  Higher-order theories of gravity are a branch of modified gravity wherein the
geometrodynamics of the four-dimensional Riemannian manifold is determined by
field equations involving derivatives of the metric tensor of order higher than
two. This paper considers a general action built with the Einstein-Hilbert term
plus additional curvature-based invariants, viz. the Starobinsky $R^{2}$-type
term, a term scaling with $R^{3}$, and a correction of the type $R\square R$.
The focus is on the background inflationary regime accommodated by these three
models. For that, the higher-order field equations are built and specified for
the FLRW line element. The dynanical analysis in the phase space is carried in
each case. This analysis shows that the Starobinsky-plus-$R^{3}$ model keeps
the good features exhibited by the pure Starobinsky inflationary model,
although the set of initial conditions for the inflaton field $\chi$ leading to
a graceful exit scenario is more contrived; the coupling constant $\alpha_{0}$
of the $R^{3}$ invariant is also constrained by the dynamical analysis. The
Starobinsky-plus-$R\square R$ model turns out being a double-field inflation
model; it consistently enables an almost-exponential primordial acceleration
followed by a radiation dominated universe if its coupling $\beta_{0}$ takes
values in the interval $0\leq\beta_{0}\leq3/4$. The models introducing
higher-order correction to Starobinsky inflation are interesting due to the
possibility of a running spectral index $n_{s}$, something that is allowed by
current CMB observations.
