= Solution
A simple zero <eigenvalue> occurs on $c=d$, except at $(1,1)$. The odd symmetry of the <vector field> gives a <pitchfork bifurcation>. Near this curve the critical <eigenvalue> is
$$
m={d-c\over2(1-d)}+O((c-d)^2).
$$
For $0<d<1$ the other two <eigenvalues> are stable: this is a <supercritical pitchfork bifurcation> as $c$ decreases through $d$. For $d>1$ it is a <subcritical pitchfork bifurcation> in the center direction, with an already unstable transverse direction.
For a <Hopf bifurcation>, factor the <characteristic polynomial> as $(m+\tau)(m^2+\omega^2)$. Comparing coefficients gives
$$
\tau=2-d/2,\qquad\omega^2=1-d,\qquad{c-d\over2}=(2-d/2)(1-d).
$$
Thus \b[the trivial branch has the local bifurcation curves]
$$
\boxed{c=d\quad(d>0),\qquad c=(d-2)^2\quad(0<d<1).}
$$
The second curve is a <Hopf bifurcation> with a stable third <eigenvalue>. The <Routh-Hurwitz stability criterion> gives the stable trivial region $0<d<1$, $d<c<(d-2)^2$. Its <Hopf bifurcation> is subcritical. To check the sign, normalize the critical right <eigenvector> by $q_x=1$ and put
$$
G={i\omega-d/2\over2i\omega(\tau+i\omega)}.
$$
The second component of the normalized left projection is $G$. There is no quadratic term at the origin, and the cubic derivative in the second component is $C_2(q,q,\bar q)=2$. The cubic Hopf coefficient is therefore $l_1=\operatorname{Re}G/\omega=4/[\omega((4-d)^2+4\omega^2)]>0$. A small repelling <periodic orbit> lies on the stable-origin side. The diagram also includes the nonzero-branch information derived next.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-51-fitzhugh-bifurcations.png]
{title=Local bifurcation curves and equilibrium stability in the positive c-d plane}
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