Chaotic exit to inflation: the dynamics of pre-inflationary universes

We show that anisotropic Bianchi type-IX models, with matter and cosmological constant have chaotic dynamics, connected to the presence of a saddle-center in phase space. The topology of cylinders emanating from unstable periodic orbits about the saddle-center provides an invariant characterization of chaos in the models. The model can be thought to describe the early stages of inflation, the way out to inflation being chaotic.