The largest variability in temperature over the last sixty thousand years is connected with Dansgaard-Oeschger events. Various prototype models have been proposed to explain these rapid climate fluctuations, but until now no observational constraint has been forwarded to choose between different theories. We assess the bimodality of the system reconstructing the topology of the multi-dimensional attractor over which the climate system evolves. Furthermore, we show that Dansgaard-Oeschger events are compatible with the crossing of a tipping point in the climate system. We use high-resolution ice core isotope data to investigate the statistical properties of the climate fluctuations in the period before the onset of the abrupt change. We find that the statistics are consistent with the switches between two different climate equilibrium states in response to a changing external forcing.