Numerical experiments are conducted on a synthetic topography with a three-dimensional thermomechanically coupled ice-sheet model, the Parallel Ice Sheet Model (PISM). Within the model, combined stress balances are connected to evolving thermodynamics and hydrology. The sensitivity of cyclic behaviour to changes in sliding-law parameters and the climate input is studied. Multiple types of oscillations were found, with strong variations in both amplitude and frequency. A physical description is given, in which these variations and transitions from one oscillation type to another are linked to the interplay of stresses, heat transport and hydrological processes. High-frequency oscillations (period 114-169 years), which are shown to have a major impact on ice velocities and a small effect on the ice volume, are related to variations in the water distribution at the base. Low-frequency cycles (period 1000+ years), which have a major impact on both velocities and ice volume, are linked to changes in the thermal regime. Oscillation characteristics are shown to be strongly sensitive to changes in sliding-law parameters and the prescribed surface temperature and mass balance. Incorporating a surface-height dependence of the mass balance is shown to provide an additional feedback, which may induce long- period oscillations.