# Qualitative Analysis of the Dynamics of a Two-Component Chiral Cosmological Model

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## Abstract

**:**

## 1. Introduction

## 2. Chiral Cosmological Model

## 3. Derivation of a Dynamical System of the 2-CCM

## 4. Critical Points of the 2-CCM

#### 4.1. Critical Points for ${h}_{22}>0$

**the main**ones.

#### 4.2. Critical Points for ${h}_{22}\ge 0$

**the special**critical points.

## 5. Asymptotical Analysis of the 2-CCM Dynamic

#### 5.1. Main Critical Points

#### 5.2. Class $\mathbf{A}$ of the Main Critical Points

#### 5.3. Class $\mathbf{B}$ of the Main Critical Points

#### 5.4. Special Critical Points

## 6. The 2-CCM with Massive Chiral Fields and ${\mathit{h}}_{\mathbf{22}}\mathbf{>}\mathbf{0}$

## 7. The 2-CCM with ${h}_{22}={\varphi}^{2}$

## 8. Example of Symmetry Braking

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Chervon, S.V. Chiral Cosmological Models: Dark Sector Fields Description. Quantum Matter
**2013**, 2, 71–82. [Google Scholar] - Perelomov, A.M. Chiral models: Geometrical aspects. Phys. Rep.
**1987**, 146, 136. [Google Scholar] [CrossRef] - Chervon, S.V.; Fomin, I.V.; Pozdeeva, E.O.; Sami, M.; Vernov, S.Y. Superpotential method for chiral cosmological models connected with modified gravity. Phys. Rev. D
**2019**, 100, 063522. [Google Scholar] [CrossRef] [Green Version] - Paliathanasis, A.; Leon, G. Asymptotic behavior of N-fields Chiral Cosmology. Eur. Phys. J. C
**2020**, 80, 847. [Google Scholar] [CrossRef] - Chervon, S.V.; Maharaj, S.D.; Beesham, A.; Kubasov, A.S. Emergent Universe Supported by Chiral Cosmological Fields in 5D Einstein-Gauss-Bonnet Gravity. Gravit. Cosmol.
**2014**, 20, 176–181. [Google Scholar] [CrossRef] [Green Version] - Beesham, A.; Chervon, S.V.; Maharaj, S.D.; Kubasov, A.S. Exact Inflationary Solutions Inspired by the Emergent Universe Scenario. Int. J. Theor. Phys.
**2015**, 54, 884–895. [Google Scholar] [CrossRef] - Chervon, S.V.; Fomin, I.V.; Beesham, A. The method of generating functions in exact scalar field inflationary cosmology. Eur. Phys. J. C
**2018**, 78, 301. [Google Scholar] [CrossRef] [Green Version] - Fomin, I.V.; Chervon, S.V. Exact and Approximate Solutions in the Friedmann Cosmology. Russ. Phys. J.
**2017**, 60, 427–440. [Google Scholar] [CrossRef] - Chervon, S.; Fomin, I.; Yurov, V.; Yurov, A. Series on the Foundations of Natural Science and Technology: Scalar Field Cosmology; World Scientific Publishing: Singapore, 2019; Volume 13. [Google Scholar]
- Abbyazov, R.R.; Chervon, S.V. Unified Dark Matter and Dark Energy Description in a Chiral Cosmological Model. Mod. Phys. Lett. A
**2013**, 28, 1350024. [Google Scholar] [CrossRef] [Green Version] - Fijii, Y.; Maeda, K.-I. The Scalar–Tensor Theory of Gravitation; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Kaiser, D.I. Conformal Transformations with Multiple Scalar Fields. Phys. Rev. D
**2010**, 81, 084044. [Google Scholar] [CrossRef] [Green Version] - Kaiser, D.I.; Sfakianakis, E.I. Multifield Inflation after Planck: The Case for Nonminimal Couplings. Phys. Rev. Lett.
**2014**, 112, 011302. [Google Scholar] [CrossRef] - Schutz, K.; Sfakianakis, E.I.; Kaiser, D.I. Multifield Inflation after Planck: Isocurvature Modes from Nonminimal Couplings. Phys. Rev. D
**2014**, 89, 064044. [Google Scholar] [CrossRef] [Green Version] - Aghanim, N.; Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Basak, S.; et al. [Planck Collaboration], Planck 2018 results. VI. Cosmological parameters. arXiv
**2018**, arXiv:1807.06209. [Google Scholar] - Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Basak, S.; et al. [Planck Collaboration], Planck 2018 results. X. Constraints on inflation. arXiv
**2018**, arXiv:1807.06211. [Google Scholar] - Gong, J.O. Multi-field inflation and cosmological perturbations. Int. J. Mod. Phys. D
**2016**, 26, 1740003. [Google Scholar] [CrossRef] - Kallosh, R.; Linde, A. Multi-field Conformal Cosmological Attractors. J. Cosmol. Astropart. Phys.
**2013**, 2013, 006. [Google Scholar] [CrossRef] [Green Version] - Abbyazov, R.R.; Chervon, S.V.; Müller, V. σCDM coupled to radiation: Dark energy and Universe acceleration. Mod. Phys. Lett. A
**2015**, 30, 1550114. [Google Scholar] [CrossRef] [Green Version] - Abbyazov, R.R.; Chervon, S.V. Interaction of chiral fields of the dark sector with cold dark matter. Gravit. Cosmol.
**2012**, 18, 262–269. [Google Scholar] [CrossRef] - Naruko, A.; Yoshida, D.; Mukohyama, S. Gravitational scalar-tensor theory. Class. Quant. Grav.
