Averaging generalized scalar field cosmologies III: Kantowski--Sachs and\n closed Friedmann--Lema\\^itre--Robertson--Walker models

Scalar field cosmologies with a generalized harmonic potential and matter\nwith energy density $\\rho_m$, pressure $p_m$, and barotropic equation of state\n(EoS) $p_m=(\\gamma-1)\\rho_m, \\; \\gamma\\in[0,2]$ in Kantowski-Sachs (KS) and\nclosed Friedmann--Lema\\^itre--Robertson--Walker (FLRW) metrics are\ninvestigated. We use methods from non--linear dynamical systems theory and\naveraging theory considering a time--dependent perturbation function $D$. We\ndefine a regular dynamical system over a compact phase space, obtaining global\nresults. That is, for KS metric the global late--time attractors of full and\ntime--averaged systems are two anisotropic contracting solutions, which are\nnon--flat locally rotationally symmetric (LRS) Kasner and Taub (flat LRS\nKasner) for $0\\leq \\gamma \\leq 2$, and flat FLRW matter--dominated universe if\n$0\\leq \\gamma \\leq \\frac{2}{3}$. For closed FLRW metric late--time attractors\nof full and averaged systems are a flat matter--dominated FLRW universe for\n$0\\leq \\gamma \\leq \\frac{2}{3}$ as in KS and Einstein-de Sitter solution for\n$0\\leq\\gamma<1$. Therefore, time--averaged system determines future asymptotics\nof full system. Also, oscillations entering the system through Klein-Gordon\n(KG) equation can be controlled and smoothed out when $D$ goes monotonically to\nzero, and incidentally for the whole $D$-range for KS and for closed FLRW (if\n$0\\leq \\gamma< 1$) too. However, for $\\gamma\\geq 1$ closed FLRW solutions of\nthe full system depart from the solutions of the averaged system as $D$ is\nlarge. Our results are supported by numerical simulations.\n