Onset of intermittent octahedral patterns in spherical B\'enard convection

The onset of convection for spherically invariant Rayleigh-B\'enard fluid flow is driven by marginal modes associated with spherical harmonics of a certain degree $\ell$, which depends upon the aspect ratio of the spherical shell. At certain critical values of the aspect ratio, marginal modes of degrees $\ell$ and $\ell+1$ coexist. Initially motivated by an experiment of electrophoretic convection between two concentric spheres carried in the International Space Station (GeoFlow project), we analyze the occurrence of intermittent dynamics near bifurcation in the case when marginal modes with $\ell=3, 4$ interact. The situation is by far more complex than in the well studied $\ell=1, 2$ mode interaction, however we show that heteroclinic cycles connecting equilibria with octahedral as well as axial symmetry can exist near bifurcation under certain conditions. Numerical simulations and continuation (using the software AUTO) on the center manifold help understanding these scenarios and show that the dynamics in these cases exhibits intermittent behaviour, even though the heteroclinic cycles may not be asymptotically stable in the usual sense.