= Solution
The four nearby portrait types in $u>0$ can be organized by increasing $\mu$ at fixed $\lambda>1/2$ close to $1/2$:
* A: $\mu<-1/4$. There is no interior <equilibrium>; trajectories approach the stable boundary <equilibrium> at the origin.
* B: $-1/4<\mu<\mu_H$. The lower interior <equilibrium> is a <saddle equilibrium> and the upper one a <sink equilibrium>. The <saddle equilibrium>'s <stable manifold> separates trajectories approaching the <sink equilibrium> from those approaching the origin.
* C: $\mu_H<\mu<\mu_G$. The upper <equilibrium> is repelling, enclosed by an attracting <periodic orbit>. The <saddle equilibrium>'s <stable manifold> separates the <periodic orbit>'s basin from the origin's basin.
* D: $\mu>\mu_G$, still in the local neighborhood with $\mu<0$. The <saddle equilibrium> and repelling upper <equilibrium> remain, but the <periodic orbit> has disappeared. Generic trajectories ultimately approach the origin.
The no-cycle region D continues around the double-zero point onto the $\lambda<1/2$ side. The sketches distinguish the boundary origin from the two interior <equilibria> and show the <periodic orbit> only in region C.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-56-quadratic-portraits.png]
{title=Four portraits near the double-zero point and the positive-parameter attracting-cycle regime}
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