# Dynamical Systems Analysis of f(Q) Gravity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}[17,18,19].

## 2. Brief Review on the Standard Approach to the Construction of a Dynamical System

## 3. Dynamical Systems Formulation

#### 3.1. General Setup with Two Fluids

#### 3.2. Fixed Points

_{n}and P

_{m}, respectively. In the following section we show that these points possess very particular fixed properties that are of interest when assessing the validity of different cosmological models.

#### 3.3. Physical Parameters of the General System

_{n}satisfying $n(Z)=2$ one immediately has ${\mathrm{\Omega}}_{f}=1$. Similarly for points P

_{m}, looking at the definition of $m(Z)$ in (15), we see that if $m(Z)\to \infty $ whilst $n(Z)$ stays finite, we also obtain ${\mathrm{\Omega}}_{f}=1$. For these two solutions, the deceleration parameter and equation of state are fixed to be $q={w}_{\mathrm{eff}}=-1$, representing a de Sitter Universe. In fact, this is a necessary requirement for any de Sitter solution given that we have assumed ${w}_{1}\ne {w}_{2}$ and ${w}_{1},{w}_{2}\ne -1$. Models where $n(Z)\ne 2$ and $m(Z)\nrightarrow \infty $ for some Z in the range $(0,1)$ cannot possess a de Sitter fixed point. This immediately rules out models such as $f(Q)\propto {Q}^{\alpha}$ for $\alpha \ne 1/2$, or $f(Q)=Q+\beta {Q}^{2}$ for $\beta \ge 0$.

## 4. Applications to $f(Q)$ Models

#### 4.1. Anagnostopoulos et al. Model

_{m}= {0,0} and ${\mathrm{P}}_{\mathrm{n}}=\{0,2\lambda /(1+2\lambda )\}$. Point A is an additional solution to the algebraic Equation (17) with properties that could not be determined in general, therefore it was left out of Table 1. The critical point ${\mathrm{P}}_{\mathrm{m}}$ satisfies $m(Z)\to \infty $ with $n(Z)=\mathrm{finite}$, whilst the point ${\mathrm{P}}_{\mathrm{n}}$ is a solution to $n(Z)=2$. Hence these two points describe de Sitter attractors, as explained in the previous section and in Table 1.

#### 4.2. Phase Space Analysis

## 5. Summary

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Stability Analysis of Anagnostopoulos et al. Model

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**Figure 1.**Phase portraits for the Anagnostopoulos et al. model (24). The physical phase space is within the bordered region whilst the red region represents accelerated expansion. (

**a**) Phase space with positive $\lambda $. (

**b**) Phase space with negative $\lambda $.

**Figure 2.**Evolution of density parameters ${\mathrm{\Omega}}_{m}$, ${\mathrm{\Omega}}_{r}$, effective equation of state ${w}_{\mathrm{eff}}$, and deceleration parameter q for GR with a positive cosmological constant (dashed) and Anagnostopoulos et al. model (solid line). (

**a**) Evolution for positive $\lambda $. (

**b**) Evolution for negative $\lambda $.

**Table 1.**Table of critical points with fixed values of deceleration parameter q and effective equation of state ${w}_{\mathrm{eff}}$.

Point | ${\mathit{X}}_{2}$ | Z | q | ${\mathit{w}}_{\mathrm{eff}}$ | Requirement |
---|---|---|---|---|---|

${\mathrm{P}}_{\mathrm{m}}$ | 0 | ${Z}^{\star}$ | $-1$ | $-1$ | $n({Z}^{\star})=2$ |

${\mathrm{P}}_{\mathrm{n}}$ | 0 | ${Z}^{\star}$ | $-1$ | $-1$ | $m({Z}^{\star})\to \infty $ |

B | ${X}_{2}^{\star}$ | 1 | $\frac{1}{2}(1+3{w}_{2})$ | ${w}_{2}$ | ${X}_{2}^{\star}$ evaluated at $Z=1$ |

C | ${X}_{2}^{\star}$ | 0 | $\frac{1}{2}(1+3{w}_{2})$ | ${w}_{2}$ | ${X}_{2}^{\star}$ evaluated at $Z=0$ |

Point | ${\mathit{X}}_{1}$ | ${\mathit{X}}_{2}$ | Z | q | ${\mathit{w}}_{\mathrm{eff}}$ | Existence Conditions | Stability |
---|---|---|---|---|---|---|---|

A | 1 | 0 | 1 | $\frac{1}{2}(1+3{w}_{1})$ | ${w}_{1}$ | none | Saddle |

B | 0 | 1 | 1 | $\frac{1}{2}(1+3{w}_{2})$ | ${w}_{2}$ | none | Unstable |

${\mathrm{P}}_{\mathrm{m}}$ | 0 | 0 | 0 | $-1$ | $-1$ | $\lambda <0$ | Nonhyperbolic |

${\mathrm{P}}_{\mathrm{n}}$ | 0 | 0 | $2\lambda /(1+2\lambda )$ | $-1$ | $-1$ | $\lambda >0$ | Stable |

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**MDPI and ACS Style**

Böhmer, C.; Jensko, E.; Lazkoz, R.
Dynamical Systems Analysis of *f*(*Q*) Gravity. *Universe* **2023**, *9*, 166.
https://doi.org/10.3390/universe9040166

**AMA Style**

Böhmer C, Jensko E, Lazkoz R.
Dynamical Systems Analysis of *f*(*Q*) Gravity. *Universe*. 2023; 9(4):166.
https://doi.org/10.3390/universe9040166

**Chicago/Turabian Style**

Böhmer, Christian, Erik Jensko, and Ruth Lazkoz.
2023. "Dynamical Systems Analysis of *f*(*Q*) Gravity" *Universe* 9, no. 4: 166.
https://doi.org/10.3390/universe9040166