# The Phase Space Analysis of Interacting K-Essence Dark Energy Models in Loop Quantum Cosmology

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## Abstract

**:**

## 1. Introduction

## 2. Model I: ${\mathcal{L}}_{\mathbf{1}}=\mathit{F}\left(\mathit{X}\right)\mathit{V}\left(\mathit{\varphi}\right)$ in Loop Quantum Cosmology

## 3. Model II: ${\mathcal{L}}_{\mathbf{2}}=\mathit{F}\left(\mathit{X}\right)-\mathit{V}\left(\mathit{\varphi}\right)$ in Loop Quantum Cosmology

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The value ranges for parameters s and $\alpha $ to create and stablize the critical point ${P}_{2}$ when $x>0$; (

**b**) The 3-dim phase space for $s=-0.005$, and $\alpha =0.01$ around the attractor ${P}_{2}=(0.709,1.996,0)$ when $x>0$.

**Figure 2.**(

**a**) The value ranges for parameters s and $\alpha $ to create and stabilize the critical point ${P}_{3}$ when $x>0$; (

**b**) The 3-dim phase space for $s=-16.5$, and $\alpha =0.015$ around the attractor ${P}_{3}=(0.738,0.142,0)$ when $x>0$.

**Figure 3.**The projection of trajectories from 3-dim phase space onto x-y plane, and z-axis is vertical to the x-y plane. (

**a**) The phase plane around ${P}_{2}=(0.709,1.996,0)$. The green curves denote EC and the black curves denote LQC; (

**b**) The phase plane around ${P}_{3}=(0.738,0.142,0)$. The green curve denotes EC and the black curve denotes LQC.

**Figure 4.**The comparison of the evolutions of ${\Omega}_{\varphi}$, ${w}_{\varphi}$ and q in EC (green) with the ones in LQC (black) around ${P}_{2}$ by the initial condition ${x}_{0}=0.8,{y}_{0}=1.5,{z}_{0}=0.1$. (

**a**–

**c**) correspond to different coupling parameters $\alpha =0,0.005,0.02$ and fixed potential parameter $s=-0.004$; alternately, (

**d**–

**f**) correspond to different potential parameters $s=-0.005,-0.003,-0.001$ and fixed coupling parameter $\alpha =0.01$.

**Figure 5.**The comparison of the evolutions of ${\Omega}_{\varphi}$, ${w}_{\varphi}$ and q in EC (green) with those in LQC (black) around ${P}_{2}$ by the initial condition ${x}_{0}=0.6,{y}_{0}=0.15,{z}_{0}=0.1$. (

**a**–

**c**) correspond to different coupling parameters $\alpha =0.012,0.015,0.018$ and fixed potential parameter $s=-16.5$; alternately, (

**d**–

**f**) correspond to different potential parameters $s=-13,-15,-17$ and fixed coupling parameter $\alpha =0.015$.

**Figure 6.**(

**a**) The evolution of H, $\dot{H}$ and $\rho $ in LQC around ${P}_{2}$, by the initial condition of ${\varphi}_{0}=20$, ${\dot{\varphi}}_{0}=0.75$, ${H}_{0}=0.07$ and ${\rho}_{0}=0.015$; (

**b**) The local figure in (

**a**) of $t\in [410,420]$ where bouncing occurs.

**Figure 7.**(

**a**) The evolutions of H, $\dot{H}$ and $\rho $ in LQC around ${P}_{3}$, by the initial condition of ${\varphi}_{0}=9$, ${\dot{\varphi}}_{0}=0.9$, ${H}_{0}=0.07$ and ${\rho}_{0}=0.015$; (

**b**) The local figure in (

**a**) of $t\in [255,260]$ where bouncing occurs.

**Figure 8.**(

**a**) The value ranges for parameters $\alpha $, ${\gamma}_{k}$ and $\Gamma $ to create and stabilize the critical point ${P}_{d}$; (

**b**) The 3-dim phase space for fixed ${\gamma}_{k}=\frac{8}{7}$ and $\Gamma =\frac{11}{10}$, corresponding to two different values $\alpha =-1$ (black) and $\alpha =-2$ (green) around ${P}_{d}=(1,0,0,\frac{30}{7})$.

**Figure 9.**(

**a**) The value ranges for parameters $\alpha $, ${\gamma}_{k}$ and $\Gamma $ to create and stabilize the critical point ${P}_{e2}$; (

**b**) The 3-dim phase space around ${P}_{e2}$ with different coupling parameter $\alpha $ by fixing ${\gamma}_{k}=\frac{6}{5}$ and $\Gamma =\frac{6}{5}$. When $\alpha =-0.3$, the black trajectories converge to ${P}_{e2}=(\frac{1}{2},0,0,4)$, while for $\alpha =-0.01$, the green trajectories converge to ${P}_{e2}=(\frac{1}{60},0,0,120)$.

