  A class of exact solutions to the Belinski-Khalatnikov-Lifshitz (BKL)
scenario is derived and tested for their stability against small perturbations.
These are the only regular solutions in the Painlev\'{e} sense. We prove that
they are unstable in the vicinity of the cosmological singularity. Regularity
of the dynamics is also examined with the dynamical systems method. Our results
confirm the conjecture of BKL that the dynamics near the singularity becomes
generically chaotic.
