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Termination of RPO8 and PO9 branches

Determine how the branches of the relative periodic orbit RPO8 and the periodic orbit PO9 terminate under Rayleigh-number continuation in the vertical convection system, including identifying the specific bifurcation or invariant solution encountered at their endpoints.

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Background

Both RPO8 and PO9 undergo saddle-node bifurcations in the studied range; one endpoint was continued to Ra≈6635, but the nature of the other endpoint is unknown.

The figure caption explicitly notes the uncertainty concerning the termination of these branches.

References

Branches RPO5, RPO8 and PO9 undergo saddle-node bifurcations and are continued until $Ra=6635$ for one of their endpoints. At the other endpoints, RPO5 possibly bifurcates from FP4 in a global bifurcation, and the termination of RPO8 and PO9 are unclear.

— Natural convection in a vertical channel. Part 3. Bifurcations of many (additional) unstable equilibria and periodic orbits (2504.05524 - Zheng et al., 7 Apr 2025) in Figure \ref{part3_sepa_BD_PO5_7_8_9_30} caption (Section 4.2.2)