  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.
