Dynamics of a Cosmological Model in f(R,T) Gravity: II. In Three-Dimensional Space

Abstract

Based on the results discussed on the invariant planes in the known literature [Universe 2022, 8, 365] for the flat FLRW space-time universe model with ideal fluid under the gravity of $f(R,T)=\xi R^{\alpha }+\zeta \sqrt{-T}$, this paper continues to describe the global dynamics of this model in the three-dimensional space containing infinity through dynamic system analysis. Moreover, the cosmological solutions of all the physical significance regions restricted by three invariant planes are also fully discussed.

1. Introduction

The present Universe is flat and undergoing accelerated expansion based on recent observations [1,2,3,4]. An obscure energy termed dark energy is associated with accelerated expansion as it can generate a late-time speeding up of the cosmological foundation. The most comprehensive theory, general relativity (GR), is an excellent match with observational data. However, failing to explain the late-time acceleration [5,6], one of the shortcomings of GR, stimulates the necessity of alternative approaches to resolving the dark side puzzle. These approaches are mainly based on two methodologies. The first methodology relates to introducing a dark energy area to vary the matter substance of the Universe, starting from either a standard scalar field, a phantom field, or an amalgamation of the two fields in a unified model and then developing into more complicated scenarios [7,8,9]. As GR, in its usual form, may not describe the Universe’s evolution accurately at astrophysical and cosmological scales, the second methodology is concerned with modifying the gravitational area. This methodology has aroused enthusiasm over the previous decades, and many novel gravity theories have been proposed. These modified gravity theories mainly incorporate $f\left(R\right)$ gravity [10,11], $f\left(\mathcal{T}\right)$ gravity [12], Gauss–Bonnet gravity [13,14], $f(R,T)$ gravity [15], $f(\mathcal{T},T)$ gravity [16], and so on, where R, T, and $\mathcal{T}$ are the Ricci scalar, the trace of the energy-momentum tensor, and the torsion scalar, respectively.

As one of the simplest modified gravity theories, $f\left(R\right)$ gravity, can create a natural connection between late-time cosmic acceleration and the inflation epoch. In this gravity theory, the Ricci scalar R is substituted by an arbitrary function $f\left(R\right)$ in Einstein–Hilbert action. Hence, the field equations linked with the standard Einstein–Hilbert action get extended in $f\left(R\right)$ gravity theory. Some $f\left(R\right)$ gravity models have been tested to explain dark energy and late-time accelerated expansion [17,18,19,20]. The inflation realized by three kinds of effects, $f\left(R\right)$ gravity, the quantum anomaly, and the $R^2$ term in loop quantum cosmology, has been studied in [21]. Sebastiani and Myrzakulov [22] analyzed many paradigms using several revised methods and obtained effective solutions for inflation in $f\left(R\right)$ gravity. Gonzalo and Diego proposed the junction condition for $f\left(R\right)$ gravity formulated in metric-affine spaces [23] through a distributional tensor approach. It has been observed that any model in $f\left(R\right)$ gravity can be recast as a cosmological fluid in generalized Robertson–Walker space-time [24]. The latest developments in modified gravity and a virtual toolbox for studying inflation, dark energy, and bouncing cosmologies in the context of modified gravity have been provided in [25]. Oikonomou investigated the phenomenology of an $f\left(R\right)$ gravity model in the presence of a primordial light axion scalar field during the inflationary era [26]. The isotropization of a pre-bounce contracting phase in $f\left(R\right)$ gravity has been investigated in [27].

Another modified gravity theory named $f(R,T)$ gravity was proposed by Harko et al. in 2011 [15], which is considered as an extension of $f\left(R\right)$ gravity. This gravity theory considers the effect of a minimal coupling between matter and geometry in gravitational action. The consequence of some imperfect exotic fluid or quantum effects originating from a conformal anomaly trace may be the reason for the dependence on trace T. In $f(R,T)$ gravity theory, geometrical contributions to the total cosmic energy density and matter particles generate cosmic acceleration. After the introduction of this theory, it attracted much attention and researchers applied it to many other cosmological scenarios, including thermodynamics [28,29], redshift drift [30], wormholes [31,32], gravitational waves [33,34], baryogenesis [35], anisotropic cosmology [36], scalar perturbations [37], and Big-Bang nucleosynthesis [38]. Recently, an anisotropic version of Tolman VII isotropic solution in $f(R,T)$ gravity has been developed in [39] via the gravitational decoupling approach, where the $f(R,T)$ model proposed is fit for describing the anisotropic nature of compact stars. Shaikh [40] studied the behaviors of general relativistic hydrodynamics in the form of perfect fluid and holographic dark energy in $f(R,T)$ gravity and the obtained solutions coincide with the observations of SNe Ia and CMB. By introducing the Hubble parameter’s inflation solutions, a mimetic $f(R,T)$ gravity theory in the presence of the swampland de Sitter conjecture has been analyzed in [41]. Gonçalves et al. [42] investigated the scalar-tensor representation of $f(R,T)$ gravity and inferred the explicit form of the functions $f(R,T)$. Various other studies related to $f(R,T)$ gravity can be found in [43,44,45,46,47,48,49].

Dynamic system analysis has been applied extensively to various models in different gravity theories [50,51,52,53,54,55,56,57] since it can depict the qualitative behavior of cosmological models where the derivation of exact solutions of evolution equations is inaccessible. Through dynamic system analysis, the dynamics of the $f(R,T)=\xi R^{\alpha }+\zeta \sqrt{-T}$ gravitational model on three invariant planes have been investigated in [57], and all the finite and infinite equilibrium points used in this work are taken from this reference. The present study mainly focuses on the dynamics of this model in three-dimensional space containing infinity, which improves the dynamic investigation of the model to a certain extent. This paper presents the dynamic behavior of the evolutionary trajectories of the cosmological model in all physically feasible regions restricted by three invariant planes and discusses the corresponding cosmological solutions.

The structure of this paper is as follows: in Section 2, we present the field equations of the model in $f(R,T)$ gravity. The dynamic equations and corresponding dynamic analysis of the model in three-dimensional space are shown in Section 3. The paper ends with a summary of the obtained results in Section 4.

