It's a dark, dark world: Background evolution of interacting $\phi$CDM models beyond simple exponential potentials

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.