  We consider separatrix solutions of the differential equations for inflaton
models with a single scalar field in a zero-curvature
Friedmann-Lema\^{\i}tre-Robertson-Walker universe. The existence and properties
of separatrices are investigated in the framework of the Hamilton-Jacobi
formalism, where the main quantity is the Hubble parameter considered as a
function of the inflaton field. A wide class of inflaton models that have
separatrix solutions (and include many of the most physically relevant
potentials) is introduced, and the properties of the corresponding separatrices
are investigated, in particular, asymptotic inflationary stages, leading
approximations to the separatrices, and full asymptotic expansions thereof. We
also prove an optimal growth criterion for potentials that do not have
separatrices.
