= Solution
At a <fixed point> $y=x$, and the first equation factors. The branches are
$$
\boxed{x=y=a^2\quad(a\in\mathbb R),\qquad x=y=\pm\sqrt a\quad(a\ge0).}
$$
The three branches meet at $a=0$, and the positive square-root branch meets the quadratic branch at $a=1$. The <Jacobian matrix> at a general <fixed point> is
$$
J=\begin{pmatrix}-2x(a^2-x)&x^2-a\\1&-1\end{pmatrix}.
$$
On $x=a^2$, its trace is $-1$ and <determinant> $a-a^4=a(1-a^3)$: the branch is attracting for $0<a<1$ and a saddle for $a<0$ or $a>1$. On $x=\pm\sqrt a$, the <eigenvalues> are $-1$ and $2a(1\mp a^{3/2})$. The negative branch is always a saddle for $a>0$; the positive branch is a saddle for $0<a<1$ and attracting for $a>1$. At the bifurcation values the zero <eigenvalue> requires the nonlinear analysis below.
For $a=1+\mu$, $u=x-1$, $v=y-x$, the equations are
$$
\dot u=(\mu-2u-u^2)(2\mu-u-v+\mu^2),\qquad\dot v=-v-\dot u,\qquad\dot\mu=0.
$$
The extended <centre manifold> has $v=h(u,\mu)$ with no linear terms. Since $h_u\dot u$ has degree at least three, its invariance equation to degree two is $0=-h_2-(2\mu^2-5\mu u+2u^2)$. Thus
$$
\boxed{v=-2\mu^2+5\mu u-2u^2+O((|u|+|\mu|)^3),\qquad\dot u=2u^2-5\mu u+2\mu^2+O(3).}
$$
The reduced quadratic factors as $(u-2\mu)(2u-\mu)$. Its two roots have reduced <eigenvalues> $3\mu$ and $-3\mu$, respectively, and the transverse <eigenvalue> remains negative. They exchange <stability> at $\mu=0$, a <transcritical bifurcation>. The exact branches have $u=2\mu+\mu^2$ and $u=\mu/2-\mu^2/8+\cdots$, consistent with this second-order evolution equation.
Near $a=0$, the slow equation has leading form $\dot x=x^3-ax$, so this is a <subcritical pitchfork bifurcation> with the parameter orientation displayed. For $a<0$ there is one nearby saddle, whose unstable direction is close to $y=x$ and stable direction close to vertical. For $a>0$ the central branch is attracting and the outer two equilibria near $\pm\sqrt a$ are saddles; their separatrices distinguish attraction toward the central equilibrium from departure outwards. The actual central location is $(a^2,a^2)$, not exactly the origin away from $a=0$.
Adding positive $\varepsilon$ breaks the pitchfork symmetry. The leading equilibrium equation near zero is $x^3-ax+\varepsilon=0$: the diagram unfolds into a single branch plus a saddle-node pair, with the latter appearing for positive $a$ of order $\varepsilon^{2/3}$. More precisely the fold is at $x=(\varepsilon/2)^{1/3}$, $a=3(\varepsilon/2)^{2/3}$ at leading order. Near one the reduced equation becomes $2u^2-5\mu u+2\mu^2+\varepsilon=0$. Its discriminant is $9\mu^2-8\varepsilon$; the crossing is replaced by \b[two saddle-node endpoints with a no-equilibrium gap] for $|\mu|<\sqrt{8\varepsilon}/3$ to leading order. Outside that gap, the lower reduced root is stable and the upper one unstable. These local approximations describe the sketch; the exact perturbed curves have higher-order corrections.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-1-bifurcations.png]
{title=Fixed-point branches, local phase portraits and the positive-forcing unfoldings of the pitchfork and transcritical bifurcations}
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-1-phase-portraits.png]
{title=Nearby planar flows on either side of the subcritical pitchfork, with exact equilibrium locations}
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