  We study general dynamical equations describing homogeneous isotropic
cosmologies coupled to a scalaron $\psi$. For flat cosmologies ($k=0$), we
analyze in detail the gauge-independent equation describing the differential,
$\chi(\alpha)\equiv\psi^\prime(\alpha)$, of the map of the metric $\alpha$ to
the scalaron field $\psi$, which is the main mathematical characteristic
locally defining a `portrait' of a cosmology in `$\alpha$-version'. In the
`$\psi$-version', a similar equation for the differential of the inverse map,
$\bar{\chi}(\psi)\equiv \chi^{-1}(\alpha)$, can be solved asymptotically or for
some `integrable' scalaron potentials $v(\psi)$. In the flat case,
$\bar{\chi}(\psi)$ and $\chi(\alpha)$ satisfy the first-order differential
equations depending only on the logarithmic derivative of the potential. Once
we know a general analytic solution for one of these $\chi$-functions, we can
explicitly derive all characteristics of the cosmological model. In the
$\alpha$-version, the whole dynamical system is integrable for $k\neq 0$ and
with any `$\alpha$-potential', $\bar{v}(\alpha)\equiv v[\psi(\alpha)]$,
replacing $v(\psi)$. There is no a priori relation between the two potentials
before deriving $\chi$ or $\bar{\chi}$, which implicitly depend on the
potential itself, but relations between the two pictures can be found by
asymptotic expansions or by inflationary perturbation theory. Explicit
applications of the results to a more rigorous treatment of the chaotic
inflation models and to their comparison with the ekpyrotic-bouncing ones are
outlined in the frame of our `$\alpha$-formulation' of isotropic scalaron
cosmologies. In particular, we establish an inflationary perturbation expansion
for $\chi$. When all the conditions for inflation are satisfied and $\chi$
obeys a certain boundary (initial) condition, we get the standard inflationary
parameters, with higher-order corrections.
