= Solution
The fixed-point branches are $x=0$ and $x=\pm\sqrt{3-6/a}$ beginning at $a=2$. Zero is a saddle before that threshold and repelling afterwards. The nonzero branches are saddles until $a=3$, attracting for $3<a<7/2$, and repelling beyond their resonant unit-circle crossing. Solid portions in the diagram mark attracting states; dashed portions mark saddles or repellers.
At $a=3$, the two asymmetric <two-cycles> split from $x=\pm1$ with components $[\pm\sqrt{3+3/a}\pm\sqrt{3-9/a}]/2$, pairing the two values with the same first sign. These branches are always unstable. In addition, the antipodal <two-cycle> occupies the constant component levels $x=\pm\sqrt3$ for every $a>0$, attracting only below $a=1/2$. It must be included for the actual printed map, even though the quadratic proposed in the previous part misses it. At a given parameter the two component levels represent one cycle, rather than two independent <fixed points>.
\Image[/past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2009-map-bifurcation.png]
{title=Fixed-point branches and both families of period-two cycles for the printed two-dimensional map}
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