Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 60 / 2 / a

Set $g(x)=x^2+x^3/3$. Equilibria satisfy ${\mu }^2=g(x)$, and their scalar linear eigenvalue isIt vanishes only at $x=0$ or $x=-2$. Substitution in the equilibrium relation gives the three nonhyperbolic parameter-state pointsThe branches with $x<-2$ and $x>0$ are stable, while $-2<x<0$ is unstable. For $0<|\mu |<2/\sqrt{3}$ there are three equilibria; beyond the two outer saddle-node bifurcations there is just the positive stable one. At $\mu =0$ there is also the distinct stable equilibrium $x=-3$.

Near the crossing, use the smooth coordinate $y=x\sqrt{1+x/3}$, whose derivative is positive near zero. The equation becomes $\dot{y}=p(y)({\mu }^2-y^2)$ with $p>0$. A positive time rescaling removes $p$, and the parameter-dependent state shift $u=y-\mu$ givesThus the crossing at $\mu =0$ is a transcritical bifurcation, with parameter $-2\mu$. The two smooth branches $y=\pm \mu$ exchange dynamical stability; labeling the stable branch by $|\mu |$ instead would obscure that exchange. The parameter-dependent coordinate shift in a transcritical bifurcation explains why the original coordinate need not display a branch identically equal to zero.

Write $h={\mu }^2+\epsilon$. The entire unfolded diagram is obtained from $g(x)=h$, with the same dynamical stability sign $-x(x+2)$ as before. A saddle-node bifurcation at $x=0$ requires $h=0$, while a saddle-node bifurcation at $x=-2$ requires $h=4/3$.

For $0<\epsilon <4/3$, the crossing separates into a positive stable branch and a negative unstable branch, with no saddle-node bifurcation near $\mu =0$. Three equilibria exist for $|\mu |<\sqrt{4/3-\epsilon }$, and the two negative branches annihilate at the outer saddle-node bifurcations $\mu =\pm \sqrt{4/3-\epsilon }$. Outside that interval only the positive stable equilibrium remains. If $\epsilon >4/3$, the diagram has just that one branch everywhere; equality is the limiting saddle-node bifurcation contact at $\mu =0,x=-2$.

For $\epsilon <0$, let $d=-\epsilon >0$. Two inner saddle-node bifurcations occur at $\mu =\pm \sqrt{d},x=0$, and two outer saddle-node bifurcations at $\mu =\pm \sqrt{4/3+d},x=-2$. For $|\mu |<\sqrt{d}$ there is a single stable equilibrium below $-3$; for $\sqrt{d}<|\mu |<\sqrt{4/3+d}$ there are three equilibria with stable–unstable–stable ordering; outside the outer saddle-node bifurcations only the positive stable equilibrium remains. The preceding figure shows representative small unfoldings. In particular this unrestricted perturbation destroys the exact transcritical bifurcation crossing.