**Figure 10.**(

**a**) Parameter region for ${P}_{h0}$; (

**b**) The phase space around ${P}_{h0}=(\frac{\sqrt[6]{2}}{3},\frac{\sqrt[3]{4}}{6},0)\approx (0.374,0.265,0)$ with $s=3$ and ${\gamma}_{k}=\frac{3}{2}$ for EC(green) and LQC(black).

**Figure 11.**The stabilities corresponding to different values of coupling parameter $\alpha $, which are shown by the projection from a 3-dim phase space onto to 2-dim x-y plane, and z-axis is vertical to the x-y plane. ${P}_{g}=(0.994,0.109,0)$ is displayed by a solid circle. (

**a**) With $\alpha =0$, trajectories converge to stable point ${P}_{h}=(0.329,0.190,0)$, just as [51] in EC; (

**b**) With $\alpha =-0.6>-0.985$, trajectories still converge to stable point ${P}_{h}=(0.666,0.121,0)$; (

**c**) With $\alpha =-1.2<-0.985$, all trajectories converge to ${P}_{g}$ which is a stable point in this case, but ${P}_{h}=(1.231,0.113,0)$ becomes unstable.

**Figure 12.**(

**a**) The x-y phase plane projected from 4-dim system of {x,y,z,$\sigma $} for both LQC(black) and EC(green dash) around ${P}_{d}$, corresponding to the parameters of ${\gamma}_{k}=\frac{8}{7}$, $\Gamma =\frac{11}{10}$ and $\alpha =-1$. The initial points are $(0.8,0.1,0.1,4)$ for LQC, and $(0.8,0.1,0,4)$ for EC. The star stands for the final state of the evolution, while the solid circle stands for the initial point. (

**b**) the evolution of the cosmological quantities of ${\Omega}_{\varphi}$, ${w}_{\varphi}$ and q for both LQC(black) and EC(green) cases around ${P}_{d}$.

**Figure 13.**(

**a**) The x-y phase plane projected from 4-dim system of {x, y, z, $\sigma $} for both LQC(black) and EC(green dash) around ${P}_{e2}$, corresponding to the parameters of ${\gamma}_{k}=\frac{6}{5}$, $\Gamma =\frac{6}{5}$ and $\alpha =-0.01$. The initial points are $(0.8,0.2,0.1,4)$ for LQC, and $(0.8,0.2,0,4)$ for EC. The star stands for the final state of the evolution, while the solid circle stands for the initial point. (

**b**) the evolution of the cosmological quantities of ${\Omega}_{\varphi}$, ${w}_{\varphi}$ and q for both LQC(black) and EC(green) cases around ${P}_{e2}$.

**Figure 14.**(

**a**) The x-y phase plane projected from 3-dim system of {x,y,z} for both LQC(black) and EC(green) around ${P}_{g}$, corresponding to the parameters of ${\gamma}_{k}=\frac{4}{3}$, $s=4$ and $\alpha =-1.2$. The initial points are $(0.5,0.2,0.1)$ for LQC, and $(0.5,0.2,0)$ for EC. The star stands for the final state of the evolution, while the solid circle stands for the initial point. (

**b**) the evolution of the cosmological quantities of ${\Omega}_{\varphi}$, ${w}_{\varphi}$ and q for both LQC(black) and EC(green) cases for ${P}_{g}$.

**Figure 15.**(

**a**) The x-y phase plane projected from 3-dim system of {x,y,z} for both LQC(black) and EC(green) around ${P}_{h}$, corresponding to the parameters of ${\gamma}_{k}=\frac{4}{3}$, $s=4$ and $\alpha =-0.6$. The initial points are $(0.5,0.2,0.1)$ for LQC, and $(0.5,0.2,0)$ for EC. The star stands for the final state of the evolution, while the solid circle stands for the initial point. (

**b**) the evolution of the cosmological quantities of ${\Omega}_{\varphi}$, ${w}_{\varphi}$ and q for both LQC(black) and EC(green) cases for ${P}_{h}$.

