Title: Bifurcation analysis of steady Rayleigh--B\'enard convection in a cubical cavity with conducting sidewalls Authors: D. Puigjaner (1), J. Herrero (2), C. Sim\'o (3) and F. Giralt (2) (1) Dept. Eng. Inform\`atica i Matem\`atiques, Univ. Rovira i Virgili, Tarragona, Catalunya, Spain. (2) Dept. Enginyeria Quimica, Univ. Rovira i Virgili, Tarragona, Catalunya, Spain. (3) Dept. Matem\`atica Aplicada i An\`alisi, Univ. de Barcelona, Barcelona, Catalunya, Spain. E-mail: dpuigja@etse.urv.es, joan.herrero@urv.cat, carles@maia.ub.es, fgiralt@urv.cat Abstract Natural convection in a cubical cavity heated from below with perfectly conducting sidewalls has been investigated numerically. A parameter continuationprocedure based on a Galerkin spectral method has been applied to obtain the bifurcation diagrams for steady flow solutions over the region Rayleigh<1.5E5 at Prandtl numbers Pr=0.71 and 130. In both cases, the bifurcation diagrams are more complex than those previously reported for adiabatic sidewalls. Four and nine different convective solutions (without taking into account the solutions obtained by symmetry) that are stable over certain ranges of Ra have been respectively identified at Pr=0.71 and 130. The dependence of the bifurcation diagrams and the topology of the flow patterns on the Prandtl number is also stronger in the case of conducting sidewalls. Most of the flow patterns investigated tend to adopt double toroid-like topologies, as Ra is increased. This is especially noticeable at Pr=130, where all flow patterns adopt double-toroid shapes that are superimposed to the characteristic topologies adopted at values of Ra slightly above the respective bifurcation points where they originate. At sufficiently high Ra the double-toroid pattern configuration prevails. This phenomenon, not observed in the case of adiabatic lateral walls, is related to the thermal activity of the sidewalls, which locally extract/supply relatively large amounts of heat from/into the fluid. Current predictions are consistent with experimental flow transitions and topologies reported in the literature. In addition, a complete bifurcation study in the two dimensional (Ra,Pr) plane has been carried out for a particular flow pattern that is stable at both Pr=0.71 and 130. Since the surface of Nusselt over the (Ra,Pr) plane presents several folds and cusps, different regions can be identified as a function of the number of particular realisations of the flow pattern, varying between zero and five. Three different regions of stability have been found for this particular flow pattern in the (Ra,Pr) plane within the range of parameters investigated, i.e., Ra<1.5E5 and 0.71