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On solving dynamical equations in general homogeneous isotropic cosmologies with scalaron

We study general dynamical equations describing homogeneous isotropic cosmologies coupled to a scalaron $ψ$. For flat cosmologies ($k=0$), we analyze in detail the gauge-independent equation describing the differential, $χ(α)\equivψ^\prime(α)$, of the map of the metric $α$ to the scalaron field $ψ$, which is the main mathematical characteristic locally defining a `portrait' of a cosmology in `$α$-version'. In the `$ψ$-version', a similar equation for the differential of the inverse map, $\barχ(ψ)\equiv χ^{-1}(α)$, can be solved asymptotically or for some `integrable' scalaron potentials $v(ψ)$. In the flat case, $\barχ(ψ)$ and $χ(α)$ 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 $χ$-functions, we can explicitly derive all characteristics of the cosmological model. In the $α$-version, the whole dynamical system is integrable for $k\neq 0$ and with any `$α$-potential', $\bar{v}(α)\equiv v[ψ(α)]$, replacing $v(ψ)$. There is no a priori relation between the two potentials before deriving $χ$ or $\barχ$, 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 `$α$-formulation' of isotropic scalaron cosmologies. In particular, we establish an inflationary perturbation expansion for $χ$. When all the conditions for inflation are satisfied and $χ$ obeys a certain boundary (initial) condition, we get the standard inflationary parameters, with higher-order corrections.