= Solution
Here $\mu>\mu_H=\lambda(\lambda-1)>0$. The origin is a <saddle equilibrium>, $P_+$ is repelling, and $P_-$ lies in $u<0$. Thus \b[there is no <saddle equilibrium> in the interior of $u>0$] to which this <periodic orbit> could become homoclinic. The origin cannot supply the missing loop: the line $u=0$ is invariant and is precisely its local <stable manifold>. An orbit in $u>0$ cannot lie on that <stable manifold>, so the positive <periodic orbit> cannot close into a <homoclinic orbit> to the origin either.
The outgoing trajectories do not escape to infinity in this regime. To see the large-amplitude turning mechanism, put $w=v/u$ once $v\ge0$. Direct differentiation gives
$$
\dot w=u(w^2-w+\lambda)-(\lambda+\mu)w,\qquad
\dot u=u[\mu+(1-w)u].
$$
Since $\lambda>1$, the quadratic $w^2-w+\lambda$ is uniformly positive. At large $u$ while $0\le w\le1$, $\dot w$ is bounded below by a positive multiple of $u$, whereas $\dot u/u\le\mu+u$. Thus $w$ rises by a fixed amount per bounded increase in $\log u$: the trajectory reaches $w>1$ and then $w>1+\mu/u$, where $\dot u<0$, before $u$ can diverge. The same estimate bounds repeated outgoing excursions; large positive $v$ also decreases when $v>u^2$. Initially negative $v$ increases until it becomes nonnegative. These nullcline and ratio estimates give bounded forward excursions.
The repelling upper <equilibrium> is therefore surrounded by recurrent trajectories, and the Poincaré-Bendixson alternative supplies periodic motion rather than a <saddle equilibrium> connection. The supercritical attracting <periodic orbit> persists in the displayed regime, with trajectories spiralling away from the upper <equilibrium> and approaching it. The last panel of the portrait figure shows this behavior; the boundary <stable manifold> of the origin is kept separate.
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