2. Field Equations of $\mathit{f}(\mathit{R},\mathit{T})$ Gravity

The $f(R,T)$ gravity theory can be described by the following action

$$S=\int \sqrt{-g}{d}^{4}x\left[\frac{1}{16\pi G}f\left(R,T^{\left(\mathrm{m}\right)}\right)+L^{\left(\mathrm{m}\right)}+L^{\left(\mathrm{rad}\right)}\right],$$

(1)

where G is the gravitational constant, g is the determinant of the metric, $L^{\left(\mathrm{m}\right)}$ and $L^{\left(\mathrm{rad}\right)}$ represent the Lagrangians of the dust matter and radiation, respectively. As usual,

$$T_{\mu \nu }\equiv -\frac{2}{\sqrt{-g}}\frac{\delta \left[\sqrt{-g}\left(L^{\left(\mathrm{m}\right)}+L^{\left(\mathrm{rad}\right)}\right)\right]}{\delta g^{\mu \nu }},$$

(2)

Moreover, assuming that $L^{\left(\mathrm{m}\right)}$ and $L^{\left(\mathrm{rad}\right)}$ depend only on the metric, we obtain

$$T_{\mu \nu }=g_{\mu \nu }\left[L^{\left(\mathrm{m}\right)}+L^{\left(\mathrm{rad}\right)}\right]-2\frac{\partial \left[L^{\left(\mathrm{m}\right)}+L^{\left(\mathrm{rad}\right)}\right]}{\partial g^{\mu \nu }}.$$

(3)

By considering a perfect fluid, we have

$$g^{\alpha \beta }\frac{\delta T^{\left(\mathrm{m}\right)}}{\delta g^{\mu \nu }}=-2T_{\mu \nu }^{\left(\mathrm{m}\right)}.$$

(4)

Then, by transforming the action S, the field equations become

$$\begin{array}{cc}\hfill f_R(R,T)& R_{\mu \nu }-\frac{1}{2}f(R,T)g_{\mu \nu }+\left(g_{\mu \nu }\square -{\nabla }_{\mu }{\nabla }_{\nu }\right)f_R(R,T)\hfill \\ \hfill & =\left(8\pi G+f_T(R,T)\right)T_{\mu \nu }^{\left(\mathrm{m}\right)}+8\pi G{T}_{\mu \nu }^{\left(\mathrm{rad}\right)},\hfill \end{array}$$

(5)

where $f_R(R,T)=\partial f(R,T)/\partial R$, $f_T(R,T)=\partial f(R,T)/\partial T$, ${\nabla }_{\mu }$ denotes the covariant derivative and $\square ={\nabla }^{\mu }{\nabla }_{\mu }$.

By contracting Equation (5), we get

$$f_R(R,T)+3\square f_R(R,T)-2f(R,T)=\left(8\pi G+f_T(R,T)\right)T.$$

(6)

Here, we are interested in studying the behavior of this theory for spatially flat FLRW space-time described by

$$d{s}^2=-d{t}^2+a^2\left(t\right)\left(d{x}^2+d{y}^2+d{z}^2\right),$$

(7)

where $a\left(t\right)$ stands for the scale factor. With the conservation of energy-momentum tensor of all matter and the relation ${\nabla }^{\mu }{T}_{\mu \nu }^{\left(\mathrm{m}\right)}=0={\nabla }^{\mu }{T}_{\mu \nu }^{\left(\mathrm{rad}\right)}$, we find

$${\dot{f}}_T(R,T)=\frac{3}{2}H\left(t\right)f_T(R,T),$$

(8)

where $H\left(t\right)=\dot{a}\left(t\right)/a\left(t\right)$ is the Hubble parameter, and a dot denotes the derivative with respect to the cosmic time t. With the metric (7), Equations (5) and (6) can be rewritten as

$$\begin{array}{cc}\hfill 3H^2{f}_R(R,& T)+\frac{1}{2}\left(f(R,T)-f_R(R,T)R\right)+3H{\dot{f}}_R(R,T)\hfill \\ \hfill & =\left(8\pi G+f_T(R,T)\right){\rho }^{\left(\mathrm{m}\right)}+8\pi G{\rho }^{\left(\mathrm{m}\right)},\hfill \end{array}$$

(9)

and

$$2\dot{H}{f}_R(R,T)+{\ddot{f}}_R(R,T)-H{\dot{f}}_R(R,T)=-\left(8\pi G+f_T(R,T)\right){\rho }^{\left(\mathrm{m}\right)}-\frac{32}{3}\pi G{\rho }^{\left(\mathrm{rad}\right)}.$$

(10)

3. Dynamic Analysis of the $\mathit{f}(\mathit{R},\mathit{T})=\mathit{\xi }{\mathit{R}}^{\mathit{\alpha }}+\mathit{\zeta }\sqrt{-\mathit{T}}$ Model

3.1. Dynamic Equations

In this section, we consider the minimally coupled $f(R,T)$ gravity, i.e., $f(R,T)=g\left(R\right)+h\left(T\right)$. In order to obtain the dynamic equations, we introduce six dimensionless independent variables [49], namely,

$$x_1\equiv -\frac{{\dot{g}}^{\prime }\left(R\right)}{H{g}^{\prime }\left(R\right)},x_2\equiv -\frac{g\left(R\right)}{6H^2{g}^{\prime }\left(R\right)},x_3\equiv \frac{R}{6H^2}=\frac{\dot{H}}{H^2}+2,$$

(11)

$$x_4\equiv -\frac{h\left(T\right)}{3H^2{g}^{\prime }\left(R\right)},x_5\equiv -\frac{8\pi G{\rho }^{\left(\mathrm{rad}\right)}}{3H^2{g}^{\prime }\left(R\right)},x_6\equiv -\frac{T{h}^{\prime }\left(T\right)}{3H^2{g}^{\prime }\left(R\right)}.$$

(12)

