Homoclinic chaos in axisymmetric Bianchi IX cosmologies reexamined

The dynamics of Bianchi IX models with two scale factors A(t) and B(t) is reexamined. The matter content of the model is assumed to be comoving dust plus a positive cosmological constant. We make a complete numerical construction of topological structures present in the phase space of the model, as the center manifold of periodic orbits and the stable and unstable cylinders of orbits that emerge from the center manifold, associated to a saddle-center critical point. We exhibit the homoclinic intersections of the unstable and stable cylinders that produce homoclinic chaos in the models. A=0 is not a singularity of the Hamiltonian dynamics and any truncation of the dynamics at A=0 is bound to a loss of information. We make an explicit construction of a set with a horseshoe structure in the A>0 region, of initial conditions corresponding to initially expanding universes. We show that the dynamics of this set is chaotic even if restricted to the region A>0 and has the Einstein universe configuration as a chaotic scatterer. The result however demands the information of the Hamiltonian dynamics in whole phase space, including the region A<0. This refutes recent claims that no homoclinic chaos is present in the dynamics of the model