  The transition route and bifurcations of the buoyant flow developing on a
heated circular horizontal surface are elaborated using direct numerical
simulations and direct stability analysis. A series of bifurcations, as a
function of Rayleigh numbers (Ra) ranging from $10^1$ to $6\times10^7$, are
found on the route to the chaos of the flow at $Pr=7$. When $Ra<1.0\times10^3$,
the buoyant flow above the heated horizontal surface is dominated by
conduction, because of which distinct thermal boundary layer and plume are not
present. At $Ra=1.1\times10^6$, a Hopf bifurcation occurs, resulting in the
flow transition from a steady state to a periodic puffing state. As Ra
increases further, the flow enters a periodic rotating state at
$Ra=1.9\times10^6$, which is a unique state that was rarely discussed in the
literature. These critical transitions, leaving from a steady state and
subsequently entering a series of periodic states (puffing, rotating, flapping
and doubling) and finally leading to chaos, are diagnosed using spectral
analysis and two-dimensional Fourier Transform (2DFT). Moreover, direct
stability analysis is conducted by introducing random numerical perturbations
into the boundary condition of the surface heating. We find that when the state
of a flow is in the vicinity of bifurcation points (e.g., $Ra=2.0\times10^6$),
the flow is conditionally unstable to perturbations, and it can bifurcate from
the rotating state to the flapping state in advance. However, for relatively
stable flow states, such as at $Ra=1.5\times10^6$, the flow remains its
periodic puffing state even though it is being perturbed.
