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The equilibria of $\dot{x}=(a^2-x)(a-y^2)$, $\dot{y}=x-y$ lie onfor every $a$, and on $x=y=\pm \sqrt{a}$ for $a\ge 0$. They meet in a subcritical pitchfork bifurcation at $a=0$; the positive square-root branch meets the $a^2$ branch and exchanges stability in a transcritical bifurcation at $a=1$.

Forthe equilibrium branches are $x=0$, $x=\pm \sqrt{2\mu }$, and $x=\pm \sqrt{\mu -a}$ wherever the square roots are real. If $a<0$, pitchforks occur at $\mu =a$ and $\mu =0$, and the two nonzero branch pairs meet in two transcritical bifurcations at $\mu =-a$. If $a=0$, all five branches meet in one degenerate fifth-order bifurcation. If $a>0$, separate pitchforks occur at $\mu =0$ and $\mu =a$. A constant perturbation produces imperfect pitchforks and saddle nodes, so none of these unperturbed bifurcations is structurally stable without the reflection symmetry.

Forthe invariant center manifold near $(x,y,k)=(0,0,0)$ is $y=0$, with reduced equation $\dot{x}=x(-k-3x+x^2)$ and hence a transcritical bifurcation. Near $(x,y,k)=(1,2,0)$, put $X=x-1$ and $v=y-2-X$. The extended center manifold beginsand its reduced equation is $\dot{X}=2(X^2-k)+O(3)$, a saddle-node bifurcation.