= Solution
We count <periodic orbits> whose trajectories remain near the two specified <homoclinic orbits>; the local hypotheses impose no bound on unrelated distant cycles. Choose the return domain small enough that $A\delta|x|^{\delta-1}$ and $B\delta|x|^{\delta-1}$ are less than one. Each admissible return branch is then a <contraction mapping>.
A right-lobe <periodic orbit> is a <fixed point> $(p,+1)$ with $p>0$ and $p=-\mu+Ap^\delta$. For small amplitudes it exists exactly when $\mu<0$, and $p=-\mu+o(|\mu|)$. Similarly, a left-lobe <periodic orbit> has $(x,y)=(-q,-1)$, $q>0$, $q=-\nu+Bq^\delta$, and exists for $\nu<0$. Their <Floquet multipliers> tend to zero, so both are attracting.
A <period-two orbit> of the section map corresponds to one flow <periodic orbit> traversing both lobes. Write its states as $(p,-1)$ and $(-q,+1)$. Its positive amplitudes satisfy
$$
\boxed{q=\mu+Ap^\delta,\qquad p=\nu+Bq^\delta.}
$$
The two-step derivative is $AB\delta^2p^{\delta-1}q^{\delta-1}$, again tending to zero. The <asymmetric planar gluing return map> therefore gives at most one such attracting two-lobe cycle in the small domain. Its boundaries are obtained by setting one amplitude to zero:
$$
\boxed{C_R:\ \mu=-A\nu^\delta\ (\nu>0),\qquad
C_L:\ \nu=-B\mu^\delta\ (\mu>0).}
$$
For the original flow these are leading equations, with corrections $o(\nu^\delta)$ or $o(\mu^\delta)$. On $C_R$, the negative unstable branch makes both excursions before returning to the positive stable branch; on $C_L$ the positive unstable branch returns to the negative stable branch. Each is a compound <homoclinic orbit>, with one extra passage near the <saddle equilibrium>. They are distinct from the primary loop curves $\mu=0$ and $\nu=0$.
The six open regions and their local cycles are:
|| Region
|| Attracting flow cycles
| $\mu<0,\ \nu<0$
| Right and left single-lobe cycles
| $\nu>0,\ \mu<-A\nu^\delta$
| Right single-lobe cycle only
| $\nu>0,\ -A\nu^\delta<\mu<0$
| Right single-lobe cycle and two-lobe cycle
| $\mu>0,\ \nu<-B\mu^\delta$
| Left single-lobe cycle only
| $\mu>0,\ -B\mu^\delta<\nu<0$
| Left single-lobe cycle and two-lobe cycle
| $\mu>0,\ \nu>0$
| Two-lobe cycle only
There is no additional small alternating cycle when both parameters are negative. Indeed the two positive-amplitude equations would imply $q<Ap^\delta$ and $p<Bq^\delta$, giving $p<BA^\delta p^{\delta^2}$, impossible for sufficiently small positive $p$. On each mixed-sign quadrant the contraction equations give the two-lobe cycle precisely on the side of the compound curve toward the positive quadrant.
On $\mu=0$, there is the primary right <homoclinic orbit>, together with a left cycle if $\nu<0$ or a two-lobe cycle if $\nu>0$. On $\nu=0$ the analogous statement holds with left and right interchanged. On $C_R$ there is a compound <homoclinic orbit> and the right single-lobe cycle; on $C_L$ there is a compound <homoclinic orbit> and the left single-lobe cycle. At $(0,0)$ the two primary loops meet at the same <saddle equilibrium>. The approach to each <homoclinic orbit> has a diverging period; it is not a saddle-node collision of two finite-period cycles.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-59-gluing-regions.png]
{title=Parameter regions and schematic phase portraits for asymmetric planar gluing}
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-59-gluing-connections.png]
{title=Primary, compound, and double homoclinic connections on every global bifurcation curve}
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