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, $\varphi$, rolls near the vicinity of a local minimum or maximum, respectively, of its potential $V(\varphi )$. Under the approximation that $(1\mathrm{/}V)(dV\mathrm{/}d\varphi )\ll 1$, [although $(1\mathrm{/}V)(d^{2}V\mathrm{/}d{\varphi }^2)$ 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 $(1\mathrm{/}V)(d^{2}V\mathrm{/}d{\varphi }^2)<3\mathrm{/}4$ at the potential minimum, the equation of state parameter $w(a)$ evolves monotonically, while for $(1\mathrm{/}V)(d^{2}V\mathrm{/}d{\varphi }^2)>3\mathrm{/}4$, $w(a)$ 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.