  A sequence of three steady - oscillatory transitions of buoyancy convection
of air in a laterally heated cube with perfectly thermally insulated horizontal
and spanwise boundaries is studied. The problem is treated by Newton and
Arnoldi methods based on Krylov subspace iteration. The finite volume grid is
gradually refined from 100^3 to 256^3 finite volumes. It is shown that the
primary instability is characterized by two competing eigenmodes, whose
temporal development results in two different oscillatory states that differ by
their symmetries. Bifurcations due to both modes are subcritical. These modes
develop into different oscillatory and then stochastic flow states, which, at
larger Grashof number, stabilize and arrive to single stable steady flow. With
further increase of the Grashof number this flow loses it stability again. It
is argued that in all the three transitions, the instabilities onsets, as well
as reinstatement of stability, take place owing to an interaction between a
destabilizing centrifugal mechanism and stabilizing effect of thermal
stratification.