We also define four dimensionless parameters [49] as follows

$$m\equiv \frac{R{g}^{″}\left(R\right)}{g^{\prime }\left(R\right)},r\equiv -\frac{R{g}^{\prime }\left(R\right)}{g\left(R\right)}=\frac{x_3}{x_2},n\equiv \frac{T{h}^{″}\left(T\right)}{h^{\prime }\left(T\right)},s\equiv \frac{T{h}^{\prime }\left(T\right)}{h\left(T\right)}=\frac{x_6}{x_4}.$$

(13)

We can rewrite Equations (9) and (10) as

$$1+\frac{g}{6H^2{g}^{\prime }}+\frac{h}{6H^2{g}^{\prime }}-\frac{R}{6H^2}+\frac{{\dot{g}}^{\prime }}{H{g}^{\prime }}=\frac{8\pi G{\rho }^{\left(\mathrm{m}\right)}}{3H^2{g}^{\prime }}+\frac{h^{\prime }{\rho }^{\left(\mathrm{m}\right)}}{3H^2{g}^{\prime }}+\frac{8\pi G{\rho }^{\left(\mathrm{rad}\right)}}{3H^2{g}^{\prime }},$$

(14)

and

$$2\frac{\dot{H}}{H^2}+\frac{{\ddot{g}}^{\prime }}{H^2{g}^{\prime }}-\frac{{\dot{g}}^{\prime }}{H{g}^{\prime }}=-\frac{8\pi G{\rho }^{\left(\mathrm{m}\right)}}{H^2{g}^{\prime }}-\frac{h^{\prime }{\rho }^{\left(\mathrm{m}\right)}}{H^2{g}^{\prime }}-\frac{32\pi G{\rho }^{\left(\mathrm{rad}\right)}}{3H^2{g}^{\prime }}.$$

(15)

Equation (8) becomes

$$T{h}^{″}=-\frac{1}{2}{h}^{\prime },$$

(16)

and then we get

$$T{h}^{\prime }-\frac{1}{2}h+C=0,$$

(17)

where C is an integration constant. Set $C=0$, which leads to $s=1/2$. Then, we obtain $x_4=2x_6$ and $h=\zeta \sqrt{-T}$, where $\zeta$ is a constant.

We study the late-time behaviors of this model, i.e., radiation does not exist, and consider the form $f(R,T)=\xi R^{\alpha }+\zeta \sqrt{-T}$, where $\xi$ is a constant. Thus, we get

$$x_3=-\alpha x_2,m=\alpha -1.$$

(18)

Setting $x_1\equiv x,x_2\equiv y,x_4\equiv z$, we obtain the autonomous dynamic system

$$\begin{array}{cc}\hfill \frac{dx}{dN}& =-1+x(x+\alpha y)+(\alpha -3)y-\frac{3}{2}z,\hfill \\ \hfill \frac{dy}{dN}& =-\frac{xy}{\alpha -1}+2y(2+\alpha y),\hfill \\ \hfill \frac{dz}{dN}& =z\left(\frac{5}{2}+x+2\alpha y\right),\hfill \end{array}$$

(19)

where $N=lna$. The density parameter of matter ${\Omega }^{\left(\mathrm{m}\right)}$ and effective equation of state ${\omega }^{\left(\mathrm{eff}\right)}$ are defined as follows

$${\Omega }^{\left(\mathrm{m}\right)}\equiv \frac{8\pi G{\rho }^{\left(\mathrm{m}\right)}}{3H^2{g}^{\prime }}=1-x+(\alpha -1)y-z,$$

(20)

$${\omega }^{\left(\mathrm{eff}\right)}\equiv -1-\frac{2\dot{H}}{3H^2}=\frac{1}{3}(1+2\alpha y).$$

(21)

3.2. Dynamic Analysis in Three-Dimensional Space

Dynamic analysis of system (19) on invariant planes has been presented in [57], where the equilibrium points used in the following are provided. According to [57], the cosmological solution matching observations can be obtained when $\alpha >1$ and $\alpha$ is approaching 1. Here, we set $\alpha =1.1$ to proceed with the dynamic analysis of the system in 3D. Other values of $\alpha$ can be taken to study this model similarly. The finite equilibrium points of system (19) are summarized in

Table 1. The finite equilibrium points of the dynamical systems.

Point	Coordinates	Scale Factor	${\mathbf{\Omega }}^{\left(\mathbf{m}\right)}$	${\mathit{\omega }}^{\left(\mathbf{eff}\right)}$	Stability
$p_1$	($1,0,0$)	$a\left(t\right)=a_0{\left(\frac{t-t_i}{t_0-t_i}\right)}^{\frac{1}{2}}$	0	$\frac{1}{3}$	saddle
$p_2$	($-1,0,0$)	$a\left(t\right)=a_0{\left(\frac{t-t_i}{t_0-t_i}\right)}^{\frac{1}{2}}$	2	$\frac{1}{3}$	saddle
$p_3$	($-\frac{5}{2},0,\frac{7}{2}$)	$a\left(t\right)=a_0{\left(\frac{t-t_i}{t_0-t_i}\right)}^{\frac{1}{2}}$	0	$\frac{1}{3}$	saddle
$p_4$	($\frac{3}{2},5,0$)	$a\left(t\right)=a_0{\left(\frac{t-t_i}{t_0-t_i}\right)}^{\frac{2}{15}}$	0	4	unstable
$p_5$	($\frac{3}{11},\frac{70}{121},0$)	$a\left(t\right)=a_0{\left(\frac{t-t_i}{t_0-t_i}\right)}^{\frac{11}{15}}$	$\frac{81}{121}$	$-\frac{1}{10}$	unstable
$p_6$	($\frac{3}{22},-\frac{145}{121},\frac{90}{121}$)	$a\left(t\right)=a_0{\left(\frac{t-t_i}{t_0-t_i}\right)}^{\frac{22}{15}}$	0	$-\frac{6}{11}$	stable

.

From Table 1, the points $p_1$, $p_3$ are radiation points with ${\Omega }^{\left(\mathrm{m}\right)}=0$ and ${\omega }^{\left(\mathrm{eff}\right)}=1/3$. The eigenvalues of $p_1$, $p_3$ are $7/2$, 2, $-6$ and $-7/2$, $-3/2$, 9, respectively, so these points are saddles.

