Three steps to accelerated expansion
  We study the dynamics of a non-minimally coupled scalar field cosmology with
a potential function. We use the framework of dynamical systems theory to
investigate all evolutional paths admissible for all initial conditions.
Additionally, we assume the presence of barotropic matter and show that the
dynamics can be formulated in terms of an autonomous dynamical system. We have
found fixed points corresponding to three main stages of the evolution of the
universe, namely, radiation, matter and quintessence domination epochs. Using
the linearization of the dynamical systems in the vicinity of the critical
points we explicitly obtain formulas determining the effective equation of
state parameter for the universe at different epochs. In our approach the form
of $w(z)$ parametrisation is derived directly from the dynamical equations
rather than postulated {\it a priori}.