**2016**, 33, 09LT01. [Google Scholar] [CrossRef] [Green Version] - Chervon, S.V.; Fomin, I.V.; Mayorova, T.I. Chiral cosmological model of f(R) gravity with a kinetic curvature scalar. Gravit. Cosmol.
**2019**, 25, 205–212. [Google Scholar] [CrossRef] - Chervon, S.V.; Nikolaev, A.V.; Mayorova, T.I. On the derivation of field equation of f(R) gravity with kinetic scalar curvature. Spacetime Fundam. Interact.
**2017**, 1, 30–37. [Google Scholar] [CrossRef] - Chervon, S.V.; Nikolaev, A.V.; Mayorova, T.I.; Odintsov, S.D.; Oikonomou, V.K. Kinetic scalar curvature extended f(R) gravity. Nucl. Phys.
**2018**, B936, 597–614. [Google Scholar] [CrossRef] - Saridakis, E.N.; Tsoukalas, M. Cosmology in new gravitational scalar-tensor theories. Phys. Rev. D
**2016**, 93, 124032. [Google Scholar] [CrossRef] [Green Version] - Chervon, S.V.; Fomin, I.V.; Mayorova, T.I.; Khapaeva, A.V. Cosmological parameters of f(R) gravity with kinetic scalar curvature. J. Phys. Conf. Ser.
**2020**, 1557, 012016. [Google Scholar] [CrossRef] - Bogoyavlensky, O.I. Methods of the Qualitative Theory of Dynamical Systems in Astrophysics and Gas Dynamics; Publishing Nauka: Moscow, Russia, 1980. [Google Scholar]
- Starobinsky, A.A. A New Type of Isotropic Cosmological Models Without Singularity. Phys. Lett. B
**1980**, 91, 99–102. [Google Scholar] [CrossRef] - Starobinsky, A.A. Dynamics of phase transition in the new inflationary universe scenario and generation of perturbations. Phys. Lett. B
**1982**, 117, 175–178. [Google Scholar] [CrossRef] - Belinsky, V.A.; Grishchuk, L.P.; Khalatnikov, I.M.; Zeldovich, Y.B. Inflationary stages in cosmological models with a scalar field. Phys. Lett. B
**1985**, 155, 232–236. [Google Scholar] [CrossRef] - Coley, A.A. Dynamical Systems and Cosmology; Springer-Science+Business Media: Dordrecht, The Netherlands, 2003. [Google Scholar]
- Coley, A.A. Proceedings of the Sixth Canadian Conference on General Relativity and Relativistic Astrophysics; Braham, S., Gegenberg, J., McKellar, R., Eds.; Fields Institute Communications Series (AMS): Providence, RI, USA, 1997; p. 19. [Google Scholar]
- Wainwright, J.; Ellis, G.F.R. Dynamical Systems in Cosmology; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Copeland, E.J.; Sami, M.; Tsujikawa, S.J. Dynamics of dark energy. Int. J. Mod. Phys. D
**2006**, 15, 1753–1936. [Google Scholar] [CrossRef] [Green Version] - Ignat’ev, Y.G.; Agathonov, A.A. Qualitative and numerical analysis of a cosmological model based on a phantom scalar field with self-interaction. Grav.Cosmol.
**2017**, 23, 230–235. [Google Scholar] [CrossRef] - Ignat’ev, Y.G.; Agathonov, A.A. Qualitative and Numerical Analysis of the Cosmological Model Based on a Phantom Scalar field with Self-Action. II. Comparative Analysis of Models of Classical and Phantom Fields. arXiv
**2017**, arXiv:1706.05619. [Google Scholar] - Ignat’ev, Y.G.; Kokh, I.A. Qualitative and Numerical Analysis of a Cosmological Model Based on an Asymmetric Scalar Doublet with Minimal connections. IV. Numerical Modeling and Types of Behavior of the Model. Russ. Phys. J.
**2019**, 3, 453–470. [Google Scholar] [CrossRef] [Green Version] - Zhuravlev, V.M. Qualitative analisys of cosmological models with scalar fields. Spacetime Fundam. Interact.
**2016**, 4, 39–51. [Google Scholar] [CrossRef] - Ignat’ev, Y.G.; Agathonov, A.A. The Peculiarities of the Cosmological Models Based on Non-Linear Classical and Phantom Fields with Minimal Interaction. I. The Cosmological Model Based on Scalar Singlet. arXiv
**2018**, arXiv:1808.04570. [Google Scholar] - Ignat’ev, Y.G.; Agathonov, A.A. The Peculiarities of the Cosmological Models Based on Nonlinear Classical and Phantom Fields with Minimal Interaction. II. The Cosmological Model Based on the Asymmetrical Scalar Doublet. arXiv
**2018**, arXiv:1810.09873. [Google Scholar] - Zhuravlev, V.M.; Podymova, T.V.; Pereskokov, E.A. Cosmological Models with a Specified Trajectory on the Energy Phase Plane. Gravit. Cosmol.
**2011**, 17, 101–109. [Google Scholar] [CrossRef] - Ignat’ev, Y.G. Standard cosmological model: Mathematical, qualitatively and numerical analysis. Spacetime Fundam. Interact.
**2016**, 3, 17–36. [Google Scholar] [CrossRef] - Chervon, S.V. On the chiral model of cosmological inflation. Russ. Phys. J.
**1995**, 38, 539. [Google Scholar] [CrossRef] - Chervon, S.V. Chiral non-linear sigma models and cosmological inflation. Gravit. Cosmol.
**1995**, 1, 91. [Google Scholar] - Chervon, S.V.; Fomin, I.V.; Kubasov, A.S. Scalar and Chiral Fields in Cosmology; Ulyanovsk State Pedagogical University: Ulyanovsk, Russia, 2015. (In Russian) [Google Scholar]