The point $p_2$ can be considered as a $\varphi$-matter-dominated epoch ($\varphi$MDE) [58] with ${\Omega }^{\left(\mathrm{m}\right)}=2$ and ${\omega }^{\left(\mathrm{eff}\right)}=1/3$. Although the value of the matter-density parameter does not match the effective equation of state, the Universe’s evolution may be related to this point in the model. The eigenvalues of $p_2$ are $3/2$, $-2$, 14, so $p_2$ is a saddle.

For point $p_4$, it has ${\Omega }^{\left(\mathrm{m}\right)}=0$ and ${\omega }^{\left(\mathrm{eff}\right)}=4$. As this point cannot describe any known matter, we consider it nonphysical. The eigenvalues of $p_4$ are $27/2$, 6, $5/4$, so $p_4$ is unstable.

The equilibrium point $p_5$ can present a matter-dominated era with ${\Omega }^{\left(\mathrm{m}\right)}=81/121$ and ${\omega }^{\left(\mathrm{eff}\right)}=-1/10$. The eigenvalues of $p_5$ are $3/2$, $-15/22\pm \beta \mathrm{i}$, where $\beta \approx 2.9845$ and ${\mathrm{i}}^2=-1$, so $p_5$ is unstable.

The point $p_6$ can be used to explain the late accelerating expansion of the Universe. At this point, matter and radiation are absent, while the Universe is dominated by dark energy. This point has ${\Omega }^{\left(\mathrm{m}\right)}=0$ and ${\omega }^{\left(\mathrm{eff}\right)}=-6/11$. The eigenvalues of $p_6$ are $3/2$, $-12/11\pm \gamma \mathrm{i}$, where $\gamma \approx 4.5144$, so $p_6$ is a stable spiral point.

For the infinite equilibrium points $e_{13}$, $E_{13}$, and a set of non-isolated points filling up the circle where the plane $y=0$ intersects the Poincaré sphere, we can only describe the trend of the trajectories around them. Because the coordinates of these infinite points are indistinct, it is challenging to study their physical properties.

The three planes $y=0$, $z=0$ and ${\Omega }^{\left(\mathrm{m}\right)}=0$ are invariant planes of system (19). Through Poincaré compactification [59], the infinite space can be closed to the finite, so the whole space can be regarded as a Poincaré ball. A Poincaré ball is divided by these three invariant planes into eight regions ${\mathrm{R}}_i(i=1,\cdots ,8)$, as shown in Figure 1. The accurate representation of regions ${\mathrm{R}}_i(i=1,\cdots ,8)$ is

$$\begin{array}{c}\hfill {\mathrm{R}}_1:x<0,{\Omega }^{\left(\mathrm{m}\right)}>0,z>0,{\mathrm{R}}_2:x>0,{\Omega }^{\left(\mathrm{m}\right)}>0,z>0,\\ \hfill {\mathrm{R}}_3:x<0,{\Omega }^{\left(\mathrm{m}\right)}>0,z<0,{\mathrm{R}}_4:x>0,{\Omega }^{\left(\mathrm{m}\right)}>0,z<0,\\ \hfill {\mathrm{R}}_5:x<0,{\Omega }^{\left(\mathrm{m}\right)}<0,z>0,{\mathrm{R}}_6:x>0,{\Omega }^{\left(\mathrm{m}\right)}<0,z>0,\\ \hfill {\mathrm{R}}_7:x<0,{\Omega }^{\left(\mathrm{m}\right)}<0,z<0,{\mathrm{R}}_8:x>0,{\Omega }^{\left(\mathrm{m}\right)}<0,z<0.\end{array}$$

The cosmological solutions in the viable $f\left(R\right)$ gravity models hold $g^{\prime }\left(R\right)>0$ which guarantees that gravity is an attractive force. According to the definition of ${\Omega }^{\left(\mathrm{m}\right)}$, these solutions must have ${\Omega }^{\left(\mathrm{m}\right)}\ge 0$. The condition ${\Omega }^{\left(\mathrm{m}\right)}\ge 0$ should hold in this model because $f\left(R\right)$ gravity can be considered as a particular case of $f(R,T)=g\left(R\right)+h\left(T\right)$ gravity with $h\left(T\right)=0$. From Figure 1, we know that ${\Omega }^{\left(\mathrm{m}\right)}>0$ in regions ${\mathrm{R}}_i(i=1,2,3,4)$, ${\Omega }^{\left(\mathrm{m}\right)}<0$ in regions ${\mathrm{R}}_i(i=5,6,7,8)$. Thus, we will focus on the dynamics of regions ${\mathrm{R}}_i(i=1,2,3,4)$ instead of ${\mathrm{R}}_i(i=5,6,7,8)$.

The dynamic behavior of system (19) inside regions ${\mathrm{R}}_i(i=1,2,3,4)$ is determined by the following five planes and one surface, which divide these four regions into several subregions:

$$\begin{array}{cc}\hfill & y=0,z=0,{\Omega }^{\left(\mathrm{m}\right)}(x,y,z)=0,\hfill \\ \hfill & b_1(x,y,z)=0,b_2(x,y)=0,b_3(x,y)=0,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}& {\Omega }^{\left(\mathrm{m}\right)}(x,y,z)=1-x+0.1y-z,\hfill \\ & b_1(x,y,z)=-1+x^2+1.1xy-1.9y-1.5z,\hfill \\ & b_2(x,y)=-10x+2.2y+4,\hfill \\ & b_3(x,y)=x+2.2y+2.5.\hfill \end{array}\end{array}$$

(22)

The region ${\mathrm{R}}_1$ is divided into seven subregions ${\mathrm{R}}_{1i}(i=1,\cdots ,7)$ (see Figure 2 and Figure 3). The dynamic behavior of system (19) in these seven subregions is presented in

Table 2. Dynamic behavior of the trajectories in the seven subregions.

