BKL proposal for dynamics near generic spacelike cosmological singularities

Let $\mathcal{S}$ be a generic spacelike initial cosmological singularity, and let $J^+(p)$ denote the causal future of a point $p\in\mathcal{S}$. BKL proposal. Sufficiently close to $\mathcal{S}$, the dynamics are described as follows: the causal future $J^+(p)$ is well described by a nonlinear system of ordinary differential equations whose solutions resemble sequences of stable or unstable Kasner-like regimes; every unstable Kasner regime transitions to a different stable or unstable Kasner regime; for stiff matter models there are finitely many Kasner inversions and the spacetime ultimately approaches a single stable Kasner metric with monotonic dynamics; for vacuum or non-stiff matter there are infinitely many Kasner inversions between unstable Kasner-like regimes in every generic $J^+(p)$. The proposal is a collection of heuristics concerning typical behavior near spacelike singularities and is often called the BKL scenario; the supplied text gives no resolution status.