  We consider cosmological models of Bianchi type. In particular, we are
interested in the alpha-limit dynamics near the Kasner circle of equilibria for
Bianchi classes VIII and IX. They correspond to cosmological models close to
the big-bang singularity.
  We prove the existence of a codimension-one family of solutions that limit,
for t to negative infinity, onto a heteroclinic 3-cycle to the Kasner circle of
equilibria. The theory extends to arbitrary heteroclinic chains that are
uniformly bounded away from the three critical Taub points on the Kasner
circle, in particular to all closed heteroclinic cycles of the Kasner map.

General Relativity and Quantum Cosmology;Dynamical Systems