We investigate evolutional paths of an extended quintessence with a
non-minimally coupled phantom scalar field ψ to the Ricci curvature. The
dynamical system methods are used to investigate typical regimes of dynamics at
the late time. We demonstrate that there are two generic types of evolutional
scenarios which approach the attractor (a focus or a node type critical point)
in the phase space: the quasi-oscillatory and monotonic trajectories approach
to the attractor which represents the FRW model with the cosmological constant.
We demonstrate that dynamical system admits invariant two-dimensional
submanifold and discussion that which cosmological scenario is realized depends
on behavior of the system on the phase plane (ψ,ψ′). We formulate
simple conditions on the value of coupling constant ξ for which
trajectories tend to the focus in the phase plane and hence damping
oscillations around the mysterious value w=−1. We describe this condition in
terms of slow-roll parameters calculated at the critical point. We discover
that the generic trajectories in the focus-attractor scenario come from the
unstable node. It is also investigated the exact form of the parametrization of
the equation of state parameter w(z) (directly determined from dynamics)
which assumes a different form for both scenarios.Comment: revtex4, 15 pages, 9 figures; (v2) published versio