Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 51 / 1 / d / Solution

The nonzero equilibrium points are paired by reflection:At either equilibrium point, the derivative of the second equation with respect to $x$ is $-1+x_{\ast }^2=2-3c/d$. The characteristic polynomial becomesIts constant term is nonzero away from the pitchfork bifurcation, so there is no further steady local bifurcation. At a Hopf bifurcation the condition $A_1{A}_2=A_3$ givesIndeed $\tau =2-d/2$ and $\tau {\omega }^2=d-c>0$ require $d<4$; within this range ${\omega }^2>0$ is equivalent to $d<1$. The apparent singularity at $d=12$ does not hide another solution of the coefficient equation. The Routh-Hurwitz stability criterion shows that both equilibrium points are stable for $0<d<1$, $c_H^{\ast }(d)<c<d$, and unstable for $c<c_H^{\ast }(d)$. For $d\ge 1$, $3c/d-d-2<1-d\le 0$, so no nonzero equilibrium point is stable.

Their Hopf bifurcations are supercritical. Here is a coefficient check that includes the effect of the quadratic term around $x_{\ast }$. Let $B_2(u,v)=2x_{\ast }{u}_x{v}_x$, $C_2(u,v,w)=2u_x{v}_x{w}_x$ and normalize $q_x=1$ as before. The inverse-linearization term has $(L^{-1}B(q,\bar{q}){)}_x=3/x_{\ast }$. AlsoThe resonant cubic Hopf coefficient isIn this simplification $x_{\ast }^2/{\omega }^2=3\tau /d$. Thus stable small periodic orbits appear around each nonzero equilibrium point on the $c<c_H^{\ast }(d)$ side. This classifies the local branches; the sketch does not infer uncomputed distant global bifurcations.