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A large number of flows with distinctive patterns have been observed inexperiments and simulations of Rayleigh-Benard convection in a water-filledcylinder whose radius is twice the height. We have adapted a time-dependentpseudospectral code, first, to carry out Newton's method and branchcontinuation and, second, to carry out the exponential power method and Arnoldiiteration to calculate leading eigenpairs and determine the stability of thesteady states. The resulting bifurcation diagram represents a compromisebetween the tendency in the bulk towards parallel rolls, and the requirementimposed by the boundary conditions that primary bifurcations be towards stateswhose azimuthal dependence is trigonometric. The diagram contains 17 branchesof stable and unstable steady states. These can be classified geometrically asroll states containing two, three, and four rolls; axisymmetric patterns withone or two tori; three-fold symmetric patterns called mercedes, mitubishi,marigold and cloverleaf; trigonometric patterns called dipole and pizza; andless symmetric patterns called CO and asymmetric three-rolls. The convectivebranches are connected to the conductive state and to each other by 16 primaryand secondary pitchfork bifurcations and turning points. In order to betterunderstand this complicated bifurcation diagram, we have partitioned itaccording to azimuthal symmetry. We have been able to determine thebifurcation-theoretic origin from the conductive state of all the branchesobserved at high Rayleigh number.

Extreme multiplicity in cylindrical Rayleigh-Benard convection: II. Bifurcation diagram and symmetry classification