  In this paper we present various convective states of zero-Prandtl number
Rayleigh-B\'enard convection using direct numerical simulations (DNS) and a
27-mode low-dimensional model containing the energetic modes of DNS. The origin
of these convective states have been explained using bifurcation analysis. The
system is chaotic at the onset itself with three coexisting chaotic attractors
that are born at two codimension-2 bifurcation points. One of the bifurcation
points with a single zero eigenvalue and a complex pair $(0,\pm i \omega)$
generates chaotic attractors and associated periodic, quasiperiodic, and
phase-locked states that are related to the wavy rolls observed in experiments
and simulations. The frequency of the wavy rolls are in general agreement with
$\omega$ of the above eigenvalue of the stability matrix. The other bifurcation
point with a double zero eigenvalue produces the other set of chaotic
attractors and ordered states such as squares, asymmetric squares, oscillating
asymmetric squares, relaxation oscillations with intermediate squares, some of
which are common to the 13-mode model of Pal et al.(2009).
