We consider forced oscillations of a nonlinear climatic oscillator. The oscillator includes radiation balance, oceanic thermal inertia, a highly simplified hydrological cycle, the mass balance and plastic flow of ice sheets, the elasticity of the earth's crust, and the viscosity of its upper mantle, as well as their various interactions. This system exhibits self-sustained periodic oscillations with amplitude of a few degrees Celsius in the absence of any periodic forcing. The oscillator's free period, depending on model parameters, lies roughly between 5000 yr and 15000 yr (5–15 ka). The model is subjected to forcing at the astronomical periodicities of precession, 19 and 23 ka, obliquity, 41 ka, and eccentricity, 100 ka and 400 ka. The forcing is assumed to act on the climatic system by variations in mean annual insolation, in the case of eccentricity, as well as by its effect on the ice-mass balance through the nonlinear precipitation-temperature feedback.
The effects investigated only cause small changes in ice mass V and global temperature T when self-sustained oscillations are absent. In their presence, nonlinearly resonant response to the forcings leads to large changes in T and V; this response obtains at the frequency of the forcing, due to the mechanism of entrainment. The nonlinear character of the response also leads to combination tones. These are linear combinations of the forcing frequencies with integer coefficients, among which the largest occur near 100 and 10 ka. Sharp peaks in spectral density at the forcing frequencies and at their combination tones are superimposed on a continuous background. The spectral power in the background decreases with increasing frequency, like random red noise. This deterministic aperiodic behavior limits the predictability of glaciation cycles.