Abstract A coupled global climate–Antarctic ice sheet model is run for 10 million years across the Eocene–Oligocene boundary ∼34 Ma. The model simulates a rapid transition from very little ice to a large continental ice sheet, forced by a gradual decline of atmospheric CO 2 and higher-frequency orbital forcing. The structure of the transition is explained in terms of height mass balance feedback (HMBF) inherent in the intersection of the ice-sheet surface with the climatic pattern of net annual accumulation minus ablation, as found in earlier simple ice sheet models. Hysteresis effects are explored by running the model in reverse, starting with a full ice sheet and gradually increasing CO 2. The effects of higher-frequency orbital forcing on the non-linear transitions are examined in simulations with and without orbital variability. Similar effects are demonstrated with a much simpler one-dimensional ice-sheet flowline model with idealized bedrock topography and parameterized mass balance forcing. It is suggested that non-linear Antarctic ice-sheet transitions and hysteresis have played important roles in many of the observed fluctuations in marine δ 18O records since 34 Ma, and that the range of atmospheric CO 2 variability needed to induce these transitions in the presence of orbital forcing is ∼2× to 4× pre-industrial level.