and a <subcritical Hopf bifurcation>, with $\lambda>0$ fixed and the cycle sufficiently small. The portraits change from a stable focus to an unstable focus surrounded by a stable cycle in the first case; in the second, a stable focus is surrounded by an unstable cycle before the focus becomes unstable and that cycle disappears.
The small-$s$ case has an additional pair of <Hopf bifurcations> at the nonzero <equilibria>:
$$
\boxed{\kappa=2\lambda,\quad\lambda<0,\qquad \omega_H=\sqrt{-2\lambda}.}
$$
Their criticality includes quadratic terms and is not read off from the positive $q^2p$ term alone. Set $q=a+x$, $a^2=-\lambda$, and $\delta=\kappa-2\lambda$. Then
$$
\ddot x=-2a^2x-3ax^2-x^3+\delta\dot x+4ax\dot x+2x^2\dot x.
$$
For an oscillator $\ddot x=-\omega_H^2x+A_2x^2+B_2x\dot x+C_3x^3+D_3x^2\dot x$, the cubic radial coefficient is $D_3/8+A_2B_2/(8\omega_H^2)$. Here it is $2/8+(-3a)(4a)/(8\cdot2a^2)=-1/2$. Thus both nonzero <Hopf bifurcations> are supercritical: for small $\delta>0$, each newly unstable well <equilibrium> is surrounded by a stable small cycle with $r^2\sim\delta$.
To constrain <periodic orbits>, use either the <divergence> or the energy
$$
\mathcal H=\frac{p^2}2+\frac\lambda2q^2-\frac\sigma4q^4,\qquad
\dot{\mathcal H}=(\kappa-2\sigma q^2)p^2,\qquad
\operatorname{div}F=\kappa-2\sigma q^2.
$$
For $\sigma=1,\kappa\le0$, the <divergence> has one strict sign except on a line, and the <Bendixson-Dulac criterion> excludes every nonconstant <periodic orbit>. For $\sigma=-1,\kappa\ge0$, the same reasoning excludes every <periodic orbit>. When the <divergence> changes sign, this criterion makes no such exclusion; the Hopf cycles above are consistent with that restriction. In the large-$s$ case a <periodic orbit> also requires $\lambda>0$: if $\lambda\le0$, a positive maximum of $q$ would have $\ddot q=-\lambda q+q^3>0$, or a negative minimum would have negative acceleration, both impossible. For $\lambda>0$ the same extremum argument confines a cycle to $|q|<\sqrt\lambda$, between the two saddle locations. The local <bifurcation> lines and representative portraits on both sides are summarized below; their global completion requires part (c).
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-85-double-zero-bifurcations.png]
{title=Local bifurcation lines and representative phase portraits for the two signs of the cubic double-zero normal form}
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