  We consider the familiar problem of a minimally coupled scalar field with
quadratic potential in flat Friedmann-Lema\^itre-Robertson-Walker cosmology to
illustrate a number of techniques and tools, which can be applied to a wide
range of scalar field potentials and problems in e.g. modified gravity. We
present a global and regular dynamical systems description that yields a global
understanding of the solution space, including asymptotic features. We
introduce dynamical systems techniques such as center manifold expansions and
use Pad\'e approximants to obtain improved approximations for the `attractor
solution' at early times. We also show that future asymptotic behavior is
associated with a limit cycle, which shows that manifest self-similarity is
asymptotically broken toward the future, and give approximate expressions for
this behavior. We then combine these results to obtain global approximations
for the attractor solution, which, e.g., might be used in the context of global
measures. In addition we elucidate the connection between slow-roll based
approximations and the attractor solution, and compare these approximations
with the center manifold based approximants.