Subregion	Corresponding Region	Dynamic Behavior
${\mathrm{R}}_{11}$	$b_1>0,b_2>0,b_3<0,y<0,z>0$	$\dot{x}>0,\dot{y}<0,\dot{z}<0$
${\mathrm{R}}_{12}$	$b_1>0,b_2<0,b_3<0,y<0,z>0$	$\dot{x}>0,\dot{y}>0,\dot{z}<0$
${\mathrm{R}}_{13}$	$b_1>0,b_2<0,b_3>0,y<0,z>0$	$\dot{x}>0,\dot{y}>0,\dot{z}>0$
${\mathrm{R}}_{14}$	$h_1<0,h_2<0,h_3>0,y<0,z>0$	$\dot{x}<0,\dot{y}>0,\dot{z}>0$
${\mathrm{R}}_{15}$	$b_1<0,b_2>0,b_3>0,y<0,z>0$	$\dot{x}<0,\dot{y}<0,\dot{z}>0$
${\mathrm{R}}_{16}$	$b_1<0,b_2>0,b_3<0,y<0,z>0$	$\dot{x}<0,\dot{y}<0,\dot{z}<0$
${\mathrm{R}}_{17}$	$b_1>0,b_2>0,b_3>0,y<0,z>0$	$\dot{x}>0,\dot{y}<0,\dot{z}>0$

.

For subregion ${\mathrm{R}}_{11}$, its front part is bounded by ${\Omega }^{\left(\mathrm{m}\right)}=0$ and $y=0$, its back is restricted by the Poincaré sphere, and its bottom is limited in $z=0$. As shown in Table 2, the trajectories of system (19) monotonically decrease in the positive y and z axes and increase in the positive x axis, which indicates that the trajectories in ${\mathrm{R}}_{11}$ can get to subregions ${\mathrm{R}}_{12}$, ${\mathrm{R}}_{17}$ and the sources of these trajectories are subregion ${\mathrm{R}}_{16}$ and infinite equilibrium points in $y=0$ restricted to ${\mathrm{R}}_{11}$.

The right of subregion ${\mathrm{R}}_{12}$ is restricted by Poincaré sphere, the left is limited by $b_3=0$, the back is bounded by ${\Omega }^{\left(\mathrm{m}\right)}=0$, the bottom is confined by $z=0$ and the front is bounded by $b_2=0$. The movement of trajectories in ${\mathrm{R}}_{12}$ increases monotonically in the positive x and y axes, and decreases in the positive z axis. Thus, ${\mathrm{R}}_{11}$ is the source of these trajectories and ${\mathrm{R}}_{13}$ is the subregion that these trajectories tend to.

The right side of ${\mathrm{R}}_{13}$ is restricted by $b_3=0$, its left is confined by $b_1=0$, and its front, bottom, and back parts are limited by the planes $b_2=0$, $z=0$, and ${\Omega }^{\left(\mathrm{m}\right)}=0$, respectively. The tendency of trajectories in subregion ${\mathrm{R}}_{13}$ presents a monotone increase along the positive three axes. The trajectories in ${\mathrm{R}}_{13}$ originate from subregions ${\mathrm{R}}_{12}$, ${\mathrm{R}}_{17}$, and the point $p_5$, and then move to ${\mathrm{R}}_{14}$ and $p_6$.

For subregion ${\mathrm{R}}_{14}$, its front and back are bounded by $b_2$ and ${\Omega }^{\left(\mathrm{m}\right)}=0$, respectively, its bottom is limited by $z=0$, and its left and right are restricted by $y=0$ and $b_1=0$, respectively. The trajectories in ${\mathrm{R}}_{14}$ monotonically decrease in the positive x axis, increase in the positive y and z axes, so the trajectories are from ${\mathrm{R}}_{13}$ and go to ${\mathrm{R}}_{15}$.

The front part of subregion ${\mathrm{R}}_{15}$ is restricted by $b_1=0$, the back is bounded by $y=0$ and $b_3=0$, the upper is limited by ${\Omega }^{\left(\mathrm{m}\right)}=0$ and $b_3=0$, and the bottom is confined by $z=0$. In subregion ${\mathrm{R}}_{15}$, trajectories are monotone decreasing in the positive x and z axes while increasing in the positive y axis. Therefore, these trajectories are from ${\mathrm{R}}_{14}$ and move to ${\mathrm{R}}_{16}$.

For ${\mathrm{R}}_{16}$, its front part is bounded by $b_3=0$ and its back part is confined by two planes ${\Omega }^{\left(\mathrm{m}\right)}=0$ and $b_3=0$. The behavior of trajectories in this subregion is decreasing monotonically in the positive three axes, so these trajectories stem from $p_3$ and ${\mathrm{R}}_{15}$ and then enter into ${\mathrm{R}}_{11}$.

The back, left, right, lower, and front parts of subregion ${\mathrm{R}}_{17}$ are bounded by the surface $b_1=0$, the planes $b_3=0$, $y=0$, $b_2=0$, and $z=0$, respectively. The trajectories in ${\mathrm{R}}_{17}$ monotonically decrease in the positive y axis and increase in the positive x and z axes. Hence, they come from subregions ${\mathrm{R}}_{11}$, ${\mathrm{R}}_{15}$, and the point $p_5$ and move towards $p_6$.

Overall, the dynamic behavior of the trajectories in the seven subregions of ${\mathrm{R}}_1$ analyzed above can be displayed as

Note: PS is a Poincaré sphere.

According to the process of dynamics represented above, the trajectories of system (19) in the region ${\mathrm{R}}_1$ possess an $\omega$-limit, i.e., future attractor, at $p_6$ and $\alpha$-limits, i.e., past attractor, at the infinite equilibrium points in $y=0$ limited to ${\mathrm{R}}_{11}$ and the finite equilibrium points $p_3$, $p_5$. Although the points $p_3$ and $p_5$ have positive real eigenvalues, they are post attractors in ${\mathrm{R}}_1$. Cosmological solutions matching observations go through a long enough matter-dominated epoch and then into an accelerated expansion, i.e., an unstable point of matter switches to a stable acceleration equilibrium point in phase space. The trajectories originated from $p_5$ can get to $p_6$ through subregions ${\mathrm{R}}_{13}$ or ${\mathrm{R}}_{17}$, which represent a class of cosmological solutions.

