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.

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)