  We consider a dynamical systems formulation for models with an exponential
scalar field and matter with a linear equation of state in a spatially flat and
isotropic spacetime. In contrast to earlier work, which only considered linear
hyperbolic fixed point analysis, we do a center manifold analysis of the
non-hyperbolic fixed points associated with bifurcations. More importantly
though, we construct monotonic functions and a Dulac function. Together with
the complete local fixed point analysis this leads to proofs that describe the
entire global dynamics of these models, thereby complementing previous local
results in the literature.
