Solving dynamical equations in general homogeneous isotropic cosmologies with a scalaron

We consider gauge-dependent dynamical equations describing homogeneous isotropic cosmologies coupled to a scalar field $\psi$ (scalaron). For flat cosmologies $(k=0)$, we analyze the gauge-independent equation describing the differential $\chi(\alpha)\equiv\psi'(\alpha)$ of the map of the metric $\alpha$ to the scalaron field $\psi$, which is the main mathematical characteristic of a cosmology and locally defines its portrait in the so-called $\alpha$ version. In the more customary $\psi$ version, the similar equation for the differential of the inverse map $\bar\chi(\psi) \equiv\chi^{-1}(\alpha)$ is solved in an asymptotic approximation for arbitrary potentials $v(\psi)$. In the flat case, $\bar\chi(\psi)$ and $\chi^{-1}(\alpha)$ satisfy first-order differential equations depending only on the logarithmic derivative of the potential, $v'(\psi)/v(\psi)$. If an analytic solution for one of the $\chi$ functions is known, then we can find all characteristics of the cosmological model. In the $\alpha$ version, the full dynamical system is explicitly integrable for $k\ne0$ with any potential $\bar v(\alpha)\equiv v[\psi(\alpha)]$ replacing $v(\psi)$. Until one of the maps, which themselves depend on the potentials, is calculated, no sort of analytic relation between these potentials can be found. Nevertheless, such relations can be found in asymptotic regions or by perturbation theory. If instead of a potential we specify a cosmological portrait, then we can reconstruct the corresponding potential. The main subject here is the mathematical structure of isotropic cosmologies. We also briefly present basic applications to a more rigorous treatment of inflation models in the framework of the $\alpha$ version of the isotropic scalaron cosmology. In particular, we construct an inflationary perturbation expansion for $\chi$. If the conditions for inflation to arise are satisfied, i.e., if $v>0$, $k=0$, $\chi^2<6$, and $\chi(\alpha)$ satisfies a certain boundary condition as $\alpha\to-\infty$, then the expansion is invariant under scaling the potential, and its first terms give the standard inflationary parameters with higher-order corrections.