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Termination of PO9 branch

Determine how the periodic orbit PO9 branch terminates, identifying the precise bifurcation or invariant solution encountered at its unresolved endpoint under Rayleigh-number continuation.

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Background

PO9 has no spatial symmetry and exhibits saddle-node bifurcations; only one endpoint was continued to Ra≈6635.

The figure caption explicitly notes that the termination of PO9 is unclear, motivating a targeted bifurcation analysis to resolve the endpoint.

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)