The aim of the paper is to examine the nonlinear dynamics of a truncated system modelling low-Prandtl-number thermal convection. The model describes situations where the primary flow is not a straight roll and the dynamics is dominated by the production of axial flow along the axis of bent rolls or of swirl along ring vortices. The physical mechanism for these processes is a spontaneous growth (i.e. bifurcation) of a vertical vorticity mode, breaking the two-dimensional symmetry of the system. A description of the model can be found in Massaguer & Mercader (1988) where the physics and the numerical results have been checked against laboratory experiments. The nonlinear dynamics of that model will be discussed in the more academic case of free boundaries, as it has been shown that for sufficiently small Prandtl numbers straight rolls cannot be expected to be the primary flow near the onset of convection (Busse & Bolton 1984). Two clearly differentiated time-dependent regimes have been found and they correspond to small and intermediate Péclet numbers. In the former regime there exists a transition to chaos with the whole scenario being dependent on a symmetry invariance common to a large number of confined flows. The route to chaos is made up of a sequence of homoclinic explosions giving rise to a cascade of period doublings, with the whole scenario being different from that of a Feigenbaum's cascade.