Superinflation, quintessence, and nonsingular cosmologies
  The dynamics of a universe dominated by a self-interacting nonminimally coupled scalar field are considered. The structure of the phase space and complete phase portraits are given. New dynamical behaviors include superinflation ($\dot{H}>0$), avoidance of big bang singularities through classical birth of the universe, and spontaneous entry into and exit from inflation. This model is promising for describing quintessence as a nonminimally coupled scalar field. 
E. Gunzig, A. Saa, L. Brenig, V. Faraoni, T.M. Rocha Filho, and A.   Figueiredo