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Bifurcation-theoretic origin of equilibrium FP13

Determine the bifurcation-theoretic origin of the equilibrium FP13 in three-dimensional vertical thermal convection governed by the Oberbeck–Boussinesq equations in a periodic box of size [Lx,Ly,Lz]=[1,8,9] with Prandtl number Pr=0.71, including identifying the parent branch or mechanism (e.g., pitchfork, Hopf, or global bifurcation) by which FP13 is created.

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Background

FP13 is an unstable equilibrium solution identified and continued in Rayleigh number within the studied domain. It exhibits Z2 symmetry (combined reflection) and undergoes a saddle-node bifurcation at Ra=6449, persisting at least beyond Ra=6800.

Despite extensive continuation, the authors could not trace FP13 back to a known branch or bifurcation mechanism, leaving its genesis within the bifurcation structure of the vertical convection system unresolved.

References

FP13 is shown in figure \ref{part3_BD-newFP}(k) and exists beyond $Ra=6800$, where we stopped the continuation; its bifurcation-theoretic origin remains 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 Section 3.3 (FP13)