= Solution
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 \b[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.
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