**Figure 1.**Isolines of the interaction potential (45) with phase trajectories for a set of 16 starting points.

**Figure 3.**Dependence of H on t (

**a**); deviation of the integral of motion $\mathsf{\Delta}\left(t\right)$ from zero (

**b**).

**Figure 4.**Isolines of the interaction potential (47) together with the phase trajectories of the model.

**Figure 6.**Dependence of H on t (

**a**); deviation of the integral of motion $\mathsf{\Delta}\left(t\right)$ from zero (

**b**).

**Figure 7.**Isolines of the interaction potential (47) together with the phase trajectories of the model.

**Figure 9.**Dependence of H on t (

**a**); deviation of the integral of motion $\mathsf{\Delta}\left(t\right)$ from zero (

**b**).

**Figure 12.**Dependence of H on t (

**a**); deviation of the integral of motion $\mathsf{\Delta}\left(t\right)$ from zero (

**b**).

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Zhuravlev, V.; Chervon, S.
Qualitative Analysis of the Dynamics of a Two-Component Chiral Cosmological Model. *Universe* **2020**, *6*, 195.
https://doi.org/10.3390/universe6110195

**AMA Style**

Zhuravlev V, Chervon S.
Qualitative Analysis of the Dynamics of a Two-Component Chiral Cosmological Model. *Universe*. 2020; 6(11):195.
https://doi.org/10.3390/universe6110195

**Chicago/Turabian Style**

Zhuravlev, Viktor, and Sergey Chervon.
2020. "Qualitative Analysis of the Dynamics of a Two-Component Chiral Cosmological Model" *Universe* 6, no. 11: 195.
https://doi.org/10.3390/universe6110195