Existence of a second heteroclinic bifurcation of PO6 from FP2

Ascertain whether the periodic orbit PO6 in the vertical convection system undergoes another heteroclinic bifurcation from the equilibrium FP2 for Rayleigh numbers slightly above Ra=6618.12, by establishing the existence of a heteroclinic cycle linking FP2 and its symmetry-related copy and determining the Rayleigh number at which the period diverges.

Background

PO6 is a pre-periodic orbit with spatio-temporal symmetry that, when continued backward in Rayleigh number, approaches a heteroclinic cycle involving two symmetry-related copies of FP2.

When continued forward, the period increases and the trajectory spends time near FP2, suggesting another heteroclinic connection slightly beyond Ra=6618.12, but this could not be confirmed due to computational limitations.

References

We suggest that PO6 may bifurcate again from FP2 in another heteroclinic bifurcation slightly beyond $Ra=6618.12$. However, since the continuation becomes computationally difficult for higher Rayleigh numbers, we have not been able to confirm this.

— 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 4.2.1 (PO6: saddle-node and global bifurcations)