  We study flat FLRW $\alpha$-attractor $\mathrm{E}$- and $\mathrm{T}$-models
by introducing a dynamical systems framework that yields regularized
unconstrained field equations on two-dimensional compact state spaces. This
results in both illustrative figures and a complete description of the entire
solution spaces of these models, including asymptotics. In particular, it is
shown that observational viability, which requires a sufficient number of
$e$-folds, is associated with a solution given by a one-dimensional center
manifold of a past asymptotic de Sitter state, where the center manifold
structure also explains why nearby solutions are attracted to this
`inflationary attractor solution.' A center manifold expansion yields a
description of the inflationary regime with arbitrary analytic accuracy, where
the slow-roll approximation asymptotically describes the tangency condition of
the center manifold at the asymptotic de Sitter state.
