  In this letter we apply dynamical system methods to study all evolutional
paths admissible for all initial conditions of the FRW cosmological model with
a non-minimally coupled to gravity scalar field and a barotropic fluid. We
choose "energy variables" as phase variables. We reduce dynamics to a
3-dimensional dynamical system for an arbitrary potential of the scalar field
in the phase space variables. After postulating the potential parameter
$\Gamma$ as a function of $\lambda$ (defined as $-V'/V$) we reduce whole
dynamics to a 3-dimensional dynamical system and study evolutional paths
leading to current accelerating expansion. If we restrict the form of the
potential then we will obtain a 2-dimensional dynamical system. We use the
dynamical system approach to find a new generic quintessence scenario of
approaching to the de Sitter attractor which appears only for the case of
non-vanishing coupling constant.