The region ${\mathrm{R}}_2$ is divided into five subregions ${\mathrm{R}}_{2i}(i=1,\cdots ,5)$, shown in Figure 4 and Figure 5, and

Table 3. Dynamic behavior of the trajectories in the five subregions.

Subregion	Corresponding Region	Dynamic Behavior
${\mathrm{R}}_{21}$	$b_1<0,b_2<0,b_3>0,y>0,z>0$	$\dot{x}<0,\dot{y}<0,\dot{z}>0$
${\mathrm{R}}_{22}$	$b_1>0,b_2>0,b_3>0,y>0,z>0$	$\dot{x}>0,\dot{y}>0,\dot{z}>0$
${\mathrm{R}}_{23}$	$b_1<0,b_2>0,b_3>0,y>0,z>0$	$\dot{x}<0,\dot{y}>0,\dot{z}>0$
${\mathrm{R}}_{24}$	$b_1>0,b_2>0,b_3>0,y>0,z>0$	$\dot{x}>0,\dot{y}>0,\dot{z}>0$
${\mathrm{R}}_{25}$	$b_1>0,b_2>0,b_3<0,y>0,z>0$	$\dot{x}>0,\dot{y}>0,\dot{z}<0$

displays the dynamic behavior of system (19) in these five subregions.

As shown in Figure 5, the front of subregion ${\mathrm{R}}_{21}$ is bounded by the plane $b_2=0$, the back is limited by ${\Omega }^{\left(\mathrm{m}\right)}=0$, the base is restricted by $z=0$, and the right is confined by $y=0$. The trajectories of system (19) in ${\mathrm{R}}_{21}$ decrease monotonically in the positive x and y axes and increase in the positive z axis. Accordingly, the trajectories in ${\mathrm{R}}_{21}$ are from $p_4$ and go to ${\mathrm{R}}_{23}$.

For subregion ${\mathrm{R}}_{22}$, its front part is restricted by Poincaré sphere and $b_1=0$, its back is bounded by ${\Omega }^{\left(\mathrm{m}\right)}=0$, and its bottom is confined by $z=0$. In this subregion, the trajectories of system (19) increase monotonically in the positive three axes directions. Therefore, these trajectories originate from $p_4$ and get to ${\mathrm{R}}_{23}$.

The front of subregion ${\mathrm{R}}_{23}$ is bounded by the Poincaré sphere, the back is limited by $y=0$ and $b_2=0$, the left and right are limited by $b_1=0$, and the bottom is confined by $z=0$. In ${\mathrm{R}}_{23}$, the trajectories are monotone decreasing in the positive y and z axes and increasing in the positive x axis, which shows that these trajectories can get to $e_{13}$ and the sources of them are $p_2$, $p_3$, $p_4$, ${\mathrm{R}}_{21}$, ${\mathrm{R}}_{22}$, and ${\mathrm{R}}_{24}$.

The front of subregion ${\mathrm{R}}_{24}$ is restricted by Poincaré sphere, the left is confined by the surface $b_1=0$, and the right, the upper, and the lower are limited by the planes $b_3=0$, ${\Omega }^{\left(\mathrm{m}\right)}=0$, and $z=0$, respectively. The trajectories in ${\mathrm{R}}_{24}$ behave the same as those in subregion ${\mathrm{R}}_{22}$. Thus, they come from ${\mathrm{R}}_{25}$ and leave for ${\mathrm{R}}_{23}$.

The front of ${\mathrm{R}}_{25}$ is restricted by Poincaré sphere and $b_3=0$, the back is limited by $y=0$, the upper is confined by ${\Omega }^{\left(\mathrm{m}\right)}=0$, and the lower is bounded by $z=0$. The tendency of the trajectories in this subregion displays a monotonic increase in the positive x and y axes, and a monotonic decrease in the positive z axis. The trajectories thereby originate from infinite equilibrium points in $y=0$ restricted to ${\mathrm{R}}_{25}$ and point to ${\mathrm{R}}_{24}$.

In summary, the dynamic behavior of the trajectories in the five subregions of ${\mathrm{R}}_2$ discussed above is shown below.

From the dynamic demonstration of ${\mathrm{R}}_2$, we find that the trajectories in ${\mathrm{R}}_2$ own an $\omega$-limit at $e_{13}$ and $\alpha$-limits at infinite equilibrium points in $y=0$ limited to ${\mathrm{R}}_{25}$ and finite equilibrium points $p_2$, $p_3$, $p_4$. In addition, $p_4$ is a past attractor for its instability. We find that there are trajectories that originate from $p_2$, $p_3$, $p_4$ and infinite point in $y=0$ restricted by ${\mathrm{R}}_{25}$ and move to $e_{13}$ in the region ${\mathrm{R}}_2$. However, cosmological solutions can not be found in this region.

The region ${\mathrm{R}}_3$ is divided into seven subregions ${\mathrm{R}}_{3i}(i=1,\cdots ,7)$ (see Figure 6 and Figure 7). The dynamic behavior of the system (19) in these seven subregions is presented in

Table 4. Dynamic behavior of the trajectories in the seven subregions.

Subregion	Corresponding Region	Dynamic Behavior
${\mathrm{R}}_{31}$	$b_1>0,b_2>0,b_3<0,y<0,z<0$	$\dot{x}>0,\dot{y}<0,\dot{z}>0$
${\mathrm{R}}_{32}$	$b_1>0,b_2>0,b_3>0,y<0,z<0$	$\dot{x}>0,\dot{y}<0,\dot{z}<0$
${\mathrm{R}}_{33}$	$b_1<0,b_2>0,b_3>0,y<0,z<0$	$\dot{x}<0,\dot{y}<0,\dot{z}<0$
${\mathrm{R}}_{34}$	$b_1<0,b_2<0,b_3>0,y<0,z<0$	$\dot{x}<0,\dot{y}>0,\dot{z}<0$
${\mathrm{R}}_{35}$	$b_1>0,b_2<0,b_3>0,y<0,z<0$	$\dot{x}>0,\dot{y}>0,\dot{z}<0$
${\mathrm{R}}_{36}$	$b_1>0,b_2<0,b_3<0,y<0,z<0$	$\dot{x}>0,\dot{y}>0,\dot{z}>0$
${\mathrm{R}}_{37}$	$b_1<0,b_2<0,b_3<0,y<0,z<0$	$\dot{x}<0,\dot{y}>0,\dot{z}>0$

.

