Axisymmetric Rayleigh–Bénard convection in a cylinder with vertical axis is studied. The nonlinear behaviour is investigated near the onset of convection using an eigenfunction expansion. It is found that initially a steady target pattern develops; the number of rolls depends only on the aspect ratio of the box. At about 5% beyond onset, an oscillatory pattern develops, in which the number of rolls oscillates between two adjacent values. The transitions between the initial steady state and this oscillatory pattern are also investigated, and fall into two main categories. As the Rayleigh number is reduced to the transition point, either the period of the travelling wave tends to infinity, whilst its amplitude stays finite, or there is a sudden transition to a vascillating pattern, the amplitude of which becomes smaller and finally vanishes, whilst the period remains finite. The results are compared with experimental work.