We use dynamical systems methods to study quintessence models in a spatially flat and isotropic spacetime with matter and a scalar field with potentials for which lambda(v) = -V,v/V is bounded, thereby going beyond the exponential potential for which lambda(v) is constant. The scalar field equation of state parameter wv plays a central role when comparing quintessence models with observations, but with the dynamical systems used to date wv is an indeterminate, discontinuous, function on the state space in the asymptotically matter dominated regime. Our first main result is the introduction of new variables that lead to a regular dynamical system on a bounded three-dimensional state space on which wv is a regular function. The solution trajectories in the state space then provide a visualization of different types of quintessence evolution, and how initial conditions affect the transition between the matter and scalar field dominated epochs; this is complemented by graphs wv(N), where N is the e-fold time, which enables characterizing different types of quintessence evolution.