In Figure 7, the right side of the subregion is limited by the plane $y=0$ and the left is restricted by $b_2=0$, the front part of ${\mathrm{R}}_{31}$ is bounded by Poincaré sphere, the back is confined by $b_3=0$ and the top is restricted by $z=0$. As shown in Table 3, the trajectories of system (19) in subregion ${\mathrm{R}}_{31}$ are monotone increasing in the positive x and z axes, decreasing in the positive y axis. Therefore, the trajectories in ${\mathrm{R}}_{31}$ stem from the infinite equilibrium points in $y=0$ restricted to ${\mathrm{R}}_{31}$ and get into subregions ${\mathrm{R}}_{32}$ and ${\mathrm{R}}_{36}$.

The bottom of subregion ${\mathrm{R}}_{32}$ is restricted by the Poincaré sphere, the top is limited by $z=0$ and $b_1=0$, the front is bounded by $b_3=0$ and the back is confined by $y=0$ and $h_2=0$. The behavior of trajectories in ${\mathrm{R}}_{32}$ is the same with those in subregion ${\mathrm{R}}_{31}$. For this reason, the trajectories in this subregion derive from ${\mathrm{R}}_{31}$ and ${\mathrm{R}}_{33}$, then enter ${\mathrm{R}}_{35}$.

For subregion ${\mathrm{R}}_{33}$, its front part is bounded by $b_1=0$, its back is restricted by $y=0$ and $b_2=0$, and the top is limited by $z=0$. In ${\mathrm{R}}_{33}$, the trajectories are monotone decreasing in the positive three axes, which shows that the source of the trajectories in ${\mathrm{R}}_{33}$ is ${\mathrm{R}}_{34}$ and these trajectories then go into ${\mathrm{R}}_{32}$.

The front part of subregion ${\mathrm{R}}_{34}$ is restricted by $b_2=0$, the upper is bounded by $z=0$, and the back is limited by $y=0$ and $h_1$. The tendency of trajectories in ${\mathrm{R}}_{34}$ is monotonic increasing in the positive y axis and decreasing in the positive x and z axes. Thus, the trajectories in this subregion originate from $p_5$ and leave for ${\mathrm{R}}_{33}$.

The upper part of subregion ${\mathrm{R}}_{35}$ is restricted by $z=0$ and $b_1=0$, the left and the right are limited by $b_2=0$ and ${\Omega }^{\left(\mathrm{m}\right)}=0$, respectively, and the front and the back are confined by $b_3=0$ and $y=0$, respectively. In subregion ${\mathrm{R}}_{35}$, the trajectories monotonically increase in the positive x and y axes, decrease in the positive z axis, which points that they come from ${\mathrm{R}}_{32}$ and ${\mathrm{R}}_{36}$, then leave for the infinite equilibrium points in $y=0$ limited to ${\mathrm{R}}_{35}$.

For ${\mathrm{R}}_{36}$, its front part is limited by $b_2=0$ and the Poincaré sphere, its top is bounded by $z=0$, and its back is restricted by $b_3=0$, ${\Omega }^{\left(\mathrm{m}\right)}=0$ and $b_1=0$. In ${\mathrm{R}}_{36}$, the dynamic behavior of trajectories is monotone increasing in positive three axes. The trajectories in this subregion are from ${\mathrm{R}}_{37}$ and move to ${\mathrm{R}}_{35}$.

The right part of ${\mathrm{R}}_{37}$ is bounded by Poincaré sphere, the back is limited by ${\Omega }^{\left(\mathrm{m}\right)}=0$, and other parts are restricted by $b_1=0$. The behavior of the trajectories in ${\mathrm{R}}_{37}$ shows a monotonic decrease in the positive x and z axes and increases in the positive y axis. For this reason, the trajectories in ${\mathrm{R}}_{37}$ originate from the infinite equilibrium point $E_{13}$ and go to ${\mathrm{R}}_{36}$.

On the whole, the behavior of the trajectories in the seven subregions of ${\mathrm{R}}_3$ discussed above can be described as

The dynamic process above indicates that the trajectories in ${\mathrm{R}}_3$ possess $\omega$-limits at the infinite equilibrium points in $y=0$ restricted to ${\mathrm{R}}_{35}$ and $\alpha$-limits at $E_{13}$ and the infinite equilibrium points in $y=0$ restricted to ${\mathrm{R}}_{31}$. We find that trajectories originating from $p_5$ can reach the infinite equilibrium points in $y=0$ limited to ${\mathrm{R}}_{35}$. However, this region contains no cosmological solutions because the infinite points that the trajectories point to are not accelerated stable points.

The region ${\mathrm{R}}_4$ is divided into five subregions ${\mathrm{R}}_{4i}(i=1,\cdots ,5)$, as shown in Figure 8 and Figure 9. The dynamic behavior of system (19) in these five subregions is displayed in

Table 5. Dynamic behavior of the trajectories in the five subregions.

Subregion	Corresponding Region	Dynamic Behavior
${\mathrm{R}}_{21}$	$b_1>0,b_2<0,b_3>0,y>0,z<0$	$\dot{x}>0,\dot{y}<0,\dot{z}<0$
${\mathrm{R}}_{22}$	$b_1<0,b_2<0,b_3>0,y>0,z<0$	$\dot{x}<0,\dot{y}<0,\dot{z}<0$
${\mathrm{R}}_{23}$	$b_1<0,b_2>0,b_3>0,y>0,z<0$	$\dot{x}<0,\dot{y}>0,\dot{z}<0$
${\mathrm{R}}_{24}$	$b_1>0,b_2>0,b_3>0,y>0,z<0$	$\dot{x}>0,\dot{y}>0,\dot{z}<0$
${\mathrm{R}}_{25}$	$b_1>0,b_2>0,b_3<0,y>0,z<0$	$\dot{x}>0,\dot{y}>0,\dot{z}>0$

.