**Figure 16.**For ${P}_{e2}$, the bouncing happens in LQC, with parameters $\eta =3$, $n=-5$, $A=0.002$, $B=0.002$ and coupling parameter $\alpha =-0.01$. (

**a**) The initial values ${\varphi}_{0}=0.6$, ${\dot{\varphi}}_{0}=1.94$, ${H}_{0}=0.21$ and ${\rho}_{0}=0.15$. (

**b**) The initial values ${\varphi}_{0}=0.5$, ${\dot{\varphi}}_{0}=2.3$, ${H}_{0}=0.3333$ and ${\rho}_{0}=0.5$.

**Figure 17.**The bouncing happens in LQC, with parameters $\eta =2$, $n=-4$, $A=0.5$, $B=0.3$, with the initial values ${\varphi}_{0}=1.85$, ${\dot{\varphi}}_{0}=0.65$, ${H}_{0}=0.212$ and ${\rho}_{0}=0.15$. (

**a**) $\alpha =0$ for stable point ${P}_{h0}$; (

**b**) $\alpha =-1.2$ for stable point ${P}_{g}$.

**Table 1.**The existence and stability conditions for six critical points; the cosmological quantities in the form of parameters s and $\alpha $ at each critical point.

Name | Existence | Stability | $0\le {\mathbf{\Omega}}_{\mathit{\varphi}}\le 1$ | ${\mathit{w}}_{\mathit{\varphi}}<-1/3$ | $\mathit{q}<0$ |
---|---|---|---|---|---|

${P}_{1}$ | none | $\begin{array}{c}\hfill 0<s<\sqrt{6},\hfill \\ \hfill \sqrt{6}s+\alpha -3<0\hfill \end{array}$ | 1 | $\frac{\sqrt{6}s-3}{3}$ | $\frac{\sqrt{6}s}{2}-1$ |

${P}_{2}$ | $\sqrt{6}-s>0$ | $\begin{array}{c}\hfill -\sqrt{6}<s<0\hfill \\ \hfill -\sqrt{6}s+\alpha -3<0\hfill \end{array}$ | 1 | $\frac{\sqrt{6}s+3}{-3}$ | $-\frac{\sqrt{6}s}{2}-1$ |

${P}_{3}$ | $\begin{array}{c}\hfill {(\alpha -3)}^{2}>-2{s}^{2}\alpha \hfill \\ \hfill s(\alpha -3)>0\hfill \end{array}$ | $\begin{array}{c}\hfill {\lambda}_{1}=\alpha -3<0\hfill \\ \hfill {\lambda}_{2}{\lambda}_{3}>0,{\lambda}_{2}+{\lambda}_{3}<0\hfill \end{array}$ | $\frac{{(\alpha -3)}^{2}}{6{s}^{2}}$ | $\frac{-2\alpha {s}^{2}}{{(\alpha -3)}^{2}}$ | $\frac{1-\alpha}{2}$ |

${P}_{4}$ | none | $\begin{array}{c}\hfill 0<s<\sqrt{6},\hfill \\ \hfill \sqrt{6}s+\alpha -3<0\hfill \end{array}$ | 1 | $\frac{\sqrt{6}s-3}{3}$ | $-\frac{\sqrt{6}s}{2}-1$ |

${P}_{5}$ | $\sqrt{6}-s>0$ | $\begin{array}{c}\hfill -\sqrt{6}<s<0\hfill \\ \hfill -\sqrt{6}s+\alpha -3<0\hfill \end{array}$ | 1 | $\frac{\sqrt{6}s+3}{-3}$ | $\frac{\sqrt{6}s}{2}-1$ |

${P}_{6}$ | $\begin{array}{c}\hfill {(\alpha -3)}^{2}>-2{s}^{2}\alpha \hfill \\ \hfill s(\alpha -3)>0\hfill \end{array}$ | $\begin{array}{c}\hfill {\lambda}_{1}=\alpha -3<0\hfill \\ \hfill {\lambda}_{2}{\lambda}_{3}>0,{\lambda}_{2}+{\lambda}_{3}<0\hfill \end{array}$ | $\frac{{(\alpha -3)}^{2}}{6{s}^{2}}$ | $\frac{-2\alpha {s}^{2}}{{(\alpha -3)}^{2}}$ | $\frac{1-\alpha}{2}$ |

**Table 2.**The existence and stability conditions for eight critical points, the cosmological quantities, and the range of the parameters ${\gamma}_{k}$, $\Gamma $ and $\alpha $ for acceleration.

