Scalar-field cosmologies with a generalized harmonic potential and matter with energy density $$\rho _m$$ , pressure $$p_m$$ , and barotropic equation of state (EoS) $$p_m=(\gamma -1)\rho _m, \; \gamma \in [0,2]$$ in Kantowski–Sachs (KS) and closed Friedmann–Lemaître–Robertson–Walker (FLRW) metrics are investigated. We use methods from non-linear dynamical systems theory and averaging theory considering a time-dependent perturbation function D. We define a regular dynamical system over a compact phase space, obtaining global results. That is, for KS metric the global late-time attractors of full and time-averaged systems are two anisotropic contracting solutions, which are non-flat locally rotationally symmetric (LRS) Kasner and Taub (flat LRS Kasner) for $$0\le \gamma \le 2$$ , and flat FLRW matter-dominated universe if $$0\le \gamma \le \frac{2}{3}$$ . For closed FLRW metric late-time attractors of full and averaged systems are a flat matter-dominated FLRW universe for $$0\le \gamma \le \frac{2}{3}$$ as in KS and Einstein–de Sitter solution for $$0\le \gamma <1$$ . Therefore, a time-averaged system determines future asymptotics of the full system. Also, oscillations entering the system through Klein–Gordon (KG) equation can be controlled and smoothed out when D goes monotonically to zero, and incidentally for the whole D-range for KS and closed FLRW (if $$0\le \gamma < 1$$ ) too. However, for $$\gamma \ge 1$$ closed FLRW solutions of the full system depart from the solutions of the averaged system as D is large. Our results are supported by numerical simulations.
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"surname": "Gonz\u00e1lez",
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"value": "Scalar-field cosmologies with a generalized harmonic potential and matter with energy density $$\\rho _m$$ <math> <msub> <mi>\u03c1</mi> <mi>m</mi> </msub> </math> , pressure $$p_m$$ <math> <msub> <mi>p</mi> <mi>m</mi> </msub> </math> , and barotropic equation of state (EoS) $$p_m=(\\gamma -1)\\rho _m, \\; \\gamma \\in [0,2]$$ <math> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>\u03b3</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>\u03c1</mi> <mi>m</mi> </msub> <mo>,</mo> <mspace width=\"0.277778em\"></mspace> <mi>\u03b3</mi> <mo>\u2208</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>]</mo> </mrow> </mrow> </math> in Kantowski\u2013Sachs (KS) and closed Friedmann\u2013Lema\u00eetre\u2013Robertson\u2013Walker (FLRW) metrics are investigated. We use methods from non-linear dynamical systems theory and averaging theory considering a time-dependent perturbation function D. We define a regular dynamical system over a compact phase space, obtaining global results. That is, for KS metric the global late-time attractors of full and time-averaged systems are two anisotropic contracting solutions, which are non-flat locally rotationally symmetric (LRS) Kasner and Taub (flat LRS Kasner) for $$0\\le \\gamma \\le 2$$ <math> <mrow> <mn>0</mn> <mo>\u2264</mo> <mi>\u03b3</mi> <mo>\u2264</mo> <mn>2</mn> </mrow> </math> , and flat FLRW matter-dominated universe if $$0\\le \\gamma \\le \\frac{2}{3}$$ <math> <mrow> <mn>0</mn> <mo>\u2264</mo> <mi>\u03b3</mi> <mo>\u2264</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </math> . For closed FLRW metric late-time attractors of full and averaged systems are a flat matter-dominated FLRW universe for $$0\\le \\gamma \\le \\frac{2}{3}$$ <math> <mrow> <mn>0</mn> <mo>\u2264</mo> <mi>\u03b3</mi> <mo>\u2264</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </math> as in KS and Einstein\u2013de Sitter solution for $$0\\le \\gamma <1$$ <math> <mrow> <mn>0</mn> <mo>\u2264</mo> <mi>\u03b3</mi> <mo><</mo> <mn>1</mn> </mrow> </math> . Therefore, a time-averaged system determines future asymptotics of the full system. Also, oscillations entering the system through Klein\u2013Gordon (KG) equation can be controlled and smoothed out when D goes monotonically to zero, and incidentally for the whole D-range for KS and closed FLRW (if $$0\\le \\gamma < 1$$ <math> <mrow> <mn>0</mn> <mo>\u2264</mo> <mi>\u03b3</mi> <mo><</mo> <mn>1</mn> </mrow> </math> ) too. However, for $$\\gamma \\ge 1$$ <math> <mrow> <mi>\u03b3</mi> <mo>\u2265</mo> <mn>1</mn> </mrow> </math> closed FLRW solutions of the full system depart from the solutions of the averaged system as D is large. Our results are supported by numerical simulations."
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