For subregion ${\mathrm{R}}_{41}$, its front part is limited by $b_2=0$, its back is bounded by $y=0$ and ${\Omega }^{\left(\mathrm{m}\right)}=0$, and its bottom and top are restricted by Poincaré sphere and $b_3=0$, respectively. The trajectories in subregion ${\mathrm{R}}_{41}$ are monotone increasing in the positive x axis, and decreasing in the positive y and z axes. So these trajectories stem from subregions ${\mathrm{R}}_{42}$ and ${\mathrm{R}}_{44}$ and destine for the infinite equilibrium points in $y=0$ limited to ${\mathrm{R}}_{41}$.

The front of subregion ${\mathrm{R}}_{42}$ is restricted by $b_2=0$, the back is bounded by $y=0$ and ${\Omega }^{\left(\mathrm{m}\right)}=0$, the top is confined by $z=0$, and the bottom is limited by $b_3=0$. The dynamic behavior of the trajectories in ${\mathrm{R}}_{42}$ is monotonically decreasing in the positive three axes directions. Hence, the trajectories in ${\mathrm{R}}_{42}$ stem from $p_4$, then leave for ${\mathrm{R}}_{41}$ and ${\mathrm{R}}_{43}$.

For ${\mathrm{R}}_{43}$, its top portion is limited by $z=0$, its back is restricted by $b_2$ and $y=0$, and the other portions are confined by $b_3=0$. The trajectories in ${\mathrm{R}}_{43}$ are monotone decreasing in the positive x and z axes, increasing in the positive y axis. These trajectories are from ${\mathrm{R}}_{42}$ and $p_2$, then head for ${\mathrm{R}}_{44}$.

The top portion of subregion ${\mathrm{R}}_{44}$ is restricted by $z=0$, $b_1$ and ${\Omega }^{\left(\mathrm{m}\right)}=0$, the right, left, front, and back are bounded by $b_3=0$, $b_2=0$, the Poincaré sphere, and $y=0$, respectively. The tendency of the trajectories in ${\mathrm{R}}_{44}$ shows monotone increasing in the positive x and y axes, monotone decreasing in the positive z axis. Thus, the trajectories in ${\mathrm{R}}_{44}$ stem from $p_2$, $p_4$, ${\mathrm{R}}_{43}$, and ${\mathrm{R}}_{45}$, and enter ${\mathrm{R}}_{41}$.

For subregion ${\mathrm{R}}_{45}$, its top is confined by the plane $z=0$, its front is limited by $b_3=0$ and the Poincaré sphere, and its back is restricted by $y=0$. In ${\mathrm{R}}_{45}$, the trajectories are monotone increasing in the positive three axes, hence they come from the infinite equilibrium points in $y=0$ restricted to ${\mathrm{R}}_{45}$, then get into ${\mathrm{R}}_{44}$.

The analysis of the dynamic behavior in the five subregions presented above can be summarized as follows:

From the dynamic flow chart above, we can obtain that the trajectories in ${\mathrm{R}}_4$ own $\omega$-limits at the infinite equilibrium points in $y=0$ restricted to ${\mathrm{R}}_{41}$ and $\alpha$-limits at $p_2$, $p_4$, and the infinite equilibrium points in $y=0$ restricted to ${\mathrm{R}}_{45}$. No cosmological solution exists in ${\mathrm{R}}_4$ because matter points and accelerated points are not located in this region.

4. Conclusions

Dynamic system analysis is a useful mathematical tool to explore the qualitative dynamic behavior of cosmological models. Due to the fact that field equations in the cosmological model include highly nonlinear terms, it is not easy to directly compare the models with the observed values. However, we can use dynamic system analysis to find the equilibrium points related to the model and the properties of these points. Since these points represent the Universe’s evolution stage, we can compare the analysis of these points with observational cosmology to obtain the rationality of the model.

In minimally coupled $f(R,T)$ gravity, the function $h\left(T\right)$ is confined to a specific form $c_1\sqrt{-T}+c_2$, where $c_1$, $c_2$ are constants. Based on Case A in Section IV of [49], we choose this particular form of $f(R,T)$ gravity, which can reduce the independent variable. Considering the late-time behavior of the model, we obtain the autonomous dynamic system with only three independent variables, which can describe the three-dimensional dynamic behavior of the system.

Through dynamic system analysis, the dynamics of the model under the background of $f(R,T)=\xi R^{\alpha }+\zeta \sqrt{-T}$ gravity in regions restricted by the three invariant planes is presented. Note that the cosmological solutions can be found in this model when $\alpha$ tends to 1 from the side greater than 1 (i.e., $\alpha \to 1^{+}$) [57], so we investigated the dynamic behavior of the system in three-dimensional space when the $\alpha$ is close to 1. As the equilibrium points $p_5$, $p_6$ are matter point and accelerated point, respectively, based on their values of ${\omega }^{\left(\mathrm{eff}\right)}$ and ${\Omega }^{\left(\mathrm{m}\right)}$, the evolution from $p_5$ to $p_6$ can describe a matter-dominated phase to an accelerated expansion. Thus, the evolution trajectories originating from $p_5$ moving to $p_6$ present a class of cosmological solutions. A Poincaré ball, which stands for the whole space, is divided into eight regions by the three invariant planes. However, as cosmological solutions hold when ${\Omega }^{\left(\mathrm{m}\right)}\ge 0$, we mainly discussed the dynamic behavior of the four regions ${\mathrm{R}}_i(i=1,2,3,4)$ in three-dimensional space and discarded the analysis of the regions ${\mathrm{R}}_i(i=5,6,7,8)$ with ${\Omega }^{\left(\mathrm{m}\right)}<0$. The dynamic behavior of each subregion in the four regions ${\mathrm{R}}_i(i=1,2,3,4)$ is presented in Table 2, Table 3, Table 4 and Table 5 and the dynamic process of all subregions are shown in the form of flow charts. The existence of cosmological solutions in the four regions has been discussed and only the region ${\mathrm{R}}_1$ has cosmological solutions which can represent the evolution from the matter-dominated epoch to the accelerated expansion.