= Solution
Use $y$ vertically and $\mu$ horizontally in the <bifurcation diagram>; the other coordinate is recovered from $x=\mu y$. The origin is the horizontal branch $y=0$. Draw it solid on $-1<\mu<0$, where it is <asymptotically stable>, and dashed on the remaining <hyperbolic> intervals, where it is unstable. Its three <nonhyperbolic fixed points> in parameter space are marked at $\mu=-1,0,1$; open markers distinguish these boundary cases from the strict <linear stability> classifications on the intervening intervals.
The nonzero <fixed point> branches are $y=\pm\sqrt{\mu(\mu-1)}$ for $\mu<0$ and $\mu>1$. Each is dashed because it is a <saddle fixed point of a map>, as proved in part (c). Near zero their shape is $y\sim\pm\sqrt{-\mu}$: the unstable branches lie on the same parameter side as the attracting origin, displaying the <subcritical pitchfork bifurcation>. The branches issuing from $\mu=1$ have $y\sim\pm\sqrt{\mu-1}$. At $\mu=-1$ no additional <fixed point> branch appears; the <Floquet multipliers> cross the <unit circle> at $\pm i$.
\Image[/past-exam-of-the-nonlinear-systems-course-of-the-university-of-oxford/2023-bifurcation.png]
{title=Fixed-point bifurcation diagram: solid denotes attracting, dashed denotes unstable, and open circles mark the three unit-circle crossings at the origin. Nonzero branches are saddles.}
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