Dark energy from a quintessence (phantom) field rolling near potential
  minimum (maximum)
  We examine dark energy models in which a quintessence or a phantom field,
$\phi$, rolls near the vicinity of a local minimum or maximum, respectively, of
its potential $V(\phi)$. Under the approximation that $(1/V)(dV/d\phi) \ll 1$,
[although $(1/V)(d^2 V/d\phi^2)$ 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 $(1/V)(d^2
V/d\phi^2)$ near the extremum, which describes the potential curvature. For
quintessence models, when $(1/V)(d^2 V/d\phi^2)<3/4$ at the potential minimum,
the equation of state parameter $w(a)$ evolves monotonically, while for
$(1/V)(d^2 V/d\phi^2)>3/4$, $w(a)$ has oscillatory behavior. For phantom
fields, the dividing line between these two types of behavior is at $(1/V)(d^2
V/d\phi^2) = -3/4$. 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.
