= Solution
With reflection symmetry, choose $\mu=\nu=m$ and $A=B$. The two primary <homoclinic orbit> curves are then encountered simultaneously at $m=0$. For $m<0$ the return has two attracting single-lobe cycles; for $m>0$ it has one attracting two-lobe cycle. This is the <symmetric homoclinic gluing bifurcation>: two cycles meet the two saddle loops and are replaced by a cycle traversing both lobes. Since $\delta>1$, the <saddle equilibrium> contracts strongly enough for the nearby cycles to be stable.
One can also use the <signed gluing-map reduction> $z=-xy$. In the symmetric return map it gives $z'=m+A\operatorname{sgn}(z)|z|^\delta$. A negative <fixed point> lifts to two reflection-related flow cycles; a positive <fixed point> lifts to one <period-two orbit> of the two-sheet section map, hence one two-lobe flow cycle. This distinction prevents counting the two section intersections as two different flow cycles.
Without reflection symmetry the two splitting parameters are independent. The primary loops occur separately, and the two compound curves bound small coexistence wedges where a single-lobe and a two-lobe cycle are both stable. Thus the six-region picture is the generic two-parameter unfolding of the symmetric gluing event. The simultaneous encounter of both primary loops is codimension two for a general planar flow but codimension one in the reflection-symmetric family. The contraction argument applies to the small return domain; it makes no assertion that a globally arbitrary $P,Q$ system has only these cycles everywhere.
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