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

Set $A=s-1\ne 0$ and rescale $q=\sqrt{|A|}u$, $p=\dot{q}$. The reflection-symmetric cubic double-zero normal form becomesIt 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 for $s\gg 1$, a pair of saddles exists when $\lambda >0$, while for $s\ll 1$, 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 $\kappa \ne 0$, the scalar centre equation there beginsFor $\kappa <0$, 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 equationConsequentlyThese 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:Their criticality includes quadratic terms and is not read off from the positive $q^{2}p$ term alone. Set $q=a+x$, $a^2=-\lambda$, and $\delta =\kappa -2\lambda$. ThenFor an oscillator $\ddot{x}=-{\omega }_H^{2}x+A_2{x}^2+B_{2}x\dot{x}+C_3{x}^3+D_3{x}^2\dot{x}$, the cubic radial coefficient is $D_3/8+A_2{B}_2/(8{\omega }_H^2)$. Here it is $2/8+(-3a)(4a)/(8\cdot 2a^2)=-1/2$. 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 energyFor $\sigma =1,\kappa \le 0$, the divergence has one strict sign except on a line, and the Bendixson-Dulac criterion excludes every nonconstant periodic orbit. For $\sigma =-1,\kappa \ge 0$, 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 $\lambda \le 0$, 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).