The last deglaciation was characterised by two abrupt warming events, at the start of the Bølling-Allerød and at the end of the Younger Dryas, but their underlying causes are unclear. Some abrupt climate changes may involve gradual forcing past a bifurcation point, in which a prevailing climate state loses its stability and the climate tips into an alternative state, providing an early warning signal in the form of slowing responses to perturbations. However, the abrupt Dansgaard-Oeschger (DO) events during the last ice age were probably triggered by stochastic fluctuations without bifurcation or early warning, and whether the onset of the Bølling-Allerød (DO event 1) was preceded by slowing down or not is debated. Here we show that the interval from the Last Glacial Maximum to the end of the Younger Dryas, as recorded in three Greenland ice cores with two different climate proxies, was accompanied by a robust slowing down in climate dynamics and an increase in climate variability, consistent with approaching bifurcation. Prior to the Bølling warming there was a robust increase in climate variability but no consistent slowing down signal, suggesting this abrupt change was probably triggered by a stochastic fluctuation. The Bølling warming marked a distinct destabilisation of the climate system, which excited an internal mode of variability in Atlantic meridional overturning circulation strength, causing multi-centennial climate fluctuations. There is some evidence for slowing down in the transition to and during the Younger Dryas. We infer that a bifurcation point was finally approached at the end of the Younger Dryas, in which the cold climate state, with weak Atlantic overturning circulation, lost its stability, and the climate tipped irreversibly into a warm interglacial state. The lack of a large triggering perturbation at the end of the Younger Dryas, and the fact that subsequent meltwater perturbations did not cause sustained cooling, support the bifurcation hypothesis.