Superinflation, quintessence, and the avoidance of the initial   singularity
  We consider the dynamics of a spatially flat universe dominated by a self-interacting nonminimally coupled scalar field. The structure of the phase space and complete phase portraits for the conformal coupling case are given. It is shown that the non-minimal coupling modifies drastically the dynamics of the universe. New cosmological behaviors are identified, including superinflation ($\dot{H}>0$), avoidance of big bang singularities through classical birth of the universe from empty Minkowski space, and spontaneous entry into and exit from inflation. The relevance of this model to the description of quintessence is discussed. 
A. Saa, E. Gunzig, L. Brenig, V. Faraoni, T.M. Rocha Filho, and A.   Figueiredo