Chaos in Preinflationary Friedmann-Robertson-Walker Universes
  The dynamics of a preinflacionary phase of the universe, and its exit to inflation, is discussed. This phase is modeled by a closed Friedmann-Robertson-Walker geometry, the matter content of which is radiation plus a scalar field minimally coupled to the gravitational field. The simple configuration, with two effective degres of freedom only, presents a very complicated dynamics connected to the existence of critical points of saddle-center type and saddle type in phase space of the system. Each of these critical points is associated to an extremum of the scalar field potential. The Topology of the phase space about the saddle-center is characterized by homoclinic cylinders emanating from unstable periodic orbits, and the transversal crossing of the cylinders, due to the non-integrability of the system, results in a chaotic dynamics. The topology of the homoclinic cylinders provides an invariant characterization of chaos. The model exhibits one or more exits to inflation, associated to one or more strong asymptotic de Sitter attractors present in phase space, but the way out from the initial singularity into any of the inflationary exits is chaotic. We discuss possible mechanisms, connected to the spectrum of inhomogeneous fluctuations in the models, which would allow us to distinguish physically the several exits to inflation. 
G. A. Monerat, H. P. de Oliveira and Ivano D. Soares