Name | Existence | Stability | $0\le {\mathbf{\Omega}}_{\mathit{\varphi}}\le 1$ | ${\mathit{w}}_{\mathit{\varphi}}<-1/3$ | $\mathit{q}<0$ |
---|---|---|---|---|---|

${P}_{a}$ | $\alpha =0$ | unstable | 0 | none | $\frac{3}{2}$ |

${P}_{b}$ | always | unstable | 1 | ${\gamma}_{k}-1$ | $\frac{3{\gamma}_{k}}{2}-1$ |

${P}_{c}$ | always | unstable | 1 | $-1$ | $-1$ |

${P}_{d}$ | always | $\begin{array}{c}\hfill 0<{\gamma}_{k}<2,\alpha +3{\gamma}_{k}<3\hfill \\ \hfill 1<\Gamma <1/{\gamma}_{k}+1/2\hfill \end{array}$ | 1 | ${\gamma}_{k}-1$ | $\frac{3{\gamma}_{k}}{2}-1$ |

${P}_{e1}$ | $\frac{\alpha}{1-{\gamma}_{k}}>0$ | unstable | $\frac{{\alpha}^{2}}{9{({\gamma}_{k}-1)}^{2}}$ | ${\gamma}_{k}-1$ | $\frac{{\alpha}^{2}+3{\gamma}_{k}-3}{6({\gamma}_{k}-1)}$ |

${P}_{e2}$ | $\frac{\alpha}{1-{\gamma}_{k}}>0$ | ${\lambda}_{1}<0,{\lambda}_{2}<0,{\lambda}_{3}<0,{\lambda}_{4}<0$ | $\frac{{\alpha}^{2}}{9{({\gamma}_{k}-1)}^{2}}$ | ${\gamma}_{k}-1$ | $\frac{{\alpha}^{2}+3{\gamma}_{k}-3}{6({\gamma}_{k}-1)}$ |

${P}_{f1}$ | none | stable | $\frac{9}{{\alpha}^{2}}$ | ${\gamma}_{k}-1$ | $-1$ |

${P}_{f2}$ | none | unstable | $\frac{9}{{\alpha}^{2}}$ | ${\gamma}_{k}-1$ | $-1$ |

**Table 3.**The existence and stability conditions for three critical points under the symmetric analysis, the cosmological quantities in form of the parameters ${\gamma}_{k}$, $\Gamma $ and $\alpha $.

Name | Existence | Stability | ${\mathbf{\Omega}}_{\mathit{\varphi}}$ | ${\mathit{w}}_{\mathbf{tot}}$ | ${\mathit{w}}_{\mathit{\varphi}}$ | q |
---|---|---|---|---|---|---|

${P}_{g}$ | $\sigma /{\gamma}_{k}>0$ | $\begin{array}{c}\hfill {\gamma}_{k}^{2}-{\sigma}^{2}>0,\hfill \\ \hfill \alpha \sigma +3{\sigma}^{2}-3{\gamma}_{k}<0\hfill \end{array}$ | 1 | $\frac{{\sigma}^{2}-{\gamma}_{k}}{{\gamma}_{k}}$ | $\frac{{\sigma}^{2}-{\gamma}_{k}}{{\gamma}_{k}}$ | $\frac{3{\sigma}^{2}-2{\gamma}_{k}}{2{\gamma}_{k}}$ |

${P}_{h}$ | $\begin{array}{c}\hfill {\gamma}_{k}>1,s>0\hfill \end{array}$ | $\begin{array}{c}\hfill \alpha \sigma +3{\sigma}^{2}-3{\gamma}_{k}>0,\hfill \\ \hfill \alpha {\gamma}_{k}+3{\gamma}_{k}\sigma -3\sigma >0\hfill \end{array}$ | $\frac{{\gamma}_{k}}{{\sigma}^{2}}$ | 0 | 0 | 1/2 |

${P}_{h0}$ | $\begin{array}{c}\hfill \alpha =0,\hfill \\ \hfill {\gamma}_{k}>1,s>0\hfill \end{array}$ | ${s}^{-2}{\gamma}_{k}{({\gamma}_{k}-1)}^{\frac{({\gamma}_{k}-2)}{{\gamma}_{k}}}<1$ | $\frac{{\gamma}_{k}}{{\sigma}^{2}}$ | 0 | 0 | 1/2 |

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**MDPI and ACS Style**

Chen, B.; Wu, Y.; Chi, J.; Liu, W.; Hu, Y.
The Phase Space Analysis of Interacting K-Essence Dark Energy Models in Loop Quantum Cosmology. *Universe* **2022**, *8*, 520.
https://doi.org/10.3390/universe8100520

**AMA Style**

Chen B, Wu Y, Chi J, Liu W, Hu Y.
The Phase Space Analysis of Interacting K-Essence Dark Energy Models in Loop Quantum Cosmology. *Universe*. 2022; 8(10):520.
https://doi.org/10.3390/universe8100520

**Chicago/Turabian Style**

Chen, Bohai, Yabo Wu, Jianan Chi, Wenzhong Liu, and Yiliang Hu.
2022. "The Phase Space Analysis of Interacting K-Essence Dark Energy Models in Loop Quantum Cosmology" *Universe* 8, no. 10: 520.
https://doi.org/10.3390/universe8100520