Cosmological dynamics of non-minimally coupled scalar field system and
  its late time cosmic relevance
  We investigate the cosmological dynamics of non-minimally coupled scalar
field system described by $F(\phi)R$ coupling with
$F(\phi)=(1-\xi\phi^N)R$($N\ge2$) and the field potential, $V(\phi)=V_0\phi^n$.
We use a generic set of dynamical variables to bring out new asymptotic regimes
of the underlying dynamics. However, our dynamical variables miss the most
important fixed point$-$ the de Sitter solution. We make use of the original
form of system of equations to investigate the issues related to this important
solution. In particular, we show that the de-Sitter solution which is a
dynamical attractor of the system lies in the region of negative effective
gravitational constant $G_N$ thereby leading to a ghost dominated universe in
future and a transient quintessence(phantom) phase with $G_N>0 $ around the
present epoch (however, as demonstrated by Starobinsky in 1981, the ghost
dominated universe, if exists, can not be accessed from the Universe we live
in, we shall say more about this important result in the last section). We also
carry out comparison of the model with other competing models of dark energy
such as galileon modified gravity and others.
