On solving dynamical equations in general homogeneous isotropic cosmologies with scalaron

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.