A rigorous understanding of the dynamical nature of spacelike singularities remains an open problem in mathematical cosmology. Since the heuristic work of Belinski–Khalatnikov–Lifshitz and Misner's Mixmaster construction, vacuum spatially homogeneous cosmological models are expected to play a key role for generic singularities. We therefore focus on this class of models. The most general cases are the Bianchi type VIII, type IX, and type VI_{-1/9}, each with a four-dimensional Hubble-normalized state space.

On one hand, we embed the types VIII and IX models into modified gravity theories and show that general relativity (GR) arises as a bifurcation point where chaotic dynamics become generic, suggesting a new approximation scheme for GR. On the other hand, we analyze the type VI_{-1/9} oscillatory regime and show that only a subset of its structure is dynamically relevant.