  We study the background cosmological dynamics with a three component source
content: a radiation fluid, a barotropic fluid to mimic the matter sector and a
single scalar field which can act as dark energy giving rise to the late-time
accelerated phase. Using the well-known dimensionless variables, we cast the
dynamical equations into an autonomous system of ordinary differential
equations (ASODE), which are studied by computing the fixed points and the
conditions for their stability. The matter fluid and the scalar field are taken
to be uncoupled at first and later, we consider a coupling between the two of
the form $Q = \sqrt{2/3}\kappa\beta\rho_m\dot{\phi}$ where $\rho_m$ is the
barotropic fluid density. The key point of our analysis is that for the closure
of ASODE, we only demand that the jerk, $\Gamma = V V"/V'^2$ is a function of
acceleration, $z = - M_p V'/ V$, that is, $\Gamma = 1+ f(z)$. In this way, we
are able to accommodate a large class of potentials that goes beyond the simple
exponential potentials. The analysis is completely generic and
\emph{independent} of the form of the potential for the scalar field. As an
illustration and confirmation of the analysis, we consider $f(z)$ of the forms
$\mu/z^2$, $\mu/z$, $(\mu-z)/z^2$ and $(\mu-z)$ to numerically compute the
evolution of cosmological parameters with and without coupling. Implications of
the approach and the results are discussed.
