  We perform a comprehensive study of a class of dark energy models - scalar
field models where the effective potential can be described by a polynomial
series - exploring their dynamical behavior using the method of flow equations
that has previously been applied to inflationary models. Using supernova,
baryon oscillation, CMB and Hubble constant data, and an implicit theoretical
prior imposed by the scalar field dynamics, we find that the LCDM model
provides an excellent fit to the data. Constraints on the generic scalar field
potential parameters are presented, along with the reconstructed w(z) histories
consistent with the data and the theoretical prior. We propose and pursue
computationally feasible algorithms to obtain estimates of the principal
components of the equation of state, as well as parameters w_0 and w_a.
Further, we use the Monte Carlo Markov Chain machinery to simulate future data
based on the Joint Dark Energy Mission, Planck and baryon acoustic oscillation
surveys and find that the inverse area figure of merit improves nearly by an
order of magnitude. Therefore, most scalar field models that are currently
consistent with data can be potentially ruled out by future experiments. We
also comment on the classification of dark energy models into "thawing'" and
"freezing" in light of the more diverse evolution histories allowed by this
general class of potentials.
