- Glacial–interglacial oscillations are often described by simple conceptual models. Relatively few models, however, are accompanied by analytical solutions, though detailed analytical investigation of climate models often leads to deeper understanding of the climate system. Here we study a simple conceptual model for glacial dynamics, a simplified version of the sea-ice-switch mechanism of Gildor and Tziperman (Paleoceanography 15:605–615, 2000), and provide a detailed analytical treatment for this model. We show that when the model is forced by a simplified insolation forcing it exhibits rich dynamics and passes through a series of bifurcations before being completely phase-locked to the insolation forcing. Our model suggests that even when the glacial cycles are self-sustained, insolation forcing has a major role on the complexity of glacial cycles: (1) it is possible to obtain glacial–interglacial oscillations for a wider parameters range when the amplitude of the insolation forcing is larger; (2) in addition, the ice-volume becomes more periodic; (3) when the period of the ice-volume is minimal the ice-volume is symmetric and for larger period is more asymmetric; (4) the ice-volume can be either periodic, higher order periodic, or quasi-periodic.