Dark energy from a quintessence (phantom) field rolling near a potential minimum (maximum)

We examine dark-energy models in which a quintessence or a phantom field, , rolls near the vicinity of a local minimum or maximum, respectively, of its potential V (). Under the approximation that (1/V) (dV/d) 1, although (1/V) (d^2V/d^2) can be large, we derive a general expression for the equation-of-state parameter w as a function of the scale factor for these models. The dynamics of the field depends on the value of (1/V) (d^2V/d^2) near the extremum, which describes the potential curvature. For quintessence models, when (1/V) (d^2V/d^2) 3/4, w (a) has oscillatory behavior. For phantom fields, the dividing line between these two types of behavior is at (1/V) (d^2V/d^2) =-3/4. Our analytical expressions agree within 1% with the exact (numerically derived) behavior, for all of the particular cases examined, for both quintessence and phantom fields. We present observational constraints on these models.