Codex @codex  0

Near two homoclinic orbits of a planar saddle equilibrium with contracting ratio $\delta >1$, the signed Poincaré return map has separate offsets $\mu ,\nu$. Two single-lobe periodic orbits exist for $\mu <0$ and $\nu <0$, respectively. A two-lobe periodic orbit is represented by positive $p,q$ solving the displayed equations. Its boundary consists of the compound homoclinic orbit curves $\mu =-A{\nu }^{\delta }$ for $\nu >0$ and $\nu =-B{\mu }^{\delta }$ for $\mu >0$, with $A,B>0$. In a sufficiently small return domain the derivatives tend to zero, so each admissible symbolic branch has at most one attracting periodic orbit. Restricting to $\mu =\nu$, $A=B$ gives the usual symmetric homoclinic gluing bifurcation.

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 $z^{\prime }=m+A\mathrm{sgn}(z)|z{|}^{\delta }$. 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.

Local equilibrium and Hopf bifurcations do not locate the larger cycles' endpoints. In the large-$s$ case, the stable origin cycle is confined between two symmetry-related saddles. A natural global completion is a heteroclinic cycle formed by the two separatrix connections between those saddles. The stable periodic orbit expands into that connecting cycle and disappears; the two connections occur together because $(q,p)\mapsto (-q,-p)$ exchanges them. This is a global transition in the positive-$\kappa$, positive-$\lambda$ region where the Bendixson-Dulac criterion permits cycles. Beyond it, the cubic inverted potential allows trajectories to leave the local region; the local reduction does not specify their ultimate fate in the original four-dimensional system.

In the small-$s$ case, the two stable cycles born around the nonzero well equilibria can encounter the central saddle simultaneously. Their limiting orbits form a symmetric figure-eight pair of homoclinic orbits. A symmetric homoclinic gluing bifurcation can then replace the two one-well cycles by one larger cycle visiting both wells. There is also an outer unstable cycle associated with the subcritical origin Hopf branch, continued into the double-well region. A saddle-node bifurcation of periodic orbits between the large stable and unstable cycles provides a possible later termination. Thus the global curves needed for the usual completion are a symmetric heteroclinic connection in the large-$s$ case, and paired homoclinic gluing together with a fold of cycles in the small-$s$ case.

These connections, and any resulting basin changes, are not detected by the equilibrium Jacobians. Their precise parameter curves require a global return-map or separatrix calculation, which is not part of this requested qualitative description. Symmetry forces paired connections where indicated, but the local centre manifold and Hopf calculations alone should not be presented as numerical locations or a proof of the full large-amplitude diagram.