  A large number of flows with distinctive patterns have been observed in
experiments and simulations of Rayleigh-Benard convection in a water-filled
cylinder whose radius is twice the height. We have adapted a time-dependent
pseudospectral code, first, to carry out Newton's method and branch
continuation and, second, to carry out the exponential power method and Arnoldi
iteration to calculate leading eigenpairs and determine the stability of the
steady states. The resulting bifurcation diagram represents a compromise
between the tendency in the bulk towards parallel rolls, and the requirement
imposed by the boundary conditions that primary bifurcations be towards states
whose azimuthal dependence is trigonometric. The diagram contains 17 branches
of stable and unstable steady states. These can be classified geometrically as
roll states containing two, three, and four rolls; axisymmetric patterns with
one or two tori; three-fold symmetric patterns called mercedes, mitubishi,
marigold and cloverleaf; trigonometric patterns called dipole and pizza; and
less symmetric patterns called CO and asymmetric three-rolls. The convective
branches are connected to the conductive state and to each other by 16 primary
and secondary pitchfork bifurcations and turning points. In order to better
understand this complicated bifurcation diagram, we have partitioned it
according to azimuthal symmetry. We have been able to determine the
bifurcation-theoretic origin from the conductive state of all the branches
observed at high Rayleigh number.
